which of the following statements about a hypothesis is correct

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Biology library

Course: biology library   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation..

• Observation: the toaster won't toast.

2. Ask a question.

• Question: Why won't my toaster toast?

3. Propose a hypothesis.

• Hypothesis: Maybe the outlet is broken.

4. Make predictions.

• Prediction: If I plug the toaster into a different outlet, then it will toast the bread.

5. Test the predictions.

• Test of prediction: Plug the toaster into a different outlet and try again.
• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• Iteration time!
• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base

Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

• Academic style
• Vague sentences
• Style consistency

See an example

For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bevans, R. (2023, June 22). Hypothesis Testing | A Step-by-Step Guide with Easy Examples. Scribbr. Retrieved April 12, 2024, from https://www.scribbr.com/statistics/hypothesis-testing/

Is this article helpful?

Rebecca Bevans

Other students also liked, choosing the right statistical test | types & examples, understanding p values | definition and examples, what is your plagiarism score.

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

7.1: Basics of Hypothesis Testing

• Last updated
• Save as PDF
• Page ID 5199

• Kathryn Kozak
• Coconino Community College

To understand the process of a hypothesis tests, you need to first have an understanding of what a hypothesis is, which is an educated guess about a parameter. Once you have the hypothesis, you collect data and use the data to make a determination to see if there is enough evidence to show that the hypothesis is true. However, in hypothesis testing you actually assume something else is true, and then you look at your data to see how likely it is to get an event that your data demonstrates with that assumption. If the event is very unusual, then you might think that your assumption is actually false. If you are able to say this assumption is false, then your hypothesis must be true. This is known as a proof by contradiction. You assume the opposite of your hypothesis is true and show that it can’t be true. If this happens, then your hypothesis must be true. All hypothesis tests go through the same process. Once you have the process down, then the concept is much easier. It is easier to see the process by looking at an example. Concepts that are needed will be detailed in this example.

Example \(\PageIndex{1}\) basics of hypothesis testing

Suppose a manufacturer of the XJ35 battery claims the mean life of the battery is 500 days with a standard deviation of 25 days. You are the buyer of this battery and you think this claim is inflated. You would like to test your belief because without a good reason you can’t get out of your contract.

What do you do?

Well first, you should know what you are trying to measure. Define the random variable.

Let x = life of a XJ35 battery

Now you are not just trying to find different x values. You are trying to find what the true mean is. Since you are trying to find it, it must be unknown. You don’t think it is 500 days. If you did, you wouldn’t be doing any testing. The true mean, \(\mu\), is unknown. That means you should define that too.

Let \(\mu\)= mean life of a XJ35 battery

You may want to collect a sample. What kind of sample?

You could ask the manufacturers to give you batteries, but there is a chance that there could be some bias in the batteries they pick. To reduce the chance of bias, it is best to take a random sample.

How big should the sample be?

A sample of size 30 or more means that you can use the central limit theorem. Pick a sample of size 30.

Example \(\PageIndex{1}\) contains the data for the sample you collected:

Now what should you do? Looking at the data set, you see some of the times are above 500 and some are below. But looking at all of the numbers is too difficult. It might be helpful to calculate the mean for this sample.

The sample mean is \(\overline{x} = 490\) days. Looking at the sample mean, one might think that you are right. However, the standard deviation and the sample size also plays a role, so maybe you are wrong.

Before going any farther, it is time to formalize a few definitions.

You have a guess that the mean life of a battery is less than 500 days. This is opposed to what the manufacturer claims. There really are two hypotheses, which are just guesses here – the one that the manufacturer claims and the one that you believe. It is helpful to have names for them.

Definition \(\PageIndex{1}\)

Null Hypothesis : historical value, claim, or product specification. The symbol used is \(H_{o}\).

Definition \(\PageIndex{2}\)

Alternate Hypothesis : what you want to prove. This is what you want to accept as true when you reject the null hypothesis. There are two symbols that are commonly used for the alternative hypothesis: \(H_{A}\) or \(H_{I}\). The symbol \(H_{A}\) will be used in this book.

In general, the hypotheses look something like this:

\(H_{o} : \mu=\mu_{o}\)

\(H_{A} : \mu<\mu_{o}\)

where \(\mu_{o}\) just represents the value that the claim says the population mean is actually equal to.

Also, \(H_{A}\) can be less than, greater than, or not equal to.

For this problem:

\(H_{o} : \mu=500\) days, since the manufacturer says the mean life of a battery is 500 days.

\(H_{A} : \mu<500\) days, since you believe that the mean life of the battery is less than 500 days.

Now back to the mean. You have a sample mean of 490 days. Is this small enough to believe that you are right and the manufacturer is wrong? How small does it have to be?

If you calculated a sample mean of 235, you would definitely believe the population mean is less than 500. But even if you had a sample mean of 435 you would probably believe that the true mean was less than 500. What about 475? Or 483? There is some point where you would stop being so sure that the population mean is less than 500. That point separates the values of where you are sure or pretty sure that the mean is less than 500 from the area where you are not so sure. How do you find that point?

Well it depends on how much error you want to make. Of course you don’t want to make any errors, but unfortunately that is unavoidable in statistics. You need to figure out how much error you made with your sample. Take the sample mean, and find the probability of getting another sample mean less than it, assuming for the moment that the manufacturer is right. The idea behind this is that you want to know what is the chance that you could have come up with your sample mean even if the population mean really is 500 days.

You want to find \(P\left(\overline{x}<490 | H_{o} \text { is true }\right)=P(\overline{x}<490 | \mu=500)\)

To compute this probability, you need to know how the sample mean is distributed. Since the sample size is at least 30, then you know the sample mean is approximately normally distributed. Remember \(\mu_{\overline{x}}=\mu\) and \(\sigma_{\overline{x}}=\dfrac{\sigma}{\sqrt{n}}\)

A picture is always useful.

Before calculating the probability, it is useful to see how many standard deviations away from the mean the sample mean is. Using the formula for the z-score from chapter 6, you find

\(z=\dfrac{\overline{x}-\mu_{o}}{\sigma / \sqrt{n}}=\dfrac{490-500}{25 / \sqrt{30}}=-2.19\)

This sample mean is more than two standard deviations away from the mean. That seems pretty far, but you should look at the probability too.

On TI-83/84:

\(P(\overline{x}<490 | \mu=500)=\text { normalcdf }(-1 E 99,490,500,25 \div \sqrt{30}) \approx 0.0142\)

\(P(\overline{x}<490 \mu=500)=\text { pnorm }(490,500,25 / \operatorname{sqrt}(30)) \approx 0.0142\)

There is a 1.42% chance that you could find a sample mean less than 490 when the population mean is 500 days. This is really small, so the chances are that the assumption that the population mean is 500 days is wrong, and you can reject the manufacturer’s claim. But how do you quantify really small? Is 5% or 10% or 15% really small? How do you decide?

Before you answer that question, a couple more definitions are needed.

Definition \(\PageIndex{3}\)

Test Statistic : \(z=\dfrac{\overline{x}-\mu_{o}}{\sigma / \sqrt{n}}\) since it is calculated as part of the testing of the hypothesis.

Definition \(\PageIndex{4}\)

p – value : probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true. It is the probability that was calculated above.

Now, how small is small enough? To answer that, you really want to know the types of errors you can make.

There are actually only two errors that can be made. The first error is if you say that \(H_{o}\) is false, when in fact it is true. This means you reject \(H_{o}\) when \(H_{o}\) was true. The second error is if you say that \(H_{o}\) is true, when in fact it is false. This means you fail to reject \(H_{o}\) when \(H_{o}\) is false. The following table organizes this for you:

Type of errors:

Definition \(\PageIndex{5}\)

Type I Error is rejecting \(H_{o}\) when \(H_{o}\) is true, and

Definition \(\PageIndex{6}\)

Type II Error is failing to reject \(H_{o}\) when \(H_{o}\) is false.

Since these are the errors, then one can define the probabilities attached to each error.

Definition \(\PageIndex{7}\)

\(\alpha\) = P(type I error) = P(rejecting \(H_{o} / H_{o}\) is true)

Definition \(\PageIndex{8}\)

\(\beta\) = P(type II error) = P(failing to reject \(H_{o} / H_{o}\) is false)

\(\alpha\) is also called the level of significance .

Another common concept that is used is Power = \(1-\beta \).

Now there is a relationship between \(\alpha\) and \(\beta\). They are not complements of each other. How are they related?

If \(\alpha\) increases that means the chances of making a type I error will increase. It is more likely that a type I error will occur. It makes sense that you are less likely to make type II errors, only because you will be rejecting \(H_{o}\) more often. You will be failing to reject \(H_{o}\) less, and therefore, the chance of making a type II error will decrease. Thus, as \(\alpha\) increases, \(\beta\) will decrease, and vice versa. That makes them seem like complements, but they aren’t complements. What gives? Consider one more factor – sample size.

Consider if you have a larger sample that is representative of the population, then it makes sense that you have more accuracy then with a smaller sample. Think of it this way, which would you trust more, a sample mean of 490 if you had a sample size of 35 or sample size of 350 (assuming a representative sample)? Of course the 350 because there are more data points and so more accuracy. If you are more accurate, then there is less chance that you will make any error. By increasing the sample size of a representative sample, you decrease both \(\alpha\) and \(\beta\).

