Section 9.1 Exercises

CLARIFYING THE CONCEPTS

Question 9.1

1. What are some characteristics of the null hypothesis? The alternative hypothesis? (p. 489)

9.1.1

The null hypothesis is assumed to be true unless the sample evidence indicates that the alternative hypothesis is true instead. It represents what has been tentatively assumed about the value of the parameter. It is the status quo hypothesis. The alternative hypothesis represents an alternative claim about the value of the parameter. The researcher concludes that the alternative hypothesis is true only if the evidence provided by the sample data indicates that it is true.

Question 9.2

2. Explain what is meant by . (p. 490)

Question 9.3

3. In the hypothesis test for the population mean, how many forms of the hypotheses are there? Write out these forms. (p. 490)

9.1.3

Form Null hypothesis Alternative hypothesis
1 vs.
2 vs.
3 vs.

Question 9.4

4. In a criminal trial, what are the two possible decision errors? What do statisticians call these errors? (p. 494)

Question 9.5

5. When does a Type I error occur? A Type II error? (p. 494)

9.1.5

A Type I error occurs when one rejects when is true. A Type II error occurs when one does not reject when is false.

Question 9.6

6. What are the two correct decisions that can be made? (p. 494)

Question 9.7

7. Say we want to test whether the population mean is less than 100, and the sample we take yields a sample mean of 90. Is this sufficient evidence that the population mean is less than 100? Explain why or why not. (p. 492)

9.1.7

No. It depends on how many standard deviations the sample mean of 90 is below the population mean of 100 and the level of significance of the test.

Question 9.8

8. True or false: If the consequences of making a Type I error are serious, then the data analyst should choose a larger level of significance. (p. 495)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 9–12 Example 3 Identifying valid and
invalid hypotheses
Exercises 13–18 Example 4 Applying the strategy
for constructing the
hypotheses about
Exercises 19–22 Example 6 Type I and Type II errors

Determine whether the following hypotheses are in a valid form or not. If not, explain why not, and put the hypotheses in a valid form.

Question 9.9

9.

9.1.9

Invalid. Statistics like should not be used in the hypotheses. So the correct form is: versus

Question 9.10

10.

Question 9.11

11.

9.1.11

Invalid. The equal sign always goes in . So the correct form is: versus

Question 9.12

12.

For Exercises 13–18, provide the null and alternative hypotheses.

Question 9.13

13. Test whether is greater than 79.

9.1.13

versus

Question 9.14

14. Test whether is less than 50.

Question 9.15

15. Test whether is different from 75.

9.1.15

versus

Question 9.16

16. Test whether or not equals 12,500.

Question 9.17

17. One person states that equals 1000. Another person states that has changed from 1000.

9.1.17

versus

Question 9.18

18. One person states that equals 32. Another person states that exceeds 32.

For Exercises 19–22, do the following:

  1. Provide the null and alternative hypotheses.
  2. Explain what it would mean to make a Type I error.
  3. Explain what it would mean to make a Type II error.

Question 9.19

19. A psychologist wants to determine whether the mean achievement gap has decreased from its previous level of 10 points.

9.1.19

(a) versus (b) Conclude that the population mean achievement gap has decreased when, in reality, it has remained the same. (c) Conclude that the population mean achievement gap has remained the same when, in reality, it has decreased.

Question 9.20

20. A manager is interested in testing whether mean throughput has increased since she came on the job, when it was 1000 units per week.

Question 9.21

21. A health maintenance organization wants to examine whether a new treatment method has decreased the mean cost from the $20,000 cost of the previous treatment.

9.1.21

(a) versus (b) Conclude that the population mean cost of the new treatment has decreased from the cost of the previous treatment when, in reality, it has remained the same. (c) Conclude that the population mean cost of the new treatment has remained the same as the cost of the previous treatment when, in reality, it has decreased.

Question 9.22

22. A sports analyst is interested in testing whether mean times in the 40-yard dash have changed from 5 seconds.

APPLYING THE CONCEPTS

For Exercises 23–35, do the following:

  1. Provide the null and alternative hypotheses.
  2. Describe the two ways a correct decision could be made in the context of the problem.
  3. Describe what a Type I error would mean in the context of the problem.
  4. Describe what a Type II error would mean in the context of the problem.

Question 9.23

23. Nurses' Study Times. The National Survey of Student Engagement reported that the mean number of hours spent studying per week for nursing majors is 18. A medical researcher is interested in investigating whether mean study time is greater than 18 hours per week.

9.1.23

(a) versus (b) Conclude that the population mean number of hours nursing majors study per week is greater than 18 hours when, in reality, it is greater than 18 hours. Conclude that the population mean number of hours nursing majors study per week is equal to 18 hours when, in reality, it is equal to 18 hours. (c) Conclude that the population mean number of hours nursing majors study per week is greater than 18 hours when, in reality, it is equal to 18 hours. (d) Conclude that the population mean number of hours nursing majors study per week is equal to 18 hours when, in reality, it is greater than 18 hours.

Question 9.24

24. Eating Trends. The NPD Group examined whether higher gasoline prices are causing consumers to dine out less than the previous mean of 700 times per year.1

Question 9.25

25. Hybrid Vehicles. The motor vehicle Web site Edmunds.com wants to test whether owners of hybrid vehicles can recoup their initial increased cost through reduced fuel consumption in less than three years, on average.

