hypothesis chapter 8

Chapter 8: Hypothesis Testing with One Sample

Introduction to Chapter 8: Hypothesis Testing with One Sample

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.

A statistician will make a decision about these claims. This process is called “ hypothesis testing .” A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.

In this chapter, you will conduct hypothesis tests on single means and single proportions. You will also learn about the errors associated with these tests.

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will:

• Set up two contradictory hypotheses.
• Collect sample data (in homework problems, the data or summary statistics will be given to you).
• Determine the correct distribution to perform the hypothesis test.
• Analyze sample data by performing the calculations that ultimately will allow you to reject or decline to reject the null hypothesis.
• Make a decision and write a meaningful conclusion.

Collaborative Exercises

Collaborative exercise.

Bring to class a newspaper, some news magazines, and some Internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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8.1 Introduction to Hypothesis Testing

One job of a statistician is to make statistical inferences about populations based on samples taken from the population.  Confidence intervals are one way to estimate a population parameter.  Another way to make a statistical inference is to make a decision about a parameter.  For instance, a car dealer advertises that its new small truck gets an average of 35 miles per gallon.  A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B.  A company says that women managers in their company earn an average of $60,000 per year.

A statistician will make a decision about whether these claims are true or false. This process is called hypothesis testing .  A hypothesis test involves collecting data from a sample and evaluating the data.  From the evidence provided by the sample data, the statistician makes a decision as to whether or not there is sufficient evidence to reject or not reject the null hypothesis.

In this chapter, you will conduct hypothesis tests on single population means and single population proportions. You will also learn about the errors associated with these tests.

Hypothesis testing consists of two contradictory hypotheses, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will:

• Set up two contradictory hypotheses.  Only one of these hypotheses is true and the hypothesis test will determine which of the hypothesis is most likely true.
• Collect sample data.  (In homework problems, the data or summary statistics will be given to you.)
• Determine the correct distribution to perform the hypothesis test.
• Analyze the sample data by performing calculations that ultimately will allow you to reject or not reject the null hypothesis.
• Make a decision and write a meaningful conclusion.

Attribution

“Chapter 9 Introduction” in Introductory Statistics by OpenStax  is licensed under a  Creative Commons Attribution 4.0 International License.

Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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8.5: Chapter 8 Exercises

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Chapter 8 Exercises

1. The plant-breeding department at a major university developed a new hybrid boysenberry plant called Stumptown 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. 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 different from the 90 days claimed. Which one is the correct set of hypotheses?

a) H 0 : p = 90% H 1 : p ≠ 90%

b) H 0 : μ = 90 H 1 : μ ≠ 90

c) H 0 : p = 92.3% H 1 : p ≠ 92.3%

d) H 0 : μ = 92.3 H 1 : μ ≠ 92.3

e) H 0 : μ ≠ 90 H 1 : μ = 90

2. Match the symbol with the correct phrase.

3. 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. Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? Which one is the correct set of hypotheses?

Federal Trade Commission, (2008). Consumer fraud and identity theft complaint data: January-December 2007 . Retrieved from website: http://www.ftc.gov/opa/2008/02/fraud.pdf .

a) H 0 : p = 23% H 1 : p < 23%

b) H 0 : μ = 23 H 1 : μ < 23

c) H 0 : p < 23% H 1 : p ≥ 23%

d) H 0 : p = 0.224 H 1 : p < 0.224

e) H 0 : μ < 0.224 H 1 : μ ≥ 0.224

4. Compute the z critical value for a right-tailed test when \(\alpha\) = 0.01.

5. Compute the z critical value for a two-tailed test when \(\alpha\) = 0.01.

6. Compute the z critical value for a left-tailed test when \(\alpha\) = 0.05.

7. Compute the z critical value for a two-tailed test when \(\alpha\) = 0.05.

8. As of 2018, the Centers for Disease Control and Protection’s (CDC) national estimate that 1 in 68 \(\approx\) 0.0147 children have been diagnosed with autism spectrum disorder (ASD). A researcher believes that the proportion of children in their county is different from the CDC estimate. Which one is the correct set of hypotheses?

