Section 10.4 Exercises

626

CLARIFYING THE CONCEPTS

Question 10.157

1. In the section, what is the test used to compare? (p. 617)

10.4.1

Two population standard deviations

Question 10.158

2. True or false: To perform the test, the random samples from the two populations may be either dependent samples or independent samples. (p. 618)

Question 10.159

3. True or false: To perform the test, the two populations must be normally distributed. (p. 618)

10.4.3

True

Question 10.160

4. What is the formula for the test statistic for the test? (p. 618)

Question 10.161

5. For the test, what distribution does the test statistic follow? What are the degrees of freedom? (p. 618)

10.4.5

An distribution with degrees of freedom in the numerator and degrees of freedom in the denominator.

Question 10.162

6. True or false: The value of the random variable can be positive or negative. (p. 618)

Question 10.163

7. What does the notation represent? (p. 619)

10.4.7

The with numerator degrees of freedom and denominator degrees of freedom, with area α to the right of .

Question 10.164

8. Briefly explain how to find the critical value for a given area to the left of it. (p. 619)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 9–20 Example 19 Finding critical values of
the distribution
Exercises 21–26 Example 20 test for comparing two
population standard
deviations: critical-value
method
Exercises 27–32 Example 21 test for comparing two
population standard
deviations: -value method

For Exercises 9–12, find the critical value for a right-tailed test with the given characteristics.

Question 10.165

9. , ,

10.4.9

.

Question 10.166

10. , ,

Question 10.167

11. , ,

10.4.11

.

Question 10.168

12. , ,

For Exercises 13–16, find the critical value for a left-tailed test with the given characteristics.

Question 10.169

13. , ,

10.4.13

.

Question 10.170

14. , ,

Question 10.171

15. , ,

10.4.15

.

Question 10.172

16. , ,

For Exercises 17–20, find the critical value for a two-tailed test with the given characteristics.

Question 10.173

17. , ,

10.4.17

Question 10.174

18. , ,

Question 10.175

19. , ,

10.4.19

Question 10.176

20. , ,

For Exercises 21–26, assume that the populations are independent and normally distributed. Perform the indicated hypothesis tests using the critical-value method. For each exercise

  1. State the hypotheses.
  2. Find the critical value(s) and state the rejection rule.
  3. Find .
  4. State the conclusion and interpretation.

Question 10.177

21.

10.4.21

(a) (b) . Reject if . (c) . (d) Since is not , we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of Population 1 is greater than the population standard deviation of Population 2.

Question 10.178

22.

Question 10.179

23.

10.4.23

(a) (b) . Reject if . (c) . (d) Since is not , we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of Population 1 is less than the population standard deviation of Population 2.

Question 10.180

24.

Question 10.181

25.

10.4.25

(a) (b) and . Reject if or if . (c) . (d) Since is , we reject . There is evidence at the level of significance that the population standard deviation of Population 1 differs from the population standard deviation of Population 2.

Question 10.182

26.

For Exercises 27–32, assume that the populations are normally distributed. Perform the indicated hypothesis tests using the -value method. For each exercise,

  1. State the hypotheses and the rejection rule.
  2. Find .
  3. Find the -value.
  4. State the conclusion and interpretation.

Question 10.183

27.

10.4.27

(a) . Reject if the .

(b) (c) . (d) Since the is not , we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of Population 1 is greater than the population standard deviation of Population 2.

Question 10.184

28.

Question 10.185

29.

10.4.29

(a) . Reject if the .

(b) (c) . (d) Since the is , we reject . There is evidence at the level of significance that the population standard deviation of Population 1 is less than the population standard deviation of Population 2.

Question 10.186

30.

Question 10.187

31.

10.4.31

(a) . Reject if the .

(b) (c) . (d) Since the is not , we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of Population 1 is different from the population standard deviation of Population 2.

Question 10.188

32.

APPLYING THE CONCEPTS

Question 10.189

33. Body Mass Index. The body mass index (BMI) is a statistical measure of an individual's body weight, based on his or her weight and height. In the Framingham Heart Study,18 a sample of 1160 males had a mean BMI of 25.5 with a sample standard deviation of 3.3, and a sample of 1571 women had a mean BMI of 24.8 with a sample standard deviation of 4.3. Test whether the population standard deviation of the males' BMI is less than that of the females, using . Assume that the samples are independent and that the two populations are normally distributed.

