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1.2: The 7-Step Process of Statistical Hypothesis Testing

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• Penn State's Department of Statistics
• The Pennsylvania State University

We will cover the seven steps one by one.

Step 1: State the Null Hypothesis

The null hypothesis can be thought of as the opposite of the "guess" the researchers made: in this example, the biologist thinks the plant height will be different for the fertilizers. So the null would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as: \[H_{0}: \ \mu_{1} = \mu_{2} = \ldots = \mu_{T}\] for \(T\) levels of an experimental treatment.

Why do we do this? Why not simply test the working hypothesis directly? The answer lies in the Popperian Principle of Falsification. Karl Popper (a philosopher) discovered that we can't conclusively confirm a hypothesis, but we can conclusively negate one. So we set up a null hypothesis which is effectively the opposite of the working hypothesis. The hope is that based on the strength of the data, we will be able to negate or reject the null hypothesis and accept an alternative hypothesis. In other words, we usually see the working hypothesis in \(H_{A}\).

Step 2: State the Alternative Hypothesis

\[H_{A}: \ \text{treatment level means not all equal}\]

The reason we state the alternative hypothesis this way is that if the null is rejected, there are many possibilities.

For example, \(\mu_{1} \neq \mu_{2} = \ldots = \mu_{T}\) is one possibility, as is \(\mu_{1} = \mu_{2} \neq \mu_{3} = \ldots = \mu_{T}\). Many people make the mistake of stating the alternative hypothesis as \(mu_{1} \neq mu_{2} \neq \ldots \neq \mu_{T}\), which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, this means that fertilizer 1 may result in plants that are really tall, but fertilizers 2, 3, and the plants with no fertilizers don't differ from one another. A simpler way of thinking about this is that at least one mean is different from all others.

Step 3: Set \(\alpha\)

If we look at what can happen in a hypothesis test, we can construct the following contingency table:

You should be familiar with type I and type II errors from your introductory course. It is important to note that we want to set \(\alpha\) before the experiment ( a priori ) because the Type I error is the more grievous error to make. The typical value of \(\alpha\) is 0.05, establishing a 95% confidence level. For this course, we will assume \(\alpha\) =0.05, unless stated otherwise.

Step 4: Collect Data

Remember the importance of recognizing whether data is collected through an experimental design or observational study.

Step 5: Calculate a test statistic

For categorical treatment level means, we use an \(F\) statistic, named after R.A. Fisher. We will explore the mechanics of computing the \(F\) statistic beginning in Chapter 2. The \(F\) value we get from the data is labeled \(F_{\text{calculated}}\).

Step 6: Construct Acceptance / Rejection regions

As with all other test statistics, a threshold (critical) value of \(F\) is established. This \(F\) value can be obtained from statistical tables or software and is referred to as \(F_{\text{critical}}\) or \(F_{\alpha}\). As a reminder, this critical value is the minimum value for the test statistic (in this case the F test) for us to be able to reject the null.

The \(F\) distribution, \(F_{\alpha}\), and the location of acceptance and rejection regions are shown in the graph below:

Step 7: Based on steps 5 and 6, draw a conclusion about H0

If the \(F_{\text{\calculated}}\) from the data is larger than the \(F_{\alpha}\), then you are in the rejection region and you can reject the null hypothesis with \((1 - \alpha)\) level of confidence.

Note that modern statistical software condenses steps 6 and 7 by providing a \(p\)-value. The \(p\)-value here is the probability of getting an \(F_{\text{calculated}}\) even greater than what you observe assuming the null hypothesis is true. If by chance, the \(F_{\text{calculated}} = F_{\alpha}\), then the \(p\)-value would exactly equal \(\alpha\). With larger \(F_{\text{calculated}}\) values, we move further into the rejection region and the \(p\) - value becomes less than \(\alpha\). So the decision rule is as follows:

If the \(p\) - value obtained from the ANOVA is less than \(\alpha\), then reject \(H_{0}\) and accept \(H_{A}\).