a) H 0 : p = 0.0147 H 1 : p ≠ 0.0147

b) H 0 : μ = 0.0147 H 1 : μ ≠ 0.0147

c) H 0 : p ≠ 0.0147 H 1 : p = 0.0147

d) H 0 : μ = 68 H 1 : μ ≠ 68

e) H 0 : = 0.0147 H 1 : ≠ 0.0147

9. Match the phrase with the correct symbol.

a. Sample Size i. α

b. Population Mean ii. n

c. Sample Variance iii. σ²

d. Sample Mean iv. s²

e. Population Standard Deviation v. s

f. P(Type I Error) vi. \(\bar{x}\)

g. Sample Standard Deviation vii. σ

h. Population Variance viii. μ

10. The Food & Drug Administration (FDA) regulates that fresh albacore tuna fish contains at most 0.82 ppm of mercury. A scientist at the FDA believes the mean amount of mercury in tuna fish for a new company exceeds the ppm of mercury. Which one is the correct set of hypotheses?

a) H 0 : p = 82% H 1 : p > 82%

b) H 0 : μ = 0.82 H 1 : μ > 0.82

c) H 0 : p > 82% H 1 : p ≤ 82%

d) H 0 : μ = 0.82 H 1 : μ ≠ 0.82

e) H 0 : μ > 0.82 H 1 : μ ≤ 0.82

11. Match the symbol with the correct phrase.

12. The plant-breeding department at a major university developed a new hybrid boysenberry plant called Stumptown 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. 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 will not market the product if the mean number of days is more than the 90 days claimed. The hypotheses are H 0 : μ = 90 H 1 : μ > 90. Which answer is the correct type I error in the context of this problem?

a) The corporation will not market the Stumptown Berry even though the berry does produce fruit within the 90 days.

b) The corporation will market the Stumptown Berry even though the berry does produce fruit within the 90 days.

c) The corporation will not market the Stumptown Berry even though the berry does produce fruit in more than 90 days.

d) The corporation will market the Stumptown Berry even though the berry does produce fruit in more than 90 days.

13. The Food & Drug Administration (FDA) regulates that fresh albacore tuna fish contains at most 0.82 ppm of mercury. A scientist at the FDA believes the mean amount of mercury in tuna fish for a new company exceeds the ppm of mercury. The hypotheses are H 0 : μ = 0.82 H 1 : μ > 0.82. Which answer is the correct type II error in the context of this problem?

a) The fish is rejected by the FDA when in fact it had less than 0.82 ppm of mercury.

b) The fish is accepted by the FDA when in fact it had less than 0.82 ppm of mercury.

c) The fish is rejected by the FDA when in fact it had more than 0.82 ppm of mercury.

d) The fish is accepted by the FDA when in fact it had more than 0.82 ppm of mercury.

14. A two-tailed z-test found a test statistic of z = 2.153. At a 1% level of significance, which would the correct decision?

a) Do not reject H 0

b) Reject H 0

c) Accept H 0

d) Reject H 1

e) Do not reject H 1

15. A left-tailed z-test found a test statistic of z = -1.99. At a 5% level of significance, what would the correct decision be?

a) Do not reject H0

b) Reject H0

c) Accept H0

d) Reject H1

e) Do not reject H1

16. A right-tailed z-test found a test statistic of z = 0.05. At a 5% level of significance, what would the correct decision be?

a) Reject H 0

b) Accept H 0

c) Reject H 1

d) Do not reject H 0

17. A two-tailed z-test found a test statistic of z = -2.19. At a 1% level of significance, which would the correct decision?