Summary of all of this:

• For a certain sample size, n , if \(\alpha\) increases, \(\beta\) decreases.
• For a certain level of significance, \(\alpha\), if n increases, \(\beta\) decreases.

Now how do you find \(\alpha\) and \(\beta\)? Well \(\alpha\) is actually chosen. There are only three values that are usually picked for \(\alpha\): 0.01, 0.05, and 0.10. \(\beta\) is very difficult to find, so usually it isn’t found. If you want to make sure it is small you take as large of a sample as you can afford provided it is a representative sample. This is one use of the Power. You want \(\beta\) to be small and the Power of the test is large. The Power word sounds good.

Which pick of \(\alpha\) do you pick? Well that depends on what you are working on. Remember in this example you are the buyer who is trying to get out of a contract to buy these batteries. If you create a type I error, you said that the batteries are bad when they aren’t, most likely the manufacturer will sue you. You want to avoid this. You might pick \(\alpha\) to be 0.01. This way you have a small chance of making a type I error. Of course this means you have more of a chance of making a type II error. No big deal right? What if the batteries are used in pacemakers and you tell the person that their pacemaker’s batteries are good for 500 days when they actually last less, that might be bad. If you make a type II error, you say that the batteries do last 500 days when they last less, then you have the possibility of killing someone. You certainly do not want to do this. In this case you might want to pick \(\alpha\) as 0.10. If both errors are equally bad, then pick \(\alpha\) as 0.05.

The above discussion is why the choice of \(\alpha\) depends on what you are researching. As the researcher, you are the one that needs to decide what \(\alpha\) level to use based on your analysis of the consequences of making each error is.

If a type I error is really bad, then pick \(\alpha\) = 0.01.

If a type II error is really bad, then pick \(\alpha\) = 0.10

If neither error is bad, or both are equally bad, then pick \(\alpha\) = 0.05

The main thing is to always pick the \(\alpha\) before you collect the data and start the test.

The above discussion was long, but it is really important information. If you don’t know what the errors of the test are about, then there really is no point in making conclusions with the tests. Make sure you understand what the two errors are and what the probabilities are for them.

Now it is time to go back to the example and put this all together. This is the basic structure of testing a hypothesis, usually called a hypothesis test. Since this one has a test statistic involving z, it is also called a z-test. And since there is only one sample, it is usually called a one-sample z-test.

Example \(\PageIndex{2}\) battery example revisited

• State the random variable and the parameter in words.
• State the null and alternative hypothesis and the level of significance.
• A random sample of size n is taken.
• The population standard derivation is known.
• The sample size is at least 30 or the population of the random variable is normally distributed.
• Find the sample statistic, test statistic, and p-value.
• Interpretation

1. x = life of battery

\(\mu\) = mean life of a XJ35 battery

2. \(H_{o} : \mu=500\) days

\(H_{A} : \mu<500\) days

\(\alpha = 0.10\) (from above discussion about consequences)

3. Every hypothesis has some assumptions that be met to make sure that the results of the hypothesis are valid. The assumptions are different for each test. This test has the following assumptions.

• This occurred in this example, since it was stated that a random sample of 30 battery lives were taken.
• This is true, since it was given in the problem.
• The sample size was 30, so this condition is met.

4. The test statistic depends on how many samples there are, what parameter you are testing, and assumptions that need to be checked. In this case, there is one sample and you are testing the mean. The assumptions were checked above.

Sample statistic:

\(\overline{x} = 490\)

Test statistic:

Using TI-83/84:

\(P(\overline{x}<490 | \mu=500)=\text { normalcdf }(-1 \mathrm{E} 99,490,500,25 / \sqrt{30}) \approx 0.0142\)

\(P(\overline{x}<490 | \mu=500)=\operatorname{pnorm}(490,500,25 / \operatorname{sqrt}(30)) \approx 0.0142\)

5. Now what? Well, this p-value is 0.0142. This is a lot smaller than the amount of error you would accept in the problem -\(\alpha\) = 0.10. That means that finding a sample mean less than 490 days is unusual to happen if \(H_{o}\) is true. This should make you think that \(H_{o}\) is not true. You should reject \(H_{o}\).

In fact, in general:

Reject \(H_{o}\) if the p-value < \(\alpha\) and

Fail to reject \(H_{o}\) if the p-value \(\geq \alpha\).

6. Since you rejected \(H_{o}\), what does this mean in the real world? That is what goes in the interpretation. Since you rejected the claim by the manufacturer that the mean life of the batteries is 500 days, then you now can believe that your hypothesis was correct. In other words, there is enough evidence to show that the mean life of the battery is less than 500 days.

Now that you know that the batteries last less than 500 days, should you cancel the contract? Statistically, there is evidence that the batteries do not last as long as the manufacturer says they should. However, based on this sample there are only ten days less on average that the batteries last. There may not be practical significance in this case. Ten days do not seem like a large difference. In reality, if the batteries are used in pacemakers, then you would probably tell the patient to have the batteries replaced every year. You have a large buffer whether the batteries last 490 days or 500 days. It seems that it might not be worth it to break the contract over ten days. What if the 10 days was practically significant? Are there any other things you should consider? You might look at the business relationship with the manufacturer. You might also look at how much it would cost to find a new manufacturer. These are also questions to consider before making any changes. What this discussion should show you is that just because a hypothesis has statistical significance does not mean it has practical significance. The hypothesis test is just one part of a research process. There are other pieces that you need to consider.

That’s it. That is what a hypothesis test looks like. All hypothesis tests are done with the same six steps. Those general six steps are outlined below.

• State the random variable and the parameter in words. This is where you are defining what the unknowns are in this problem. x = random variable \(\mu\) = mean of random variable, if the parameter of interest is the mean. There are other parameters you can test, and you would use the appropriate symbol for that parameter.
• State the null and alternative hypotheses and the level of significance \(H_{o} : \mu=\mu_{o}\), where \(\mu_{o}\) is the known mean \(H_{A} : \mu<\mu_{o}\) \(H_{A} : \mu>\mu_{o}\), use the appropriate one for your problem \(H_{A} : \mu \neq \mu_{o}\) Also, state your \(\alpha\) level here.
• State and check the assumptions for a hypothesis test. Each hypothesis test has its own assumptions. They will be stated when the different hypothesis tests are discussed.
• Find the sample statistic, test statistic, and p-value. This depends on what parameter you are working with, how many samples, and the assumptions of the test. The p-value depends on your \(H_{A}\). If you are doing the \(H_{A}\) with the less than, then it is a left-tailed test, and you find the probability of being in that left tail. If you are doing the \(H_{A}\) with the greater than, then it is a right-tailed test, and you find the probability of being in the right tail. If you are doing the \(H_{A}\) with the not equal to, then you are doing a two-tail test, and you find the probability of being in both tails. Because of symmetry, you could find the probability in one tail and double this value to find the probability in both tails.
• Conclusion This is where you write reject \(H_{o}\) or fail to reject \(H_{o}\). The rule is: if the p-value < \(\alpha\), then reject \(H_{o}\). If the p-value \(\geq \alpha\), then fail to reject \(H_{o}\).
• Interpretation This is where you interpret in real world terms the conclusion to the test. The conclusion for a hypothesis test is that you either have enough evidence to show \(H_{A}\) is true, or you do not have enough evidence to show \(H_{A}\) is true.

Sorry, one more concept about the conclusion and interpretation. First, the conclusion is that you reject \(H_{o}\) or you fail to reject \(H_{o}\). Why was it said like this? It is because you never accept the null hypothesis. If you wanted to accept the null hypothesis, then why do the test in the first place? In the interpretation, you either have enough evidence to show \(H_{A}\) is true, or you do not have enough evidence to show \(H_{A}\) is true. You wouldn’t want to go to all this work and then find out you wanted to accept the claim. Why go through the trouble? You always want to show that the alternative hypothesis is true. Sometimes you can do that and sometimes you can’t. It doesn’t mean you proved the null hypothesis; it just means you can’t prove the alternative hypothesis. Here is an example to demonstrate this.

Example \(\PageIndex{3}\) conclusion in hypothesis tests

In the U.S. court system a jury trial could be set up as a hypothesis test. To really help you see how this works, let’s use OJ Simpson as an example. In the court system, a person is presumed innocent until he/she is proven guilty, and this is your null hypothesis. OJ Simpson was a football player in the 1970s. In 1994 his ex-wife and her friend were killed. OJ Simpson was accused of the crime, and in 1995 the case was tried. The prosecutors wanted to prove OJ was guilty of killing his wife and her friend, and that is the alternative hypothesis

\(H_{0}\): OJ is innocent of killing his wife and her friend

\(H_{A}\): OJ is guilty of killing his wife and her friend

In this case, a verdict of not guilty was given. That does not mean that he is innocent of this crime. It means there was not enough evidence to prove he was guilty. Many people believe that OJ was guilty of this crime, but the jury did not feel that the evidence presented was enough to show there was guilt. The verdict in a jury trial is always guilty or not guilty!

The same is true in a hypothesis test. There is either enough or not enough evidence to show that alternative hypothesis. It is not that you proved the null hypothesis true.

When identifying hypothesis, it is important to state your random variable and the appropriate parameter you want to make a decision about. If count something, then the random variable is the number of whatever you counted. The parameter is the proportion of what you counted. If the random variable is something you measured, then the parameter is the mean of what you measured. (Note: there are other parameters you can calculate, and some analysis of those will be presented in later chapters.)