If you are not familiar with this material, we suggest that you review course materials from your basic statistics course.

Resources: Course Assignments

Module 10 Assignment: Hypothesis Testing for the Population Mean

The purpose of this activity is to give you guided practice in going through the process of a t-test for the population mean, and teach you how to carry out this test using statistical software.

Background:

A group of 75 college students from a certain liberal arts college were randomly sampled and asked about the number of alcoholic drinks they have in a typical week. The file containing the data is linked below. The purpose of this  study  was to compare the drinking habits of the students at the college to the drinking habits of college students in general. In particular, the dean of students, who initiated this study, would like to check whether the mean number of alcoholic drinks that students at his college have in a typical week differs from the mean of U.S. college students in general, which is estimated to be 4.73.

Question 1:

Let μ be the mean number of alcoholic beverages that students in the college drink in a typical week. State the hypotheses that are being tested in this problem.

Question 2:

Here is a histogram of the data. Can we safely use the t-test with this data?

Instructions

Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.

R  |  StatCrunch  |  Minitab  |  Excel  |  TI Calculator

Question 3:

State the test statistic, interpret its value and show how it was found.

Question 4:

Based on the P-value, draw your conclusions in context.

Question 5:

What would your conclusions be if the dean of students suspected that the mean number of alcoholic drinks that students in the college consume in a typical week is  lower  than the mean of U.S. college students in general? In other words, if this were a test of the hypotheses:

H 0 : μ = 4.73 drinks per week

H a : μ < 4.73 drinks per week

Question 6:

Now suppose that instead of the 75 students having been randomly selected from the entire student body, the 75 students had been randomly selected  only  from the engineering classes at the college (for the sake of convenience).

Address the following two issues regarding the effect of such a change in the study design:

a. Would we still be mathematically justified in using the T-test for obtaining conclusions, as we did previously?

b. Would the resulting conclusions still address the question of interest (which, remember, was to investigate the drinking habits of the students at the college as whole)?

Concepts in Statistics Copyright © 2023 by CUNY School of Professional Studies is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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Chapter 10: Inference for Means

Assignment: hypothesis testing for the population mean.

The purpose of this activity is to give you guided practice in going through the process of a t-test for the population mean, and teach you how to carry out this test using statistical software.

Background:

A group of 75 college students from a certain liberal arts college were randomly sampled and asked about the number of alcoholic drinks they have in a typical week. The file containing the data is linked below. The purpose of this study was to compare the drinking habits of the students at the college to the drinking habits of college students in general. In particular, the dean of students, who initiated this study, would like to check whether the mean number of alcoholic drinks that students at his college have in a typical week differs from the mean of U.S. college students in general, which is estimated to be 4.73.

Question 1:

Let μ be the mean number of alcoholic beverages that students in the college drink in a typical week. State the hypotheses that are being tested in this problem.

Question 2:

Here is a histogram of the data. Can we safely use the t-test with this data?

Instructions

Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.

R | StatCrunch | Minitab | Excel 2007 | TI Calculator

Question 3:

State the test statistic, interpret its value and show how it was found.

Question 4:

Based on the P-value, draw your conclusions in context.

Question 5:

What would your conclusions be if the dean of students suspected that the mean number of alcoholic drinks that students in the college consume in a typical week is lower than the mean of U.S. college students in general? In other words, if this were a test of the hypotheses:

H 0 : μ = 4.73 drinks per week

H a : μ < 4.73 drinks per week

Question 6:

Now suppose that instead of the 75 students having been randomly selected from the entire student body, the 75 students had been randomly selected only from the engineering classes at the college (for the sake of convenience).

Address the following two issues regarding the effect of such a change in the study design:

a. Would we still be mathematically justified in using the T-test for obtaining conclusions, as we did previously?

b. Would the resulting conclusions still address the question of interest (which, remember, was to investigate the drinking habits of the students at the college as whole)?

• Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution

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10: hypothesis testing, lesson overview section  .

In Lesson 2 we saw the value of random assignment in designed experiments. Random assignment alleviates the bias that might cause a systematic difference between groups unrelated to the treatment itself. Precautions like blinding that ensure that the subjects are treated the same during the experiment then leave us with just two possibilities for the cause of differences seen between groups. Either:

• the treatment was effective in producing the changes (the research hypothesis), or
• differences were just the result of the luck of the draw (the null hypothesis).

This shows the importance of addressing the concept of statistical significance. If it is very unlikely that the results of a randomized experiment are just the result of random chance, then we are left with the treatment itself as the probable cause of any relationship seen. Even in an observational study, being able to show that random chance is a poor explanation of the data is still good evidence for a true association in the population (even though it is poor evidence of causality).

This lesson focuses on Statistical hypothesis testing. In a significance test, you carry out a probability calculation assuming the null hypothesis is true to see if random chance is a plausible explanation for the data. Let's illustrate the process with an example.

Example 10.1 Section  

Physical theory suggests that when a coin is spun on a table (rather than flipped in the air) the probability it lands heads up is less than 0.5. We are hesitant to believe this without proof.

To test the theory we carry out an experiment and independently spin a penny 100 times getting 37 heads and 63 tails. Thus, the observed proportion of heads is 37 / 100 = 0.37

We have two possible explanations for the data:

Null Hypothesis : The data is merely a reflection of chance variation. The probability of heads when a penny is spun is really p = 0.5

Alternative Hypothesis : The probability of heads when a penny is spun is really < 0.5.

A statistical hypothesis test is designed to answer the question: "Does the Null Hypothesis provide a reasonable explanation of the data?”

To answer this question we carry out a probability calculation. First, we can calculate a

Test Statistic = a measure of the difference between the data and what is expected when the null hypothesis is true.

In our example, the null hypothesis says the number of heads in 100 spins would closely follow the normal distribution with p = 0.5. So, if the null hypothesis is true, we expect half (0.5) heads give or take a standard deviation of

\[\sqrt{\frac{0.5(1-0.5)}{100}}=0.05\]

Further, we can see how unusual our data is if the null hypothesis is true by finding the standard score z for the test statistic and using the normal curve:

\[z = (0.37-0.5)/0.05 = -2.6\]

How unusual is the value we got, assuming the null hypothesis (i.e., the real proportion is 0.5) is true? We know that standard scores of -2.6 or lower only happen about 0.5% of the time. So the null hypothesis provides a poor explanation for our data. This would seem to provide strong evidence that spinning a coin has less than a 50% chance of landing heads.

• Formulate appropriate null and alternative hypotheses.
• Identify the type 1 and the type 2 error in the context of the problem.
• Use  the four basic steps to carry out a significance test in some basic situations.
• Interpret a p -value in terms of the problem.
• State an appropriate conclusion for a hypothesis test.

• Mastering Hypothesis Testing in Econometrics: A Comprehensive Guide for University Assignments

Hypothesis Testing in Econometrics: A Critical Tool for Your Homework

Econometrics, as a specialized branch of economics, serves as the intersection between economic theory and statistical methods, facilitating the testing of hypotheses and the prediction of future trends. At its core, econometric analysis relies heavily on the crucial tool of hypothesis testing, an integral process empowering both seasoned researchers and aspiring students to extract meaningful conclusions from raw data. This proficiency in hypothesis testing is a cornerstone for making informed decisions and accurate predictions, forming the bedrock of economic research and analysis. As we embark on an exploration of the significance of hypothesis testing in econometrics, it becomes evident that this methodological approach is not merely an academic exercise but a practical and essential skill set for anyone involved in economic inquiries. For students navigating the labyrinth of university assignments, understanding and effectively utilizing hypothesis testing can be the key to unlocking the analytical potential required to excel in econometrics coursework. By grasping the intricacies of formulating hypotheses, discerning between null and alternative hypotheses, and judiciously choosing the significance level, students lay a solid foundation for robust econometric analyses. Selecting an appropriate test statistic, be it a t-test, F-test, or chi-square test, becomes the linchpin in the hypothesis testing process, demanding a keen understanding of the data at hand and the specific economic relationship under scrutiny. Armed with the proficiency to collect, clean, and prepare relevant data, students can seamlessly transition into the realm of conducting hypothesis tests using sophisticated econometric software, transforming theoretical concepts into tangible results. The interpretation of these results is the final frontier, where students showcase not only their statistical acumen but also their ability to contextualize findings within the broader economic landscape. In the crucible of university assignments, the practical application of hypothesis testing becomes a proving ground for students, allowing them to demonstrate their mastery over econometric principles. If you need assistance with your Econometrics homework , understanding and effectively applying hypothesis testing principles are essential for tackling assignments and conducting robust econometric analyses.