18. 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. Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? The hypotheses are H0: p = 23% H1: p < 23%. Which answer is the correct type I error in the context of this problem?

a) It is believed that less than 23% of Alaskans had identity theft and there really was 23% or less that experienced identity theft.

b) It is believed that more than 23% of Alaskans had identity theft and there really was 23% or more that experience identity theft.

c) It is believed that less than 23% of Alaskans had identity theft even though there really was 23% or more that experienced identity theft.

d) It is believed that more than 23% of Alaskans had identity theft even though there really was less than 23% that experienced identity theft

19. A hypothesis test was conducted during a clinical trial to see if a new COVID-19 vaccination reduces the risk of contracting the virus. What is the Type I and II errors in terms of approving the vaccine for use?

20. A manufacturer of rechargeable laptop batteries claims its batteries have, on average, 500 charges. A consumer group decides to test this claim by assessing the number of times 30 of their laptop batteries can be recharged and finds a p-value is 0.1111; thus, the null hypothesis is not rejected. What is the Type II error for this situation?

21. A commonly cited standard for one-way length (duration) of school bus rides for elementary school children is 30 minutes. A local government office in a rural area conducts a study to determine if elementary schoolers in their district have a longer average one-way commute time. If they determine that the average commute time of students in their district is significantly higher than the commonly cited standard they will invest in increasing the number of school buses to help shorten commute time. What would a Type II error mean in this context?

22. The Centers for Disease Control and Prevention (CDC) 2018 national estimate that 1 in 68 \(\approx\) 0.0147 children have been diagnosed with autism spectrum disorder (ASD). A researcher believes that the proportion of children in their county is different from the CDC estimate. The hypotheses are H 0 : p = 0.0147 H 1 : p ≠ 0.0147. Which answer is the correct type II error in the context of this problem?

a) The proportion of children diagnosed with ASD in the researcher’s county is believed to be different from the national estimate, even though the proportion is the same.

b) The proportion of children diagnosed with ASD in the researcher’s county is believed to be different from the national estimate and the proportion is different.

c) The proportion of children diagnosed with ASD in the researcher’s county is believed to be the same as the national estimate, even though the proportion is different.

d) The proportion of children diagnosed with ASD in the researcher’s county is believed to be the same as the national estimate and the proportion is the same.

23. The Food & Drug Administration (FDA) regulates that fresh albacore tuna fish contains at most 0.82 ppm of mercury. A scientist at the FDA believes the mean amount of mercury in tuna fish for a new company exceeds the ppm of mercury. A test statistic was found to be 2.576 and a critical value was found to be 1.645, what is the correct decision and summary?

a) Reject H 0 , there is enough evidence to support the claim that the amount of mercury in the new company’s tuna fish exceeds the FDA limit of 0.82 ppm.

b) Accept H 0 , there is not enough evidence to reject the claim that the amount of mercury in the new company’s tuna fish exceeds the FDA limit of 0.82 ppm.

c) Reject H 1 , there is not enough evidence to reject the claim that the amount of mercury in the new company’s tuna fish exceeds the FDA limit of 0.82 ppm.

d) Reject H 0 , there is not enough evidence to support the claim that the amount of mercury in the new company’s tuna fish exceeds the FDA limit of 0.82 ppm.

e) Do not reject H 0 , there is not enough evidence to support the claim that the amount of mercury in the new company’s tuna fish exceeds the FDA limit of 0.82 ppm.

24. The plant-breeding department at a major university developed a new hybrid boysenberry plant called Stumptown 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. 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 corporation wants to know if the mean number of days is different from the 90 days claimed. A random sample was taken and the following test statistic was z = -2.15 and critical values of z = ±1.96 was found. What is the correct decision and summary?

a) Do not reject H 0 , there is not enough evidence to support the corporation’s claim that the mean number of days until a berry is produced is different from the 90 days claimed by the university.

b) Reject H 0 , there is enough evidence to support the corporation’s claim that the mean number of days until a berry is produced is different from the 90 days claimed by the university.

c) Accept H 0 , there is enough evidence to support the corporation’s claim that the mean number of days until a berry is produced is different from the 90 days claimed by the university.

d) Reject H 1 , there is not enough evidence to reject the corporation’s claim that the mean number of days until a berry is produced is different from the 90 days claimed by the university.

e) Reject H 0 , there is not enough evidence to support the corporation’s claim that the mean number of days until a berry is produced is different from the 90 days claimed by the university.