Example \(\PageIndex{4}\) stating hypotheses

Identify the hypotheses necessary to test the following statements:

• The average salary of a teacher is more than $30,000.
• The proportion of students who like math is less than 10%.
• The average age of students in this class differs from 21.

a. x = salary of teacher

\(\mu\) = mean salary of teacher

The guess is that \(\mu>\$ 30,000\) and that is the alternative hypothesis.

The null hypothesis has the same parameter and number with an equal sign.

\(\begin{array}{l}{H_{0} : \mu=\$ 30,000} \\ {H_{A} : \mu>\$ 30,000}\end{array}\)

b. x = number od students who like math

p = proportion of students who like math

The guess is that p < 0.10 and that is the alternative hypothesis.

\(\begin{array}{l}{H_{0} : p=0.10} \\ {H_{A} : p<0.10}\end{array}\)

c. x = age of students in this class

\(\mu\) = mean age of students in this class

The guess is that \(\mu \neq 21\) and that is the alternative hypothesis.

\(\begin{array}{c}{H_{0} : \mu=21} \\ {H_{A} : \mu \neq 21}\end{array}\)

Example \(\PageIndex{5}\) Stating Type I and II Errors and Picking Level of Significance

• The plant-breeding department at a major university developed a new hybrid raspberry plant called YumYum Berry. Based on research data, the claim is made that from the time shoots are planted 90 days on average are required to obtain the first berry with a standard deviation of 9.2 days. A corporation that is interested in marketing the product tests 60 shoots by planting them and recording the number of days before each plant produces its first berry. The sample mean is 92.3 days. The corporation wants to know if the mean number of days is more than the 90 days claimed. State the type I and type II errors in terms of this problem, consequences of each error, and state which level of significance to use.
• A concern was raised in Australia that the percentage of deaths of Aboriginal prisoners was higher than the percent of deaths of non-indigenous prisoners, which is 0.27%. State the type I and type II errors in terms of this problem, consequences of each error, and state which level of significance to use.

a. x = time to first berry for YumYum Berry plant

\(\mu\) = mean time to first berry for YumYum Berry plant

\(\begin{array}{l}{H_{0} : \mu=90} \\ {H_{A} : \mu>90}\end{array}\)

Type I Error: If the corporation does a type I error, then they will say that the plants take longer to produce than 90 days when they don’t. They probably will not want to market the plants if they think they will take longer. They will not market them even though in reality the plants do produce in 90 days. They may have loss of future earnings, but that is all.

Type II error: The corporation do not say that the plants take longer then 90 days to produce when they do take longer. Most likely they will market the plants. The plants will take longer, and so customers might get upset and then the company would get a bad reputation. This would be really bad for the company.

Level of significance: It appears that the corporation would not want to make a type II error. Pick a 10% level of significance, \(\alpha = 0.10\).

b. x = number of Aboriginal prisoners who have died

p = proportion of Aboriginal prisoners who have died

\(\begin{array}{l}{H_{o} : p=0.27 \%} \\ {H_{A} : p>0.27 \%}\end{array}\)

Type I error: Rejecting that the proportion of Aboriginal prisoners who died was 0.27%, when in fact it was 0.27%. This would mean you would say there is a problem when there isn’t one. You could anger the Aboriginal community, and spend time and energy researching something that isn’t a problem.

Type II error: Failing to reject that the proportion of Aboriginal prisoners who died was 0.27%, when in fact it is higher than 0.27%. This would mean that you wouldn’t think there was a problem with Aboriginal prisoners dying when there really is a problem. You risk causing deaths when there could be a way to avoid them.

Level of significance: It appears that both errors may be issues in this case. You wouldn’t want to anger the Aboriginal community when there isn’t an issue, and you wouldn’t want people to die when there may be a way to stop it. It may be best to pick a 5% level of significance, \(\alpha = 0.05\).

Hypothesis testing is really easy if you follow the same recipe every time. The only differences in the various problems are the assumptions of the test and the test statistic you calculate so you can find the p-value. Do the same steps, in the same order, with the same words, every time and these problems become very easy.

Exercise \(\PageIndex{1}\)

For the problems in this section, a question is being asked. This is to help you understand what the hypotheses are. You are not to run any hypothesis tests and come up with any conclusions in this section.

• Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? State the random variable, population parameter, and hypotheses.
• According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? State the random variable, population parameter, and hypotheses.
• The Kyoto Protocol was signed in 1997, and required countries to start reducing their carbon emissions. The protocol became enforceable in February 2005. In 2004, the mean CO2 emission was 4.87 metric tons per capita. Is there enough evidence to show that the mean CO2 emission is lower in 2010 than in 2004? State the random variable, population parameter, and hypotheses.
• The FDA regulates that fish that is consumed is allowed to contain 1.0 mg/kg of mercury. In Florida, bass fish were collected in 53 different lakes to measure the amount of mercury in the fish. The data for the average amount of mercury in each lake is in Example \(\PageIndex{5}\) ("Multi-disciplinary niser activity," 2013). Do the data provide enough evidence to show that the fish in Florida lakes has more mercury than the allowable amount? State the random variable, population parameter, and hypotheses.
• Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the manufacturer, and the appropriate alpha level to use. State why you picked this alpha level.
• According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the state of Arizona, and the appropriate alpha level to use. State why you picked this alpha level.
• The Kyoto Protocol was signed in 1997, and required countries to start reducing their carbon emissions. The protocol became enforceable in February 2005. In 2004, the mean CO2 emission was 4.87 metric tons per capita. Is there enough evidence to show that the mean CO2 emission is lower in 2010 than in 2004? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the agency overseeing the protocol, and the appropriate alpha level to use. State why you picked this alpha level.
• The FDA regulates that fish that is consumed is allowed to contain 1.0 mg/kg of mercury. In Florida, bass fish were collected in 53 different lakes to measure the amount of mercury in the fish. The data for the average amount of mercury in each lake is in Example \(\PageIndex{5}\) ("Multi-disciplinary niser activity," 2013). Do the data provide enough evidence to show that the fish in Florida lakes has more mercury than the allowable amount? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the FDA, and the appropriate alpha level to use. State why you picked this alpha level.

1. \(H_{o} : p=0.11, H_{A} : p>0.11\)

3. \(H_{o} : \mu=4.87 \text { metric tons per capita, } H_{A} : \mu<4.87 \text { metric tons per capita }\)

5. See solutions

7. See solutions

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

3.14: Experiments and Hypotheses

• Last updated
• Save as PDF
• Page ID 43504

Now we’ll focus on the methods of scientific inquiry. Science often involves making observations and developing hypotheses. Experiments and further observations are often used to test the hypotheses.

A scientific experiment is a carefully organized procedure in which the scientist intervenes in a system to change something, then observes the result of the change. Scientific inquiry often involves doing experiments, though not always. For example, a scientist studying the mating behaviors of ladybugs might begin with detailed observations of ladybugs mating in their natural habitats. While this research may not be experimental, it is scientific: it involves careful and verifiable observation of the natural world. The same scientist might then treat some of the ladybugs with a hormone hypothesized to trigger mating and observe whether these ladybugs mated sooner or more often than untreated ones. This would qualify as an experiment because the scientist is now making a change in the system and observing the effects.

Forming a Hypothesis

When conducting scientific experiments, researchers develop hypotheses to guide experimental design. A hypothesis is a suggested explanation that is both testable and falsifiable. You must be able to test your hypothesis, and it must be possible to prove your hypothesis true or false.

For example, Michael observes that maple trees lose their leaves in the fall. He might then propose a possible explanation for this observation: “cold weather causes maple trees to lose their leaves in the fall.” This statement is testable. He could grow maple trees in a warm enclosed environment such as a greenhouse and see if their leaves still dropped in the fall. The hypothesis is also falsifiable. If the leaves still dropped in the warm environment, then clearly temperature was not the main factor in causing maple leaves to drop in autumn.

In the Try It below, you can practice recognizing scientific hypotheses. As you consider each statement, try to think as a scientist would: can I test this hypothesis with observations or experiments? Is the statement falsifiable? If the answer to either of these questions is “no,” the statement is not a valid scientific hypothesis.

Practice Questions

Determine whether each following statement is a scientific hypothesis.

Air pollution from automobile exhaust can trigger symptoms in people with asthma.

• No. This statement is not testable or falsifiable.
• No. This statement is not testable.
• No. This statement is not falsifiable.
• Yes. This statement is testable and falsifiable.

[reveal-answer q=”429550″] Show Answer [/reveal-answer] [hidden-answer a=”429550″]d: Yes. This statement is testable and falsifiable. This could be tested with a number of different kinds of observations and experiments, and it is possible to gather evidence that indicates that air pollution is not linked with asthma.

[/hidden-answer]

Natural disasters, such as tornadoes, are punishments for bad thoughts and behaviors.

[reveal-answer q=”74245″]Show Answer[/reveal-answer] [hidden-answer a=”74245″]

a: No. This statement is not testable or falsifiable. “Bad thoughts and behaviors” are excessively vague and subjective variables that would be impossible to measure or agree upon in a reliable way. The statement might be “falsifiable” if you came up with a counterexample: a “wicked” place that was not punished by a natural disaster. But some would question whether the people in that place were really wicked, and others would continue to predict that a natural disaster was bound to strike that place at some point. There is no reason to suspect that people’s immoral behavior affects the weather unless you bring up the intervention of a supernatural being, making this idea even harder to test.