Whether unraveling the impact of government policies on economic growth or scrutinizing the intricate connections between consumer spending and GDP, students can wield hypothesis testing as a powerful tool to unveil the economic narratives hidden within the data. As this blog post unfolds, it serves as a comprehensive guide, offering students insights into the multifaceted realm of hypothesis testing in econometrics. By internalizing the nuances of each step—from defining variables and hypotheses to conducting tests and drawing conclusions—students can navigate their assignments with confidence, transforming their analyses into contributions that enrich the broader field of economics. In essence, hypothesis testing emerges not merely as an academic exercise but as a critical instrument that empowers students to unravel the complexities of economic relationships and make meaningful contributions to the ever-evolving tapestry of econometrics.

Understanding Hypothesis Testing in Econometrics

In the intricate domain of econometrics, hypothesis testing stands as a pivotal process, serving as the bedrock for making insightful inferences about population parameters from sample data. This methodological approach involves the systematic formulation of hypotheses concerning specific economic relationships or phenomena, followed by the rigorous application of statistical methods to scrutinize the validity of these propositions. The essence of hypothesis testing lies in its ability to distill complex economic theories into testable statements, allowing researchers to draw meaningful conclusions from empirical data. As econometricians grapple with questions about the impact of variables on economic outcomes or the validity of economic models, hypothesis testing emerges as the compass guiding them through the intricate maze of data analysis. This systematic methodology not only illuminates the relationships between economic variables but also provides a structured framework for assessing the robustness of economic theories in the face of empirical evidence. Consequently, understanding hypothesis testing in econometrics is tantamount to wielding a powerful analytical tool that empowers researchers to navigate the uncertainties inherent in economic phenomena, ensuring that the inferences drawn are not only theoretically grounded but also statistically sound.

Formulating Hypotheses

Before delving into the intricate world of hypothesis testing, it is imperative to grasp the art of formulating hypotheses accurately. In econometrics, hypotheses typically revolve around the relationships between variables. For instance, one might hypothesize that there exists a substantial relationship between the unemployment rate and inflation rate, setting the stage for a nuanced exploration of these economic indicators.

Null and Alternative Hypotheses

At the core of hypothesis testing lie two pivotal hypotheses—the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis often embodies the status quo or the absence of an effect, while the alternative hypothesis posits a change or difference. Applying this to the unemployment and inflation scenario, the null hypothesis could assert that there is no significant relationship between the two, while the alternative hypothesis would counter that proposition.

Choosing the Significance Level

A critical aspect in hypothesis testing is the selection of the significance level, denoted as α, which serves as the threshold for rejecting the null hypothesis. Commonly adopted values for α include 0.05 or 5%, yet researchers possess the flexibility to opt for different levels based on contextual considerations and their desired level of confidence in the resulting conclusions.

Selecting a Test Statistic

The choice of a suitable test statistic hinges on the nature of the data and the specific hypothesis under scrutiny. Econometricians confront a plethora of options, including t-tests, F-tests, and chi-square tests, necessitating a judicious selection aligning with the intricacies of their research questions.