25. You are conducting a study to see if the accuracy rate for fingerprint identification is significantly different from 0.34. Thus, you are performing a two-tailed test. Your sample data produce the test statistic z = 2.504. Use your calculator to find the p-value and state the correct decision and summary.

26. The SAT exam in previous years is normally distributed with an average score of 1,000 points and a standard deviation of 150 points. The test writers for this upcoming year want to make sure that the new test does not have a significantly different mean score. They have a random sample of 20 students take the SAT and their mean score was 1,050 points.

a) Test to see if the mean time has significantly changed using a 5% level of significance. Show all your steps using the critical value method.

b) What is a type I error for this problem?

c) What is a type II error for this problem?

27. A sample of 45 body temperatures of athletes had a mean of 98.8˚F. Assume the population standard deviation is known to be 0.62˚F. Test the claim that the mean body temperature for all athletes is more than 98.6˚F. Use a 1% level of significance. Show all your steps using the p-value method.

28. Compute the t critical value for a left-tailed test when \(\alpha\) = 0.10 and df = 10.

29. Compute the t critical value for a two-tailed test when \(\alpha\) = 0.05 with a sample size of 18.

30. Using a t-distribution with df = 25, find the P(t ≥ 2.185).

31. A student is interested in becoming an actuary. They know that becoming an actuary takes a lot of schooling and they will have to take out student loans. They want to make sure the starting salary will be higher than $55,000/year. They randomly sample 30 starting salaries for actuaries and find a p-value of 0.0392. Use \(\alpha\) = 0.05.

a) Choose the correct hypotheses.

i. H 0 : μ = 55,000 H 1 : μ < 55,000

ii. H 0 : μ > 55,000 H 1 : μ ≤ 55,000

iii. H 0 : μ = 55,000 H 1 : μ > 55,000

iv. H 0 : μ < 55,000 H 1 : μ ≥ 55,000

v. H 0 : μ = 55,000 H 1 : μ ≠ 55,000

b) Should the student pursue an actuary career?

i. Yes, since we reject the null hypothesis.

ii. Yes, since we reject the claim.

iii. No, since we reject the claim.

iv. No, since we reject the null hypothesis.

32. The workweek for adults in the United States work full-time is normally distributed with a mean of 47 hours. A newly hired engineer at a start-up company believes that employees at start-up companies work more on average then working adults in the U.S. She asks 12 engineering friends at start-ups for the lengths in hours of their workweek. Their responses are shown in the table below. Test the claim using a 5% level of significance. Show all 5 steps using the p-value method.

33. The average number of calories from a fast food meal for adults in the United States is 842 calories. A nutritionist believes that the average is higher than reported. They sample 11 meals that adults ordered and measure the calories for each meal shown below. Test the claim using a 5% level of significance. Assume that fast food calories are normally distributed. Show all 5 steps using the p-value method.

34. Honda advertises the 2018 Honda Civic as getting 32 mpg for city driving. A skeptical consumer about to purchase this model believes the mpg is less than the advertised amount and randomly selects 35 2018 Honda Civic owners and asks them what their car’s mpg is. Use a 1% significance level. They find a p-value of 0.0436.

i. H 0 : μ = 32 H 1 : μ < 32

ii. H 0 : μ < 32 H 1 : μ ≥ 32

iii. H 0 : μ = 32 H 1 : μ > 32

iv. H 0 : μ = 35 H 1 : μ ≠ 35

v. H 0 : μ = 32 H 1 : μ ≠ 32

b) Choose the correct decision based off the reported p-value.

i. Reject H 0

ii. Do not reject H 0

iii. Do not reject H 1

iv. Reject H 1

For exercises 35-40, show all 5 steps for hypothesis testing:

a) State the hypotheses.

b) Compute the test statistic.

c) Compute the critical value or p-value.

d) State the decision.

e) Write a summary.