Testing a Vaccine

Let’s examine the scientific process by discussing an actual scientific experiment conducted by researchers at the University of Washington. These researchers investigated whether a vaccine may reduce the incidence of the human papillomavirus (HPV). The experimental process and results were published in an article titled, “ A controlled trial of a human papillomavirus type 16 vaccine .”

Preliminary observations made by the researchers who conducted the HPV experiment are listed below:

• Human papillomavirus (HPV) is the most common sexually transmitted virus in the United States.
• There are about 40 different types of HPV. A significant number of people that have HPV are unaware of it because many of these viruses cause no symptoms.
• Some types of HPV can cause cervical cancer.
• About 4,000 women a year die of cervical cancer in the United States.

Practice Question

Researchers have developed a potential vaccine against HPV and want to test it. What is the first testable hypothesis that the researchers should study?

• HPV causes cervical cancer.
• People should not have unprotected sex with many partners.
• People who get the vaccine will not get HPV.
• The HPV vaccine will protect people against cancer.

[reveal-answer q=”20917″] Show Answer [/reveal-answer] [hidden-answer a=”20917″]Hypothesis A is not the best choice because this information is already known from previous studies. Hypothesis B is not testable because scientific hypotheses are not value statements; they do not include judgments like “should,” “better than,” etc. Scientific evidence certainly might support this value judgment, but a hypothesis would take a different form: “Having unprotected sex with many partners increases a person’s risk for cervical cancer.” Before the researchers can test if the vaccine protects against cancer (hypothesis D), they want to test if it protects against the virus. This statement will make an excellent hypothesis for the next study. The researchers should first test hypothesis C—whether or not the new vaccine can prevent HPV.[/hidden-answer]

Experimental Design

You’ve successfully identified a hypothesis for the University of Washington’s study on HPV: People who get the HPV vaccine will not get HPV.

The next step is to design an experiment that will test this hypothesis. There are several important factors to consider when designing a scientific experiment. First, scientific experiments must have an experimental group. This is the group that receives the experimental treatment necessary to address the hypothesis.

The experimental group receives the vaccine, but how can we know if the vaccine made a difference? Many things may change HPV infection rates in a group of people over time. To clearly show that the vaccine was effective in helping the experimental group, we need to include in our study an otherwise similar control group that does not get the treatment. We can then compare the two groups and determine if the vaccine made a difference. The control group shows us what happens in the absence of the factor under study.

However, the control group cannot get “nothing.” Instead, the control group often receives a placebo. A placebo is a procedure that has no expected therapeutic effect—such as giving a person a sugar pill or a shot containing only plain saline solution with no drug. Scientific studies have shown that the “placebo effect” can alter experimental results because when individuals are told that they are or are not being treated, this knowledge can alter their actions or their emotions, which can then alter the results of the experiment.

Moreover, if the doctor knows which group a patient is in, this can also influence the results of the experiment. Without saying so directly, the doctor may show—through body language or other subtle cues—his or her views about whether the patient is likely to get well. These errors can then alter the patient’s experience and change the results of the experiment. Therefore, many clinical studies are “double blind.” In these studies, neither the doctor nor the patient knows which group the patient is in until all experimental results have been collected.

Both placebo treatments and double-blind procedures are designed to prevent bias. Bias is any systematic error that makes a particular experimental outcome more or less likely. Errors can happen in any experiment: people make mistakes in measurement, instruments fail, computer glitches can alter data. But most such errors are random and don’t favor one outcome over another. Patients’ belief in a treatment can make it more likely to appear to “work.” Placebos and double-blind procedures are used to level the playing field so that both groups of study subjects are treated equally and share similar beliefs about their treatment.

The scientists who are researching the effectiveness of the HPV vaccine will test their hypothesis by separating 2,392 young women into two groups: the control group and the experimental group. Answer the following questions about these two groups.

• This group is given a placebo.
• This group is deliberately infected with HPV.
• This group is given nothing.
• This group is given the HPV vaccine.

[reveal-answer q=”918962″] Show Answers [/reveal-answer] [hidden-answer a=”918962″]

• a: This group is given a placebo. A placebo will be a shot, just like the HPV vaccine, but it will have no active ingredient. It may change peoples’ thinking or behavior to have such a shot given to them, but it will not stimulate the immune systems of the subjects in the same way as predicted for the vaccine itself.
• d: This group is given the HPV vaccine. The experimental group will receive the HPV vaccine and researchers will then be able to see if it works, when compared to the control group.

Experimental Variables

A variable is a characteristic of a subject (in this case, of a person in the study) that can vary over time or among individuals. Sometimes a variable takes the form of a category, such as male or female; often a variable can be measured precisely, such as body height. Ideally, only one variable is different between the control group and the experimental group in a scientific experiment. Otherwise, the researchers will not be able to determine which variable caused any differences seen in the results. For example, imagine that the people in the control group were, on average, much more sexually active than the people in the experimental group. If, at the end of the experiment, the control group had a higher rate of HPV infection, could you confidently determine why? Maybe the experimental subjects were protected by the vaccine, but maybe they were protected by their low level of sexual contact.

To avoid this situation, experimenters make sure that their subject groups are as similar as possible in all variables except for the variable that is being tested in the experiment. This variable, or factor, will be deliberately changed in the experimental group. The one variable that is different between the two groups is called the independent variable. An independent variable is known or hypothesized to cause some outcome. Imagine an educational researcher investigating the effectiveness of a new teaching strategy in a classroom. The experimental group receives the new teaching strategy, while the control group receives the traditional strategy. It is the teaching strategy that is the independent variable in this scenario. In an experiment, the independent variable is the variable that the scientist deliberately changes or imposes on the subjects.

Dependent variables are known or hypothesized consequences; they are the effects that result from changes or differences in an independent variable. In an experiment, the dependent variables are those that the scientist measures before, during, and particularly at the end of the experiment to see if they have changed as expected. The dependent variable must be stated so that it is clear how it will be observed or measured. Rather than comparing “learning” among students (which is a vague and difficult to measure concept), an educational researcher might choose to compare test scores, which are very specific and easy to measure.

In any real-world example, many, many variables MIGHT affect the outcome of an experiment, yet only one or a few independent variables can be tested. Other variables must be kept as similar as possible between the study groups and are called control variables . For our educational research example, if the control group consisted only of people between the ages of 18 and 20 and the experimental group contained people between the ages of 30 and 35, we would not know if it was the teaching strategy or the students’ ages that played a larger role in the results. To avoid this problem, a good study will be set up so that each group contains students with a similar age profile. In a well-designed educational research study, student age will be a controlled variable, along with other possibly important factors like gender, past educational achievement, and pre-existing knowledge of the subject area.

What is the independent variable in this experiment?

• Sex (all of the subjects will be female)
• Presence or absence of the HPV vaccine
• Presence or absence of HPV (the virus)

[reveal-answer q=”68680″]Show Answer[/reveal-answer] [hidden-answer a=”68680″]Answer b. Presence or absence of the HPV vaccine. This is the variable that is different between the control and the experimental groups. All the subjects in this study are female, so this variable is the same in all groups. In a well-designed study, the two groups will be of similar age. The presence or absence of the virus is what the researchers will measure at the end of the experiment. Ideally the two groups will both be HPV-free at the start of the experiment.

List three control variables other than age.

[practice-area rows=”3″][/practice-area] [reveal-answer q=”903121″]Show Answer[/reveal-answer] [hidden-answer a=”903121″]Some possible control variables would be: general health of the women, sexual activity, lifestyle, diet, socioeconomic status, etc.

What is the dependent variable in this experiment?

• Sex (male or female)
• Rates of HPV infection
• Age (years)

[reveal-answer q=”907103″]Show Answer[/reveal-answer] [hidden-answer a=”907103″]Answer b. Rates of HPV infection. The researchers will measure how many individuals got infected with HPV after a given period of time.[/hidden-answer]

Contributors and Attributions

• Revision and adaptation. Authored by : Shelli Carter and Lumen Learning. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
• Scientific Inquiry. Provided by : Open Learning Initiative. Located at : https://oli.cmu.edu/jcourse/workbook/activity/page?context=434a5c2680020ca6017c03488572e0f8 . Project : Introduction to Biology (Open + Free). License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike

What is a scientific hypothesis?

It's the initial building block in the scientific method.

Hypothesis basics

What makes a hypothesis testable.

• Types of hypotheses
• Hypothesis versus theory

Additional resources

Bibliography.

A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research. 

The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).

A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.

A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .

Here are some examples of hypothesis statements:

• If garlic repels fleas, then a dog that is given garlic every day will not get fleas.
• If sugar causes cavities, then people who eat a lot of candy may be more prone to cavities.
• If ultraviolet light can damage the eyes, then maybe this light can cause blindness.

A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."

An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.

Types of scientific hypotheses

In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .

For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."

If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (​​BCcampus, 2015). 

There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.

Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley . 

A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.

Scientific theory vs. scientific hypothesis

The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.

"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts." 

• Read more about writing a hypothesis, from the American Medical Writers Association.
• Find out why a hypothesis isn't always necessary in science, from The American Biology Teacher.
• Learn about null and alternative hypotheses, from Prof. Essa on YouTube .

Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis

Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.

California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm  

Karl Popper, "Conjectures and Refutations," Routledge, 1963.

Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.‌

University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf  

William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/  

University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf  

University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19

Sign up for the Live Science daily newsletter now

Get the world’s most fascinating discoveries delivered straight to your inbox.

Part of the San Andreas fault may be gearing up for an earthquake

Antarctica is covered in volcanoes, could they erupt?

'Exceptional' prosthesis of gold, silver and wool helped 18th-century man live with cleft palate

Most Popular

• 2 Eclipse from space: See the moon's shadow race across North America at 1,500 mph in epic satellite footage
• 3 Superfast drone fitted with new 'rotating detonation rocket engine' approaches the speed of sound
• 4 NASA spacecraft snaps mysterious 'surfboard' orbiting the moon. What is it?
• 5 Neolithic women in Europe were tied up and buried alive in ritual sacrifices, study suggests
• 2 No, you didn't see a solar flare during the total eclipse — but you may have seen something just as special
• 3 Decomposing globster washes ashore in Malaysia, drawing crowds
• 4 Eclipse from space: See the moon's shadow race across North America at 1,500 mph in epic satellite footage
• 5 Superfast drone fitted with new 'rotating detonation rocket engine' approaches the speed of sound

Choose Your Test

Sat / act prep online guides and tips, what is a hypothesis and how do i write one.

General Education

Think about something strange and unexplainable in your life. Maybe you get a headache right before it rains, or maybe you think your favorite sports team wins when you wear a certain color. If you wanted to see whether these are just coincidences or scientific fact, you would form a hypothesis, then create an experiment to see whether that hypothesis is true or not.

But what is a hypothesis, anyway? If you’re not sure about what a hypothesis is--or how to test for one!--you’re in the right place. This article will teach you everything you need to know about hypotheses, including: 

• Defining the term “hypothesis” 
• Providing hypothesis examples 
• Giving you tips for how to write your own hypothesis

So let’s get started!

What Is a Hypothesis?

Merriam Webster defines a hypothesis as “an assumption or concession made for the sake of argument.” In other words, a hypothesis is an educated guess . Scientists make a reasonable assumption--or a hypothesis--then design an experiment to test whether it’s true or not. Keep in mind that in science, a hypothesis should be testable. You have to be able to design an experiment that tests your hypothesis in order for it to be valid. 

As you could assume from that statement, it’s easy to make a bad hypothesis. But when you’re holding an experiment, it’s even more important that your guesses be good...after all, you’re spending time (and maybe money!) to figure out more about your observation. That’s why we refer to a hypothesis as an educated guess--good hypotheses are based on existing data and research to make them as sound as possible.

Hypotheses are one part of what’s called the scientific method .  Every (good) experiment or study is based in the scientific method. The scientific method gives order and structure to experiments and ensures that interference from scientists or outside influences does not skew the results. It’s important that you understand the concepts of the scientific method before holding your own experiment. Though it may vary among scientists, the scientific method is generally made up of six steps (in order):

• Observation
• Asking questions
• Forming a hypothesis
• Analyze the data
• Communicate your results

You’ll notice that the hypothesis comes pretty early on when conducting an experiment. That’s because experiments work best when they’re trying to answer one specific question. And you can’t conduct an experiment until you know what you’re trying to prove!

Independent and Dependent Variables 

After doing your research, you’re ready for another important step in forming your hypothesis: identifying variables. Variables are basically any factor that could influence the outcome of your experiment . Variables have to be measurable and related to the topic being studied.

There are two types of variables:  independent variables and dependent variables. I ndependent variables remain constant . For example, age is an independent variable; it will stay the same, and researchers can look at different ages to see if it has an effect on the dependent variable. 

Speaking of dependent variables... dependent variables are subject to the influence of the independent variable , meaning that they are not constant. Let’s say you want to test whether a person’s age affects how much sleep they need. In that case, the independent variable is age (like we mentioned above), and the dependent variable is how much sleep a person gets. 

Variables will be crucial in writing your hypothesis. You need to be able to identify which variable is which, as both the independent and dependent variables will be written into your hypothesis. For instance, in a study about exercise, the independent variable might be the speed at which the respondents walk for thirty minutes, and the dependent variable would be their heart rate. In your study and in your hypothesis, you’re trying to understand the relationship between the two variables.

Elements of a Good Hypothesis

The best hypotheses start by asking the right questions . For instance, if you’ve observed that the grass is greener when it rains twice a week, you could ask what kind of grass it is, what elevation it’s at, and if the grass across the street responds to rain in the same way. Any of these questions could become the backbone of experiments to test why the grass gets greener when it rains fairly frequently.

As you’re asking more questions about your first observation, make sure you’re also making more observations . If it doesn’t rain for two weeks and the grass still looks green, that’s an important observation that could influence your hypothesis. You'll continue observing all throughout your experiment, but until the hypothesis is finalized, every observation should be noted.

Finally, you should consult secondary research before writing your hypothesis . Secondary research is comprised of results found and published by other people. You can usually find this information online or at your library. Additionally, m ake sure the research you find is credible and related to your topic. If you’re studying the correlation between rain and grass growth, it would help you to research rain patterns over the past twenty years for your county, published by a local agricultural association. You should also research the types of grass common in your area, the type of grass in your lawn, and whether anyone else has conducted experiments about your hypothesis. Also be sure you’re checking the quality of your research . Research done by a middle school student about what minerals can be found in rainwater would be less useful than an article published by a local university.

Writing Your Hypothesis

Once you’ve considered all of the factors above, you’re ready to start writing your hypothesis. Hypotheses usually take a certain form when they’re written out in a research report.

When you boil down your hypothesis statement, you are writing down your best guess and not the question at hand . This means that your statement should be written as if it is fact already, even though you are simply testing it.

The reason for this is that, after you have completed your study, you'll either accept or reject your if-then or your null hypothesis. All hypothesis testing examples should be measurable and able to be confirmed or denied. You cannot confirm a question, only a statement! 

In fact, you come up with hypothesis examples all the time! For instance, when you guess on the outcome of a basketball game, you don’t say, “Will the Miami Heat beat the Boston Celtics?” but instead, “I think the Miami Heat will beat the Boston Celtics.” You state it as if it is already true, even if it turns out you’re wrong. You do the same thing when writing your hypothesis.

Additionally, keep in mind that hypotheses can range from very specific to very broad.  These hypotheses can be specific, but if your hypothesis testing examples involve a broad range of causes and effects, your hypothesis can also be broad.  

The Two Types of Hypotheses

Now that you understand what goes into a hypothesis, it’s time to look more closely at the two most common types of hypothesis: the if-then hypothesis and the null hypothesis.

#1: If-Then Hypotheses

First of all, if-then hypotheses typically follow this formula:

If ____ happens, then ____ will happen.

The goal of this type of hypothesis is to test the causal relationship between the independent and dependent variable. It’s fairly simple, and each hypothesis can vary in how detailed it can be. We create if-then hypotheses all the time with our daily predictions. Here are some examples of hypotheses that use an if-then structure from daily life: 

• If I get enough sleep, I’ll be able to get more work done tomorrow.
• If the bus is on time, I can make it to my friend’s birthday party. 
• If I study every night this week, I’ll get a better grade on my exam. 

In each of these situations, you’re making a guess on how an independent variable (sleep, time, or studying) will affect a dependent variable (the amount of work you can do, making it to a party on time, or getting better grades). 

You may still be asking, “What is an example of a hypothesis used in scientific research?” Take one of the hypothesis examples from a real-world study on whether using technology before bed affects children’s sleep patterns. The hypothesis read s:

“We hypothesized that increased hours of tablet- and phone-based screen time at bedtime would be inversely correlated with sleep quality and child attention.”

It might not look like it, but this is an if-then statement. The researchers basically said, “If children have more screen usage at bedtime, then their quality of sleep and attention will be worse.” The sleep quality and attention are the dependent variables and the screen usage is the independent variable. (Usually, the independent variable comes after the “if” and the dependent variable comes after the “then,” as it is the independent variable that affects the dependent variable.) This is an excellent example of how flexible hypothesis statements can be, as long as the general idea of “if-then” and the independent and dependent variables are present.

#2: Null Hypotheses

Your if-then hypothesis is not the only one needed to complete a successful experiment, however. You also need a null hypothesis to test it against. In its most basic form, the null hypothesis is the opposite of your if-then hypothesis . When you write your null hypothesis, you are writing a hypothesis that suggests that your guess is not true, and that the independent and dependent variables have no relationship .

One null hypothesis for the cell phone and sleep study from the last section might say: 

“If children have more screen usage at bedtime, their quality of sleep and attention will not be worse.” 

In this case, this is a null hypothesis because it’s asking the opposite of the original thesis! 

Conversely, if your if-then hypothesis suggests that your two variables have no relationship, then your null hypothesis would suggest that there is one. So, pretend that there is a study that is asking the question, “Does the amount of followers on Instagram influence how long people spend on the app?” The independent variable is the amount of followers, and the dependent variable is the time spent. But if you, as the researcher, don’t think there is a relationship between the number of followers and time spent, you might write an if-then hypothesis that reads:

“If people have many followers on Instagram, they will not spend more time on the app than people who have less.”