Collecting and Analyzing Data

With hypotheses formulated and a test statistic chosen, the next pivotal phase involves the collection and analysis of data. The utilization of econometric software such as R, Python, or statistical packages like STATA and SAS streamlines this process. The overarching goal is to discern whether the observed data furnishes adequate evidence to warrant the rejection of the null hypothesis.

Interpreting Results

Upon completing the hypothesis test, researchers confront the task of interpreting the results. If the p-value, denoting the probability, falls below the chosen significance level, the null hypothesis is discarded. This signifies compelling evidence in favor of the alternative hypothesis. Conversely, if the p-value surpasses the designated significance level, the null hypothesis remains unscathed. This interpretative step is pivotal, requiring researchers to synthesize statistical outcomes with economic intuition to draw meaningful and contextually relevant conclusions from their analyses.

Practical Application in University Assignments

The transition from theoretical understanding to practical application is paramount for students seeking excellence in their university assignments centered around hypothesis testing in econometrics. Armed with a foundation in the intricacies of formulating hypotheses, discerning between null and alternative hypotheses, and choosing significance levels and appropriate test statistics, students are poised to tackle real-world economic questions. Selecting relevant topics aligned with their coursework interests, such as analyzing the impact of government policies on economic growth or investigating the relationship between consumer spending and GDP, provides students with a meaningful context for their assignments. The process of defining variables and hypotheses becomes a critical precursor, as students articulate clear research questions, ensuring the subsequent collection and preparation of data is purposeful and conducive to robust analyses. Leveraging econometric software like R, Python, or statistical packages such as STATA and SAS, students can seamlessly execute hypothesis tests, transforming raw data into meaningful results. The interpretation of these results is the pinnacle of practical application, requiring students to not only understand statistical outcomes but also to contextualize their findings within the broader economic landscape. Thus, university assignments become a dynamic platform for students to showcase their adeptness in hypothesis testing, providing a conduit for the integration of theoretical knowledge with real-world problem-solving skills essential for success in the field of econometrics. In this practical realm, students not only demonstrate their mastery of the hypothesis testing process but also contribute meaningfully to the ongoing discourse in economics by applying their skills to unravel the complexities of economic relationships and phenomena.

Selecting Relevant Topics

Embarking on a successful journey in econometrics assignments begins with the pivotal task of selecting a relevant and intriguing topic. Aligning your choice with personal interests and coursework not only fosters engagement but also lays the foundation for a well-crafted assignment. Whether delving into the intricate analysis of government policies on economic growth or unraveling the complex relationship between consumer spending and GDP, the chosen topic sets the stage for a nuanced exploration that can significantly contribute to the field of econometrics.

Defining Variables and Hypotheses

The subsequent step involves the meticulous definition of variables and the formulation of hypotheses, which collectively serve as the bedrock for the entire hypothesis testing process. This critical phase requires clarity and precision, as researchers articulate the key elements they will study and craft hypotheses that align with their overarching research questions. By investing time in this foundational step, students ensure the coherence and validity of their subsequent analyses.

Data Collection and Preparation

With a well-defined research framework, the focus shifts to data collection and preparation. The quality of the analysis hinges on the accuracy, representativeness, and suitability of the gathered data for the chosen hypothesis testing method. Rigorous data cleaning and preparation are indispensable steps to guarantee the reliability of subsequent results, ensuring that the data reflects the nuances of the economic phenomena under investigation.

Choosing the Right Test Statistic

The nature of hypotheses and data dictates the choice of an appropriate test statistic, a decision that necessitates careful consideration and justification. In this section of the assignment, students demonstrate their understanding of various statistical methods and their ability to apply them judiciously to the specific context of their research. Clear and concise justifications enhance the credibility of the subsequent hypothesis testing process.

Conducting the Hypothesis Test

The utilization of econometric software becomes paramount as students transition to the actual hypothesis testing phase. Executing the test involves providing detailed steps, code, and output in the assignment, showcasing the rigor and transparency of the analysis. Additionally, researchers should discuss any assumptions made during the testing process, acknowledging the limitations and potential implications on the results.