35. The total of individual pounds of garbage discarded by 17 households in one week is shown below. The current waste removal system company has a weekly maximum weight policy of 36 pounds. Test the claim that the average weekly household garbage weight is less than the company's weekly maximum. Use a 5% level of significance.

36. The world’s smallest mammal is the bumblebee bat (also known as Kitti’s hog-nosed bat or Craseonycteris thonglongyai). Such bats are roughly the size of a large bumblebee. A sample of 10 bats weighed in grams are shown below. Test the claim that mean weight for all bumblebee bats is not equal to 2.1 g using a 1% level of significance. Assume that the bat weights are normally distributed.

37. The average age of an adult's first vacation without a parent or guardian was reported to be 23 years old. A travel agent believes that the average age is different from what was reported. They sample 28 adults and they asked their age in years when they first vacationed as an adult without a parent or guardian, data shown below. Test the claim using a 10% level of significance.

38. Test the claim that the proportion of people who own dogs is less than 32%. A random sample of 1,000 people found that 28% owned dogs. Do the sample data provide convincing evidence to support the claim? Test the relevant hypotheses using a 10% level of significance.

39. The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.3% of American adults suffer from depression or a depressive illness. Suppose that in a survey of 2,000 people in a certain city, 11.1% of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that city suffering from depression or a depressive illness is more than the 9.3% in the general adult American population. Test the relevant hypotheses using a 5% level of significance.

40. The United States Department of Energy reported that 48% of homes were heated by natural gas. A random sample of 333 homes in Oregon found that 149 were heated by natural gas. Test the claim that the proportion of homes in Oregon that were heated by natural gas is different from what was reported. Use a 1% significance level.

41. A 2019 survey by the Bureau of Labor Statistics reported that 92% of Americans working in large companies have paid leave. In January 2021, a random survey of workers showed that 89% had paid leave. The resulting p-value is 0.009; thus, the null hypothesis is rejected. It is concluded that there has been a decrease in the proportion of people, who have paid leave from 2019 to January 2021. What type of error is possible in this situation?

a) Type I Error

b) Type II Error

c) Standard Error

d) Margin of Error

e) No error was made.

For exercises 42-44, show all 5 steps for hypothesis testing:

42. Nationwide 40.1% of employed teachers are union members. A random sample of 250 Oregon teachers showed that 110 belonged to a union. At \(\alpha\) = 0.10, is there sufficient evidence to conclude that the proportion of union membership for Oregon teachers is higher than the national proportion?

43. You are conducting a study to see if the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.31. A random sample of 735 men over the age of 50 found that 208 have their prostate regularly examined. Do the sample data provide convincing evidence to support the claim? Test the relevant hypotheses using a 5% level of significance.

44. Nationally the percentage of adults that have their teeth cleaned by a dentist yearly is 64%. A dentist in Portland, Oregon believes that regionally the percent is higher. A sample of 2,000 Portlanders found that 1,312 had their teeth cleaned by a dentist in the last year. Test the relevant hypotheses using a 10% level of significance.

5) ±2.5758

7) ±1.96

9) a) ii. b) viii. c) iv. d) vi. e) vii. f) i. g) v. h) iii.

11) 100(1 – α)% = Confidence Level 1 – β = Power β = P(Type II Error) µ = Parameter α = Significance Level

19) The implication of a Type I error from the clinical trial is that the vaccination will be approved when it indeed does not reduce the risk of contracting the virus. The implication of a Type II error from the clinical trial is that the vaccination will not be approved when it indeed does reduce the risk of contracting the virus.