In this case, the if-then suggests there isn’t a relationship between the variables. In that case, one of the null hypothesis examples might say:

“If people have many followers on Instagram, they will spend more time on the app than people who have less.”

You then test both the if-then and the null hypothesis to gauge if there is a relationship between the variables, and if so, how much of a relationship. 

4 Tips to Write the Best Hypothesis

If you’re going to take the time to hold an experiment, whether in school or by yourself, you’re also going to want to take the time to make sure your hypothesis is a good one. The best hypotheses have four major elements in common: plausibility, defined concepts, observability, and general explanation.

#1: Plausibility

At first glance, this quality of a hypothesis might seem obvious. When your hypothesis is plausible, that means it’s possible given what we know about science and general common sense. However, improbable hypotheses are more common than you might think. 

Imagine you’re studying weight gain and television watching habits. If you hypothesize that people who watch more than  twenty hours of television a week will gain two hundred pounds or more over the course of a year, this might be improbable (though it’s potentially possible). Consequently, c ommon sense can tell us the results of the study before the study even begins.

Improbable hypotheses generally go against  science, as well. Take this hypothesis example: 

“If a person smokes one cigarette a day, then they will have lungs just as healthy as the average person’s.” 

This hypothesis is obviously untrue, as studies have shown again and again that cigarettes negatively affect lung health. You must be careful that your hypotheses do not reflect your own personal opinion more than they do scientifically-supported findings. This plausibility points to the necessity of research before the hypothesis is written to make sure that your hypothesis has not already been disproven.

#2: Defined Concepts

The more advanced you are in your studies, the more likely that the terms you’re using in your hypothesis are specific to a limited set of knowledge. One of the hypothesis testing examples might include the readability of printed text in newspapers, where you might use words like “kerning” and “x-height.” Unless your readers have a background in graphic design, it’s likely that they won’t know what you mean by these terms. Thus, it’s important to either write what they mean in the hypothesis itself or in the report before the hypothesis.

Here’s what we mean. Which of the following sentences makes more sense to the common person?

If the kerning is greater than average, more words will be read per minute.

If the space between letters is greater than average, more words will be read per minute.

For people reading your report that are not experts in typography, simply adding a few more words will be helpful in clarifying exactly what the experiment is all about. It’s always a good idea to make your research and findings as accessible as possible. 

Good hypotheses ensure that you can observe the results. 

#3: Observability

In order to measure the truth or falsity of your hypothesis, you must be able to see your variables and the way they interact. For instance, if your hypothesis is that the flight patterns of satellites affect the strength of certain television signals, yet you don’t have a telescope to view the satellites or a television to monitor the signal strength, you cannot properly observe your hypothesis and thus cannot continue your study.

Some variables may seem easy to observe, but if you do not have a system of measurement in place, you cannot observe your hypothesis properly. Here’s an example: if you’re experimenting on the effect of healthy food on overall happiness, but you don’t have a way to monitor and measure what “overall happiness” means, your results will not reflect the truth. Monitoring how often someone smiles for a whole day is not reasonably observable, but having the participants state how happy they feel on a scale of one to ten is more observable. 

In writing your hypothesis, always keep in mind how you'll execute the experiment.

#4: Generalizability 

Perhaps you’d like to study what color your best friend wears the most often by observing and documenting the colors she wears each day of the week. This might be fun information for her and you to know, but beyond you two, there aren’t many people who could benefit from this experiment. When you start an experiment, you should note how generalizable your findings may be if they are confirmed. Generalizability is basically how common a particular phenomenon is to other people’s everyday life.

Let’s say you’re asking a question about the health benefits of eating an apple for one day only, you need to realize that the experiment may be too specific to be helpful. It does not help to explain a phenomenon that many people experience. If you find yourself with too specific of a hypothesis, go back to asking the big question: what is it that you want to know, and what do you think will happen between your two variables?

Hypothesis Testing Examples

We know it can be hard to write a good hypothesis unless you’ve seen some good hypothesis examples. We’ve included four hypothesis examples based on some made-up experiments. Use these as templates or launch pads for coming up with your own hypotheses.

Experiment #1: Students Studying Outside (Writing a Hypothesis)

You are a student at PrepScholar University. When you walk around campus, you notice that, when the temperature is above 60 degrees, more students study in the quad. You want to know when your fellow students are more likely to study outside. With this information, how do you make the best hypothesis possible?

You must remember to make additional observations and do secondary research before writing your hypothesis. In doing so, you notice that no one studies outside when it’s 75 degrees and raining, so this should be included in your experiment. Also, studies done on the topic beforehand suggested that students are more likely to study in temperatures less than 85 degrees. With this in mind, you feel confident that you can identify your variables and write your hypotheses:

If-then: “If the temperature in Fahrenheit is less than 60 degrees, significantly fewer students will study outside.”

Null: “If the temperature in Fahrenheit is less than 60 degrees, the same number of students will study outside as when it is more than 60 degrees.”

These hypotheses are plausible, as the temperatures are reasonably within the bounds of what is possible. The number of people in the quad is also easily observable. It is also not a phenomenon specific to only one person or at one time, but instead can explain a phenomenon for a broader group of people.

To complete this experiment, you pick the month of October to observe the quad. Every day (except on the days where it’s raining)from 3 to 4 PM, when most classes have released for the day, you observe how many people are on the quad. You measure how many people come  and how many leave. You also write down the temperature on the hour. 

After writing down all of your observations and putting them on a graph, you find that the most students study on the quad when it is 70 degrees outside, and that the number of students drops a lot once the temperature reaches 60 degrees or below. In this case, your research report would state that you accept or “failed to reject” your first hypothesis with your findings.

Experiment #2: The Cupcake Store (Forming a Simple Experiment)

Let’s say that you work at a bakery. You specialize in cupcakes, and you make only two colors of frosting: yellow and purple. You want to know what kind of customers are more likely to buy what kind of cupcake, so you set up an experiment. Your independent variable is the customer’s gender, and the dependent variable is the color of the frosting. What is an example of a hypothesis that might answer the question of this study?

Here’s what your hypotheses might look like: 

If-then: “If customers’ gender is female, then they will buy more yellow cupcakes than purple cupcakes.”

Null: “If customers’ gender is female, then they will be just as likely to buy purple cupcakes as yellow cupcakes.”

This is a pretty simple experiment! It passes the test of plausibility (there could easily be a difference), defined concepts (there’s nothing complicated about cupcakes!), observability (both color and gender can be easily observed), and general explanation ( this would potentially help you make better business decisions ).

Experiment #3: Backyard Bird Feeders (Integrating Multiple Variables and Rejecting the If-Then Hypothesis)

While watching your backyard bird feeder, you realized that different birds come on the days when you change the types of seeds. You decide that you want to see more cardinals in your backyard, so you decide to see what type of food they like the best and set up an experiment. 

However, one morning, you notice that, while some cardinals are present, blue jays are eating out of your backyard feeder filled with millet. You decide that, of all of the other birds, you would like to see the blue jays the least. This means you'll have more than one variable in your hypothesis. Your new hypotheses might look like this: 

If-then: “If sunflower seeds are placed in the bird feeders, then more cardinals will come than blue jays. If millet is placed in the bird feeders, then more blue jays will come than cardinals.”

Null: “If either sunflower seeds or millet are placed in the bird, equal numbers of cardinals and blue jays will come.”

Through simple observation, you actually find that cardinals come as often as blue jays when sunflower seeds or millet is in the bird feeder. In this case, you would reject your “if-then” hypothesis and “fail to reject” your null hypothesis . You cannot accept your first hypothesis, because it’s clearly not true. Instead you found that there was actually no relation between your different variables. Consequently, you would need to run more experiments with different variables to see if the new variables impact the results.

Experiment #4: In-Class Survey (Including an Alternative Hypothesis)

You’re about to give a speech in one of your classes about the importance of paying attention. You want to take this opportunity to test a hypothesis you’ve had for a while: 

If-then: If students sit in the first two rows of the classroom, then they will listen better than students who do not.

Null: If students sit in the first two rows of the classroom, then they will not listen better or worse than students who do not.

You give your speech and then ask your teacher if you can hand out a short survey to the class. On the survey, you’ve included questions about some of the topics you talked about. When you get back the results, you’re surprised to see that not only do the students in the first two rows not pay better attention, but they also scored worse than students in other parts of the classroom! Here, both your if-then and your null hypotheses are not representative of your findings. What do you do?

This is when you reject both your if-then and null hypotheses and instead create an alternative hypothesis . This type of hypothesis is used in the rare circumstance that neither of your hypotheses is able to capture your findings . Now you can use what you’ve learned to draft new hypotheses and test again! 

Key Takeaways: Hypothesis Writing

The more comfortable you become with writing hypotheses, the better they will become. The structure of hypotheses is flexible and may need to be changed depending on what topic you are studying. The most important thing to remember is the purpose of your hypothesis and the difference between the if-then and the null . From there, in forming your hypothesis, you should constantly be asking questions, making observations, doing secondary research, and considering your variables. After you have written your hypothesis, be sure to edit it so that it is plausible, clearly defined, observable, and helpful in explaining a general phenomenon.

Writing a hypothesis is something that everyone, from elementary school children competing in a science fair to professional scientists in a lab, needs to know how to do. Hypotheses are vital in experiments and in properly executing the scientific method . When done correctly, hypotheses will set up your studies for success and help you to understand the world a little better, one experiment at a time.