Interpreting Results and Drawing Conclusions

The final stretch of the assignment entails the interpretation of results and the drawing of meaningful conclusions. This stage demands a synthesis of statistical outcomes with economic intuition, showcasing the researcher's comprehension of the broader implications of their findings. The discussion revolves around whether the null hypothesis is rejected, providing evidence in favor of the alternative hypothesis, or if it remains intact, influencing subsequent economic interpretations. In essence, this is the pinnacle where students exhibit their ability to translate statistical results into meaningful insights with profound economic implications.

To excel in hypothesis testing assignments in econometrics, consider the following tips:

• Understand the Economic Context: A foundational step involves gaining a thorough understanding of the economic context surrounding the research question. This understanding is crucial in formulating meaningful hypotheses that align with the intricacies of economic relationships and dynamics.
• Practice with Real Data: Theoretical knowledge alone is insufficient. To enhance practical skills in data collection, analysis, and interpretation, students must actively engage with real-world data. This hands-on experience not only reinforces theoretical concepts but also prepares students for the complexities of applying these concepts to actual economic scenarios.
• Seek Guidance: Hypothesis testing can be challenging, and seeking guidance is a proactive approach to overcoming difficulties. Professors, classmates, and online resources can provide valuable insights, clarifications, and additional resources to navigate through the intricacies of hypothesis testing.
• Stay Updated: The field of econometrics is dynamic, with continuous advancements and evolving techniques. To ensure long-term success, students should stay abreast of new developments and emerging methodologies in econometrics. This commitment to staying updated ensures that students' skills remain relevant and adaptable to the evolving landscape of economic research and analysis.

In conclusion, mastering the intricacies of hypothesis testing in econometrics is not merely an academic endeavor; it is a gateway to unlocking the analytical potential needed for success in university assignments and beyond. Selecting relevant topics, defining variables, meticulous data collection, and judicious interpretation of results constitute a comprehensive approach that transforms theoretical knowledge into practical skills. As students navigate the dynamic landscape of economic relationships, the application of hypothesis testing becomes a beacon, illuminating the path toward informed decision-making and insightful conclusions. With each assignment, students not only demonstrate their competence in statistical methodologies but also contribute meaningfully to the ongoing discourse in economics, solidifying their role as adept analysts in the ever-evolving field of econometrics.

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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

Key Terms of Hypothesis Testing

• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

• i,j are the rows and columns index respectively.

Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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ExcelR Data Science Assignment No 3

shanuhalli/Assignment-Hypothesis-Testing

Folders and files, repository files navigation, assignment03-hypothesis-testing, hypothesis testing :.

---> Hypothesis testing is a part of statistics in which we make assumptions about the population parameter. So, hypothesis testing mentions a proper procedure by analysing a random sample of the population to accept or reject the assumption. Hypothesis testing is the way of trying to make sense of assumptions by looking at the sample data.

Type of Hypothesis :

---> The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected. There are two types of statistical hypotheses.

• Null Hypothesis :

The null hypothesis, denoted by Ho, is usually the hypothesis that sample observations result purely from chance.

• Alternative Hypothesis :

The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

This assignment will study following Questions:

A F&B manager wants to determine whether there is any significant difference in the diameter of the cutlet between two units. A randomly selected sample of cutlets was collected from both units and measured? Analyze the data and draw inferences at 5% significance level. Please state the assumptions and tests that you carried out to check validity of the assumptions.

Sales of products in four different regions is tabulated for males and females. Find if male-female buyer rations are similar across regions.

TeleCall uses 4 centers around the globe to process customer order forms. They audit a certain % of the customer order forms. Any error in order form renders it defective and has to be reworked before processing. The manager wants to check whether the defective % varies by centre. Please analyze the data at 5% significance level and help the manager draw appropriate inferences

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