21) The local government decides that the data do not provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact higher than 30 minutes.

27) H 0: µ = 98.6; H 1 : µ > 98.6; z = 2.1639; p-value = 0.0152; Do not reject H 0 . There is not enough evidence to support the claim that the mean body temperature for all athletes is more than 98.6˚F.

29) ±2.1098

31) a) iii. b) i.

33) H 0 : µ = 842; H 1 : µ > 842; t = 0.8218; p-value = 0.2152; Do not reject H 0 . We do not have evidence to support the claim the average calories from a fast food meal is higher than reported.

35) H 0 : µ = 36; H 1 : µ < 36; t = -1.9758; p-value = 0.0438; Reject H 0 . There is enough evidence to support the claim the average weekly household garbage weight is less than the company’s weekly 36 lb. maximum.

37) H 0 : µ = 23; H 1 : µ ≠ 23; t = 1.4224; p-value = 0.1664; Do not reject H 0 . We do not have enough evidence to support the claim that the mean age adults travel without a parent or guardian differs from 23.

39) H 0 : p = 0.093; H 1 : p > 0.093; t = 2.7116; p-value = 0.0027; Reject H 0 . There is enough evidence to support the claim the population proportion of American adults that suffer from depression or a depressive illness is more than 9.3%.

43) H 0 : p = 0.31; H 1 : p < 0.31; t = -1.5831; p-value = 0.0567; Do not reject H 0 . There is not enough evidence to support the claim the population proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.31

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Understandable Statistics, Concepts and Methods

Charles henry brase, corrinne pellillo brase, hypothesis testing - all with video answers.

Introduction to Statistical Tests

Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you answer the following questions. (a) What is a null hypothesis $H_{0}$ ? (b) What is an alternate hypothesis $H_{1} ?$ (c) What is a type I error? a type II error? (d) What is the level of significance of a test? What is the probability of a type II error?

In a statistical test, we have a choice of a left-tailed test, a right-tailed test, or a two-tailed test. Is it the null hypothesis or the alternate hypothesis that determines which type of test is used? Explain your answer.

If we fail to reject (i.e., "accept") the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.

If we reject the null hypothesis, does this mean that we have proved it to be false beyond all doubt? Explain your answer.

What terminology do we use for the probability of rejecting the null hypothesis when it is true? What symbol do we use for this probability? Is this the probability of a type I or a type II error?

What terminology do we use for the probability of rejecting the null hypothesis when it is, in fact, false?

If the $P$ -value in a statistical test is greater than the level of significance for the test, do we reject or fail to reject $H_{0} ?$

If the $P$ -value in a statistical test is less than or equal to the level of significance for the test, do we reject or fail to reject $H_{0}$ ?

Suppose the $P$ -value in a right-tailed test is 0.0092. Based on the same population, sample, and null hypothesis, what is the $P$ -value for a corresponding two-tailed test?

Suppose the $P$ -value in a two-tailed test is 0.0134. Based on the same population, sample, and null hypothesis, and assuming the test statistic $z$ is negative, what is the $P$ -value for a corresponding left-tailed test?

Suppose you want to test the claim that a population mean equals $40 .$ (a) State the null hypothesis. (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from $40 .$ (c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may exceed $40 .$ (d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be less than $40 .$

Suppose you want to test the claim that a population mean equals $30 .$ (a) State the null hypothesis. (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from $30 .$ (c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be greater than $30 .$ (d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may not be as large as $30 .$

A random sample of size 20 from a normal distribution with $\sigma=4$ produced a sample mean of 8 (a) Is the $\bar{x}$ distribution normal? Explain. (b) Compute the sample test statistic $z$ under the null hypothesis $H_{0}: \mu=7$ (c) For $H_{1}: \mu \neq 7$, estimate the $P$ -value of the test statistic. (d) For a level of significance of 0.05 and the hypotheses of parts (b) and (c), do you reject or fail to reject the null hypothesis? Explain.