What’s Next?

If you’re studying for the science portion of the ACT, there’s definitely a lot you need to know. We’ve got the tools to help, though! Start by checking out our ultimate study guide for the ACT Science subject test. Once you read through that, be sure to download our recommended ACT Science practice tests , since they’re one of the most foolproof ways to improve your score. (And don’t forget to check out our expert guide book , too.)

If you love science and want to major in a scientific field, you should start preparing in high school . Here are the science classes you should take to set yourself up for success.

If you’re trying to think of science experiments you can do for class (or for a science fair!), here’s a list of 37 awesome science experiments you can do at home

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

Student and Parent Forum

Our new student and parent forum, at ExpertHub.PrepScholar.com , allow you to interact with your peers and the PrepScholar staff. See how other students and parents are navigating high school, college, and the college admissions process. Ask questions; get answers.

Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!

Improve With Our Famous Guides

• For All Students

The 5 Strategies You Must Be Using to Improve 160+ SAT Points

How to Get a Perfect 1600, by a Perfect Scorer

Series: How to Get 800 on Each SAT Section:

Score 800 on SAT Math

Score 800 on SAT Reading

Score 800 on SAT Writing

Series: How to Get to 600 on Each SAT Section:

Score 600 on SAT Math

Score 600 on SAT Reading

Score 600 on SAT Writing

Free Complete Official SAT Practice Tests

What SAT Target Score Should You Be Aiming For?

15 Strategies to Improve Your SAT Essay

The 5 Strategies You Must Be Using to Improve 4+ ACT Points

How to Get a Perfect 36 ACT, by a Perfect Scorer

Series: How to Get 36 on Each ACT Section:

36 on ACT English

36 on ACT Math

36 on ACT Reading

36 on ACT Science

Series: How to Get to 24 on Each ACT Section:

24 on ACT English

24 on ACT Math

24 on ACT Reading

24 on ACT Science

What ACT target score should you be aiming for?

ACT Vocabulary You Must Know

ACT Writing: 15 Tips to Raise Your Essay Score

How to Get Into Harvard and the Ivy League

How to Get a Perfect 4.0 GPA

How to Write an Amazing College Essay

What Exactly Are Colleges Looking For?

Is the ACT easier than the SAT? A Comprehensive Guide

Should you retake your SAT or ACT?

When should you take the SAT or ACT?

Stay Informed

Get the latest articles and test prep tips!

Looking for Graduate School Test Prep?

Check out our top-rated graduate blogs here:

GRE Online Prep Blog

GMAT Online Prep Blog

TOEFL Online Prep Blog

Holly R. "I am absolutely overjoyed and cannot thank you enough for helping me!”

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.1: Statements and Conditional Statements

• Last updated
• Save as PDF
• Page ID 7034

• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2\(x\)+5 = 10 is not a statement since we do not know what \(x\) represents. If we substitute a specific value for \(x\) (such as \(x\) = 3), then the resulting equation, 2\(\cdot\)3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

• There exists a real number \(x\) such that 2\(x\)+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely \(x\) = 2.5.
• For each real number \(x\), \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\). This is a statement since either the sentence \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\) is true when any real number is substituted for \(x\) (in which case, the statement is true) or there is at least one real number that can be substituted for \(x\) and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
• Solve the equation \(x^2 - 7x +10 =0\). This is not a statement since it is a directive. It does not assert that something is true.
• \((a+b)^2 = a^2+b^2\) is not a statement since it is not known what \(a\) and \(b\) represent. However, the sentence, “There exist real numbers \(a\) and \(b\) such that \((a+b)^2 = a^2+b^2\)" is a statement. In fact, this is a true statement since there are such integers. For example, if \(a=1\) and \(b=0\), then \((a+b)^2 = a^2+b^2\).
• Compare the statement in the previous item to the statement, “For all real numbers \(a\) and \(b\), \((a+b)^2 = a^2+b^2\)." This is a false statement since there are values for \(a\) and \(b\) for which \((a+b)^2 \ne a^2+b^2\). For example, if \(a=2\) and \(b=3\), then \((a+b)^2 = 5^2 = 25\) and \(a^2 + b^2 = 2^2 +3^2 = 13\).

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

• 2\(\cdot\)7 + 8 = 22.
• \((x-1) = \sqrt(x + 11)\).
• \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(23x + 27y = 52\).
• Given a line \(L\) and a point \(P\) not on that line, there is a unique line through \(P\) that does not intersect \(L\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) for all real numbers \(a\) and \(b\).
• The derivative of \(f(x) = \sin x\) is \(f' (x) = \cos x\).
• Does the equation \(3x^2 - 5x - 7 = 0\) have two real number solutions?
• If \(ABC\) is a right triangle with right angle at vertex \(B\), and if \(D\) is the midpoint of the hypotenuse, then the line segment connecting vertex \(B\) to \(D\) is half the length of the hypotenuse.
• There do not exist three integers \(x\), \(y\), and \(z\) such that \(x^3 + y^2 = z^3\).

Add texts here. Do not delete this text first.

How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

• Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that \(\sin(2x) = 2 \sin(x)\), for all real numbers \(x\), we can test this conjecture by substituting specific values for \(x\). One way to do this is to choose values of \(x\) for which \(\sin(x)\)is known. Using \(x = \frac{\pi}{4}\), we see that

\(\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,\) and

\(2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2\).

Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If \(x\) and \(y\) are odd integers, then \(x + y\) is an even integer.

We can do lots of calculation, such as \(3 + 7 = 10\) and \(5 + 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

• Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that \(\sin (2x) = 2 \sin(x)\), for all real numbers \(x\), we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for \(\sin (2x)\), but if we do not, we can always look it up. We should recall (or find) that for all real numbers \(x\), \[\sin(2x) = 2 \sin(x)\cos(x).\]
• We could use this identity to argue that the conjecture “for all real numbers \(x\), \(\sin (2x) = 2 \sin(x)\)” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
• Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

• \((a + b)^2 = a^2 + b^2\), for all real numbers a and b.
• There are integers \(x\) and \(y\) such that \(2x + 5y = 41\).
• If \(x\) is an even integer, then \(x^2\) is an even integer.
• If \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements \(P\) and \(Q\), a statement of the form “If \(P\) then \(Q\)” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If \(P\) then \(Q\)” depends on the truth values of \(P\) and \(Q\). The statement “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. The statement \(P\) is called the hypothesis of the conditional statement, and the statement \(Q\) is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If \(P\) then \(Q\),” where \(P\) and \(Q\) are sentences. For this conditional statement, \(P\) is called the hypothesis and \(Q\) is called the conclusion .

Intuitively, “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If \(P\) then \(Q\).” We will use the notation \(P \to Q\) to represent “If \(P\) then \(Q\).” When \(P\) and \(Q\) are statements, it seems reasonable that the truth value (true or false) of the conditional statement \(P \to Q\) depends on the truth values of \(P\) and \(Q\). There are four cases to consider:

• \(P\) is true and \(Q\) is true.
• \(P\) is false and \(Q\) is true.
• \(P\) is true and \(Q\) is false.
• \(P\) is false and \(Q\) is false.

The conditional statement \(P \to Q\) means that \(Q\) is true whenever \(P\) is true. It says nothing about the truth value of \(Q\) when \(P\) is false. Using this as a guide, we define the conditional statement \(P \to Q\) to be false only when \(P\) is true and \(Q\) is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, \(P \to Q\) is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement \(P \to Q\). (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for \(P \to Q\)

The important thing to remember is that the conditional statement \(P \to Q\) has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for \(P\) and \(Q\), but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as \(P \to Q\) where \(P\) is the statement, “It is not raining” and \(Q\) is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement \(P \to Q\) as being false to mean that I lied and think of the statement \(P \to Q\) as being true to mean that I did not lie. We will now check the truth value of \(P \to Q\) based on the truth values of \(P\) and \(Q\).

• Suppose that both \(P\) and \(Q\) are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that\(P \to Q\) is true.
• Suppose that \(P\) is true and \(Q\) is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement \(P \to Q\) is false.
• Now suppose that \(P\) is false and \(Q\) is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
• Finally, suppose that both \(P\) and \(Q\) are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was \(P \to Q\), I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement \(P \to Q\) cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable \(x\). From the context of this sentence, it seems that we can substitute any positive real number for \(x\). We can also substitute 0 for \(x\) or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if \(x = -3\), then \(x^2 + 8x = -15\), which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if \(x = 4\), then \(x^2 + 8x = 48\), which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If \(n\) is a positive integer, then \(n^2 - n +41\) is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for \(n = 1\), \(n = 2\), \(n = 3\), \(n = 4\), \(n = 5\), and \(n = 10\). Then record the results for at least four other values of \(n\). Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

• If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
• If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
• If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number.

Perhaps for all of the values you tried for \(n\), \((n^2 - n + 41)\) turned out to be a prime number. However, if we try \(n = 41\), we ge \(n^2 - n + 41 = 41^2 - 41 + 41\) \(n^2 - n + 41 = 41^2\) So in the case where \(n = 41\), the hypothesis is true (41 is a positive integer) and the conclusion is false \(41^2\) is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number” is false. There are other counterexamples (such as \(n = 42\), \(n = 45\), and \(n = 50\)), but only one counterexample is needed to prove that the statement is false.

• Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for \(x\) for the conditional statement “If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.” For every positive real number used for \(x\), we saw that \(x^2 + 8x\) was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for \(x\). So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function \(f\) is differentiable at \(a\), then the function \(f\) is continuous at \(a\).

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

• It is known that the function \(f\), where \(f(x) = \sin x\), is differentiable at 0.
• It is known that the function \(f\), where \(f(x) = \sqrt[3]x\), is not differentiable at 0.
• It is known that the function \(f\), where \(f(x) = |x|\), is continuous at 0.
• It is known that the function \(f\), where \(f(x) = \dfrac{|x|}{x}\) is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are \(\sqrt2\), \(\pi\) and \(e\). We usually use the symbol \(\mathbb{Q}\) to represent the set of all rational numbers. (The letter \(\mathbb{Q}\) is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol \(\mathbb{N}\) to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If \(n\) is an integer, we can write \(n = \dfrac{n}{1}\). So each integer is a rational number and hence also a real number.

We will use the letter \(\mathbb{Z}\) to stand for the set of integers. (The letter \(\mathbb{Z}\) is from the German word, \(Zahlen\), for numbers.) Three of the basic properties of the integers are that the set \(\mathbb{Z}\) is closed under addition , the set \(\mathbb{Z}\) is closed under multiplication , and the set of integers is closed under subtraction. This means that

• If \(x\) and \(y\) are integers, then \(x + y\) is an integer;
• If \(x\) and \(y\) are integers, then \(x \cdot y\) is an integer; and
• If \(x\) and \(y\) are integers, then \(x - y\) is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

• In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If \(x\) and \(y\) are natural numbers, then \(x - y\) is a natural number. However, since 5 and 8 are natural numbers, \(5 - 8 = -3\), which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
• We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are rational numbers (so \(a\), \(b\), \(c\), and \(d\) are integers and \(b\) and \(d\) are not zero), then \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.\) Since the integers are closed under multiplication, we know that \(ac\) and \(bd\) are integers and since \(b \ne 0\) and \(d \ne 0\), \(bd \ne 0\). Hence, \(\dfrac{ac}{bd}\) is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

• Is the set of rational numbers closed under addition? Explain.
• Is the set of integers closed under division? Explain.
• Is the set of rational numbers closed under subtraction? Explain.
• Which of the following sentences are statements? (a) \(3^2 + 4^2 = 5^2.\) (b) \(a^2 + b^2 = c^2.\) (c) There exists integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2.\) (d) If \(x^2 = 4\), then \(x = 2.\) (e) For each real number \(x\), if \(x^2 = 4\), then \(x = 2.\) (f) For each real number \(t\), \(\sin^2t + \cos^2t = 1.\) (g) \(\sin x < \sin (\frac{\pi}{4}).\) (h) If \(n\) is a prime number, then \(n^2\) has three positive factors. (i) 1 + \(\tan^2 \theta = \text{sec}^2 \theta.\) (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
• Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^2\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p = 2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \ne 2\) or \(p\) is an odd number. (e) \(p \ne 2\) or \(p\) is a even number, then \(p\) is not prime.
• Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
• Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
• Let \(P\) be the statement “Student X passed every assignment in Calculus I,” and let \(Q\) be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for \(P\) to be true? What does it mean for \(Q\) to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for \(P \to Q\)?

Theorem If f is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and a < 0, then the function f has a maximum value when \(x = \dfrac{-b}{2a}\). Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem If \(f\) is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and ac < 0, then the function \(f\) has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem A. If \(f\) is a cubic function of the form \(f (x) = x^3 - x + b\) and b > 1, then the function \(f\) has exactly one \(x\)-intercept. Following is another theorem about \(x\)-intercepts of functions: Theorem B . If \(f\) and \(g\) are functions with \(g (x) = k \cdot f (x)\), where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\)-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f (x) = x^3 -x + 7\) (b) \(g (x) = x^3 + x +7\) (c) \(h (x) = -x^3 + x - 5\) (d) \(k (x) = 2x^3 + 2x + 3\) (e) \(r (x) = x^4 - x + 11\) (f) \(F (x) = 2x^3 - 2x + 7\)

• (a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
• Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of \(2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)\) and examine the units digits of these numbers, we could make the following conjectures (among others): \(\bullet\) If \(n\) is a natural number, then the units digit of \(2^n\) must be 2, 4, 6, or 8. \(\bullet\) The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of \(4 (4^1, 4^2, 4^3, 4^4, 4^5,...)\)? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form \(7^n - 2^n\), where \(n\) is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If \(n\) is a natural number, then ... .” (c) Let \(f (x) = e^(2x)\). Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

O.J. Simpson, NFL star whose murder trial gripped the nation, dies of cancer at 76

O.J. Simpson , the former NFL star who was acquitted of murdering his ex-wife and her friend in a televised trial that gripped the nation, has died of cancer, according to his family.

"He was surrounded by his children and grandchildren," the family said in a statement posted on X . "During this time of transition, his family asks that you please respect their wishes for privacy and grace."

Reports circulated in February that Simpson had been diagnosed with prostate cancer and was in hospice care as he underwent chemotherapy. He denied that he was in hospice in a video posted on X, but did not address whether he'd been diagnosed with cancer.

“Hospice? Hospice? You talking ‘bout hospice?” he said in the video with a laugh, adding that he doesn’t know who started the rumors. 

Orenthal James Simpson played 11 seasons in the National Football League and was known as "The Juice" to his fans, but his sports legacy was tarnished forever in the 1990s after his ex-wife Nicole Brown Simpson and her friend Ronald Goldman were killed.

Brown Simpson, 35, and Goldman, 25, were found stabbed to death outside her Los Angeles home in 1994.

On June 13, 1994, Goldman was returning sunglasses that the mother of Brown Simpson had left at a restaurant where he worked. The two were stabbed and slashed dozens of times, and their bodies were found the next day.

When Los Angeles police officers went to Simpson's home to speak to him about the slayings, Simpson did not answer the door but officers noticed a trail of blood leading to his car, as well as blood on his car.

Once a revered athlete, Simpson went from a Hall of Fame icon to a murder suspect.

Days later, officials charged Simpson with the murders and he attempted to evade arrest, resulting in an infamous hourslong police chase along Southern California's highways in his white Ford Bronco .

Simpson's case went to trial in 1995 and was broadcast to millions of viewers across the nation. The court case was dubbed the "trial of the century" as it dragged on for months and transformed into a public spectacle.

Feelings over the trial have remained mixed over the years, with many accusing the Los Angeles Police Department of racism in its handling of the case. Others believe that Simpson's ability to retain high-powered attorneys allowed him to get away with murder.

The trial made prosecutors Christopher Darden and Marcia Clark household names, in addition to Simpson's defense attorneys Johnnie Cochran, Alan Dershowitz and Robert Kardashian.

He was acquitted of both murders in a controversial verdict. Two years later, he was found civilly liable for wrongful death in the double homicide case.

Despite his acquittal in the criminal trial, many still believed Simpson was guilty, a belief bolstered by a jury ordering him to pay $33 million to Goldman's family in the civil case — damages that were never paid in full.

Goldman's father, Fred Goldman, spoke to NBC News by phone Thursday and described Simpson's death as "no great loss."

“The only thing I have to say is it’s just further reminder of Ron being gone all these years," he said. "It’s no great loss to the world. It’s a further reminder of Ron’s being gone.”

Bob Costas, the sports broadcaster who worked with Simpson for years at NBC Sports covering the NFL, said Simpson leaves behind “a complicated legacy, to put it mildly.”

“I can’t think of anyone historical or someone that we may have known where the first chapter and the second chapter of their lives are such a stark contrast … revered and then reviled,” Costas said on NBC’s “TODAY” show Friday.

In 2007, Simpson led an armed robbery attempt of a sports memorabilia dealer in Las Vegas. He argued in court that he was recovering his own stolen items, but his defense failed to sway the jury.

He was convicted and sentenced to 33 years in prison, of which he served only nine before he was released on parole .

Simpson spoke to The Associated Press by phone in 2019 , telling them that he was healthy and happy living in Las Vegas. He maintained that he believed his robbery conviction was unfair, but said: “I believe in the legal system and I honored it. I served my time.”

The Simpson murder trial was re-enacted and relitigated decades later in FX’s “The People v. O.J. Simpson,” an installment of the network’s popular “American Crime Story” series in 2016. Released that same year was the Academy Award-winning documentary "O.J.: Made in America," detailing Simpson's rise and fall.

Simpson was born in San Francisco and raised in public housing, going to a local community college before transferring to the University of Southern California. He was part of the school's national championship in 1967 and earned the Heisman Trophy the next year.

He was drafted by the Buffalo Bills in 1969 as a No. 1 overall pick.

According to NBC Sports , Simpson was the first player in the league to rush for 2,000 or more yards in a season and is considered the best running back of his era.

Simpson had three children from his first marriage to Marguerite Whitley, one of whom died in a drowning accident as a toddler.

He also shared two children with Brown Simpson.

Following her murder and his acquittal, Simpson won custody of their shared children and moved to Miami with them. His custody fight with his former-in-laws also drew headlines as the children's grandparents took him to court in a bitter legal battle.

Doha Madani is a senior breaking news reporter for NBC News. Pronouns: she/her.