9.1.25

(a) versus (b) Conclude that the population mean number years it takes for owners of hybrid vehicles to recoup their initial increased cost through reduced fuel consumption is less than 3 years when, in reality, it is less than 3 years. Conclude that the population mean number years it takes for owners of hybrid vehicles to recoup their initial increased cost through reduced fuel consumption is equal to 3 years when, in reality, it is equal to 3 years. (c) Conclude that the population mean number years it takes for owners of hybrid vehicles to recoup their initial increased cost through reduced fuel consumption is less than 3 years when, in reality, it is equal to 3 years. (d) Conclude that the population mean number years it takes for owners of hybrid vehicles to recoup their initial increased cost through reduced fuel consumption is equal to 3 years when, in reality, it is less than 3 years.

497

Question 9.26

26. Traffic Light Cameras. The Ministry of Transportation in the province of Ontario wants to determine whether the installation of cameras that take pictures at traffic lights has decreased the mean number of fatal and injury collisions to below the previous mean of 339.1 per year.

Question 9.27

27. Price of Milk. The U.S. Department of Agriculture is examining whether the mean price for a gallon of milk has increased since January 2011, when it was $3.34.

9.1.27

(a) (b) Conclude that the mean is greater than $3.24 when it actually is greater than $3.24, and conclude that the mean is equal to $3.24 when it actually is equal to $3.24. (c) Concluding that the mean is greater than $3.24 when it actually is equal to $3.24 (d) Concluding that the mean is equal to $3.24 when it actually is greater than $3.24

Question 9.28

28. Americans' Height. Dr. Richard Steckel, professor of economics and anthropology at The Ohio State University, wants to investigate whether the mean height of Americans has changed from 175 centimeters.2

Question 9.29

29. Caffeine in Starbucks Coffee. The medical information Web site MayoClinic.org reports that a 16-ounce Starbucks Park Place brewed coffee contains 350 milligrams (mg) of caffeine. Suppose a local health organization is interested in whether the mean amount of caffeine in this coffee has changed from 350 mg.

9.1.29

(a) (b) Conclude that the population mean amount of caffeine in a 16-ounce Starbucks Park Place brewed coffee has changed from 350 milligrams when, in reality, it has changed from 350 milligrams. Conclude that the population mean amount of caffeine in a 16-ounce Starbucks Park Place brewed coffee has not changed from 350 milligrams when, in reality, it has not changed from 350 milligrams.

(c) Conclude that the population mean amount of caffeine in a 16-ounce Starbucks Park Place brewed coffee has changed from 350 milligrams when, in reality, it has not changed from 350 milligrams. (d) Conclude that the population mean amount of caffeine in a 16-ounce Starbucks Park Place brewed coffee has not changed from 350 milligrams when, in reality, it has changed from 350 milligrams.

Question 9.30

30. Mobile Apps. The Nielsen organization is preparing a study to determine whether the mean number of apps used per month by iPhone and Android users has increased from 28.6, its 2014 level.

image Chapter 9 Case Study: Clothing Store Sales.

The business research questions in Exercises 31–35 pertain to our Chapter 9 Case Study, Clothing Store Sales.

Question 9.31

31. Purchase Visits. The store manager wants to know whether her customers have made on average more than five visits during the six-month period.

9.1.31

(a) (b) Conclude that the population mean number of visits that customers make to the store in the six-month period is greater than 5 when, in reality, it is greater than 5. Conclude that the population mean number of visits that customers make to the store in the six-month period is equal to 5 when, in reality, it is equal to 5. (c) Conclude that the population mean number of visits that customers make to the store in the six-month period is greater than 5 when, in reality, it is equal to 5. (d) Conclude that the population mean number of visits that customers make to the store in the six-month period is equal to 5 when, in reality, it is greater than 5.

Question 9.32

32. Total Net Sales. The district manager has set a goal to achieve a mean of more than $413 total sales per customer.

Question 9.33

33. Items Purchased. The assistant manager wants to determine whether the mean number of items customers are buying is less than 20.

9.1.33

(a) (b) Conclude that the population mean number of items that customers are buying is less than 20 when, in reality, it is less than 20. Conclude that the population mean number of items that customers are buying is equal to 20 when, in reality, it is equal to 20. (c) Conclude that the population mean number of items that customers are buying is less than 20 when, in reality, it is equal to 20. (d) Conclude that the population mean number of items that customers are buying is equal to 20 when, in reality, it is less than 20.

Question 9.34

34. Coupons. The sales department is interested in finding out if the mean number of coupons used differs from 0.75.

Question 9.35

35. Days Since Purchase. The marketing manager wants to make sure that the mean number of days since purchase is less than 150.

9.1.35

(a) (b) Concluding that the population mean number of days since purchase is less than 150 when, in reality, it is less than 150. Concluding that the population mean number of days since purchase is equal to 150 when, in reality, it is equal to 150. (c) Concluding that the population mean number of days since purchase is less than 150 when, in reality, it is equal to 150. (d) Concluding that the population mean number of days since purchase is equal to 150 when, in reality, it is less than 150.