A random sample of size 16 from a normal distribution with $\sigma=3$ produced a sample mean of 4.5 (a) Check Requirements Is the $\bar{x}$ distribution normal? Explain. (b) Compute the sample test statistic $z$ under the null hypothesis $H_{0}: \mu=6.3$ (c) For $H_{1}: \mu<6.3,$ estimate the $P$ -value of the test statistic. (d) For a level of significance of 0.01 and the hypotheses of parts (b) and (c), do you reject or fail to reject the null hypothesis? Explain.

The body weight of a healthy 3-month-old colt should be about $\mu=60 \mathrm{kg}$ (Source: The Merck Veterinary Manual, a standard reference manual used in most veterinary colleges). (a) If you want to set up a statistical test to challenge the claim that $\mu=60 \mathrm{kg},$ what would you use for the null hypothesis $H_{0} ?$ (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than $60 \mathrm{kg}$. What would you use for the alternate hypothesis $H_{1} ?$ (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than $60 \mathrm{kg}$. What would you use for the alternate hypothesis? (d) Suppose you want to test the claim that the average weight of such a wild colt is different from $60 \mathrm{kg}$. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the $P$ -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

How much customers buy is a direct result of how much time they spend in a store. A study of average shopping times in a large national housewares store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill): Women with female companion: 8.3 min. Women with male companion: 4.5 min. Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of 8.3 minutes shopping in such a store. (a) What would you use for the null and alternate hypotheses if you believe the average shopping time is less than 8.3 minutes? Is this a right-tailed, left-tailed, or two-tailed test? (b) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 8.3 minutes? Is this a righttailed, left-tailed, or two-tailed test? Stores that sell mainly to women should figure out a way to engage the interest of men-perhaps comfortable seats and a big TV with sports programs! Suppose such an entertainment center was installed and you now wish to challenge the claim that a woman with a male friend spends only 4.5 minutes shopping in a housewares store. (c) What would you use for the null and alternate hypotheses if you believe the average shopping time is more than 4.5 minutes? Is this a right-tailed, left-tailed, or two-tailed test? (d) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 4.5 minutes? Is this a righttailed, left-tailed, or two-tailed test?

Weatherwise magazine is published in association with the American Meteorological Society. Volume $46,$ Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of 16.4 feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. (a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height? (b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis? (c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis? (d) Suppose you do not know whether the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the $P$ -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Consumer Reports stated that the mean time for a Chrysler Concorde to go from 0 to 60 miles per hour is 8.7 seconds. (a) If you want to set up a statistical test to challenge the claim of 8.7 seconds, what would you use for the null hypothesis? (b) The town of Leadville, Colorado, has an elevation over 10,000 feet. Suppose you wanted to test the claim that the average time to accelerate from 0 to 60 miles per hour is longer in Leadville (because of less oxygen). What would you use for the alternate hypothesis? (c) Suppose you made an engine modification and you think the average time to accelerate from 0 to 60 miles per hour is reduced. What would you use for the alternate hypothesis? (d) For each of the tests in parts (b) and (c), would the $P$ -value area be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the $z$ value of the sample test statistic. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Let $x$ be a random variable representing dividend yield of Australian bank stocks. We may assume that $x$ has a normal distribution with $\sigma=2.4 \% .$ A random sample of 10 Australian bank stocks gave the following yields. The sample mean is $\bar{x}=5.38 \% .$ For the entire Australian stock market, the mean dividend yield is $\mu=4.7 \%$ (Reference: Forbes). Do these data indicate that the dividend yield of all Australian bank stocks is higher than $4.7 \% ?$ Use $\alpha=0.01$

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the $z$ value of the sample test statistic. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in $\mathrm{mg} / 100 \mathrm{ml}$ ). The sample mean is $\bar{x} \approx 93.8 .$ Let $x$ be a random variable representing glucose readings taken from Gentle Ben. We may assume that $x$ has a normal distribution, and we know from past experience that $\sigma=12.5 .$ The mean glucose level for horses should be $\mu=85 \mathrm{mg} / 100 \mathrm{ml}$ (Reference: Merck Veterinary Manual). Do these data indicate that Gentle Ben has an overall average glucose level higher than $85 ?$ Use $\alpha=0.05$

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the $z$ value of the sample test statistic. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna) (Reference: Hummingbirds by K. Long and W. Alther). Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were $$3.7 \quad 2.9 \quad 3.8 \quad 4.2 \quad 4.8 \quad 3.1$$ The sample mean is $\bar{x}=3.75$ grams. Let $x$ be a random variable representing weights of Anna's hummingbirds in this part of the Grand Canyon. We assume that $x$ has a normal distribution and $\sigma=0.70$ gram. It is known that for the population of all Anna's hummingbirds, the mean weight is $\mu=4.55$ grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.55 grams? Use $\alpha=0.01$.

The price-to-earnings (P/E) ratio is an important tool in financial work. A random sample of 14 large U.S. banks (J.P. Morgan, Bank of America, and others) gave the following $\mathrm{P} / \mathrm{E}$ ratios (Reference: Forbes ). The sample mean is $\bar{x} \approx 17.1 .$ Generally speaking, a low $\mathrm{P} / \mathrm{E}$ ratio indicates a "value" or bargain stock. A recent copy of The Wall Street Journal indicated that the P/E ratio of the entire $\mathrm{S} \& \mathrm{P}$ 500 stock index is $\mu=19 .$ Let $x$ be a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that $x$ has a normal distribution and $\sigma=4.5 .$ Do these data indicate that the P/E ratio of all U.S. bank stocks is less than $19 ?$ Use $\alpha=0.05$

Nationally, about $11 \%$ of the total U.S. wheat crop is destroyed each year by hail (Reference: Agricultural Statistics, U.S. Department of Agriculture). An insurance company is studying wheat hail damage claims in Weld County, Colorado. A random sample of 16 claims in Weld County gave the following data (\% wheat crop lost to hail). $$\begin{array}{cccccccc} 15 & 8 & 9 & 11 & 12 & 20 & 14 & 11 \\ 7 & 10 & 24 & 20 & 13 & 9 & 12 & 5 \end{array}$$ The sample mean is $\bar{x}=12.5 \% .$ Let $x$ be a random variable that represents the percentage of wheat crop in Weld County lost to hail. Assume that $x$ has a normal distribution and $\sigma=5.0 \% .$ Do these data indicate that the percentage of wheat crop lost to hail in Weld County is different (either way) from the national mean of $11 \% ?$ Use $\alpha=0.01$.

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the $z$ value of the sample test statistic. (c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Total blood volume (in ml) per body weight (in $\mathrm{kg}$ ) is important in medical research. For healthy adults, the red blood cell volume mean is about $\mu=28 \mathrm{ml} / \mathrm{kg}$ (Reference: Laboratory and Diagnostic Tests by F. Fischbach). Red blood cell volume that is too low or too high can indicate a medical problem (see reference). Suppose that Roger has had seven blood tests, and the red blood cell volumes were $$\begin{array}{rrrrrr} 32 & 25 & 41 & 35 & 30 & 37 & 29 \end{array}$$ The sample mean is $\bar{x} \approx 32.7 \mathrm{ml} / \mathrm{kg} .$ Let $x$ be a random variable that represents Roger's red blood cell volume. Assume that $x$ has a normal distribution and $\sigma=4.75 .$ Do the data indicate that Roger's red blood cell volume is different (either way) from $\mu=28 \mathrm{ml} / \mathrm{kg} ?$ Use a 0.01 level of significance.