10.4.33

Critical-value method: . . Reject if . . Since is ≤ 0.9009, we reject . There is evidence at the level of significance that the population standard deviation of the males' BMI is less than the population standard deviation of the females' BMI. method: . Reject if the . . Since the is ≤ 0.05, we reject . There is evidence at the level of significance that the population standard deviation of the males' BMI is less than the population standard deviation of the females' BMI.

Question 10.190

34. Smoking. A research paper19 described two common types of smokers: (a) Morning Highs: those who tend to smoke more in the morning than the rest of the day, and (b) Flatlines: those who tend to smoke at the same rate all day. A sample of 43 smokers from the Morning High group had a mean of 30.7 cigarettes smoked per day, with a sample standard deviation of 10.5 cigarettes. A sample of 247 smokers from the Flatline group had a mean of 22.0 cigarettes smoked per day, with a sample standard deviation of 7.7 cigarettes. Test whether the population standard deviation of the Morning High group is greater than that of the Flatline group, using . Assume that the samples are independent and that the two populations are normally distributed.

627

Question 10.191

35. Autism and Asperger's Syndrome. A study20 compared the IQ scores of children with autism and children with Asperger's syndrome. A sample of 85 children with autism had a mean IQ of 72.71 with a sample standard deviation of 15.08. A sample of 341 children with Asperger's syndrome had a mean IQ of 95.76 with a sample standard deviation of 16.56. Test whether the population standard deviations of the two groups differ, using . Assume that the samples are independent and that the two populations are normally distributed.

10.4.35

Critical-value method: and . Reject if or if . . Since is not ≤ 0.6623 and not ≥1.39, we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of the IQ scores of children with autism differs from the population standard deviation of the IQ scores of children with Asperger's syndrome. method: . Reject if the . . . Since the is not ≤ 0.10, we do not reject . There is insufficient evidence at the level of significance that the population standard deviation of the IQ scores of children with autism differs from the population standard deviation of the IQ scores of children with Asperger's syndrome.

Question 10.192

36. Blood Pressure. A recent study21 compared the systolic blood pressure readings of two groups of adults: those ages 30–44 and those ages 70-79. A random sample of 100 individuals in the younger group had a sample mean systolic blood pressure reading of 111, with a sample standard deviation of 5. A random sample of 100 individuals in the older group had a sample mean systolic blood pressure reading of 134, with a sample standard deviation of 13. Test whether the population standard deviation of the younger group is less than that of the older group, using . Assume that the samples are independent and that the two populations are normally distributed.

Question 10.193

37. Gold and Silver Prices. We are interested in testing whether the variability in gold prices exceeds that of silver prices. Refer to the following data, which represent random samples of the prices taken between June 2009 and July 2010, and the following normal probability plots.

gold

silver

  1. Do the probability plots indicate violation of the normality assumption?
    Gold
    1121.5 1128 1086.8 1098.25 1114 1104.25 1123.5
    1154.5 1185 1236 1227.75 1225 1234 1208
    image
    Silver
    17.17 18.52 16.23 15.57 16.50 17.02 17.69
    18.27 18.71 19.11 18.30 18.42 18.65 18.42
    image
  2. Test whether the population standard deviation of gold prices is greater than that of silver prices, using . Assume that the samples are independent.

10.4.37

(a) No (b) Critical-value method: . . Reject if . . Since is ≥ 4.16, we reject . There is evidence at the level of significance that the population standard deviation of gold prices is greater than the population standard deviation of silver prices. method: . Reject if the . . Since the is ≤ 0.01, we reject . There is evidence at the level of significance that the population standard deviation of gold prices is greater than the population standard deviation of silver prices.

Question 10.194

38. Miles per Gallon. The following data represent the combined (city and highway) fuel efficiency in miles per gallon (mpg) for independent random samples of models manufactured by Honda and Lexus.

Honda car mpg Lexus car mpg
Accord 24 GX 470 15
Odyssey 18 LS 460 18
Civic Hybrid 42 RX 350 19
Fit 31 IS 350 20
CR-V 23 GS 450 23
Ridgeline 17 IS 250 24
S2000 21
  1. The boxplots represent the miles per gallon for each manufacturer. Based on this graph, would you expect that the standard deviations of the two manufacturers are the same or different? Why?
  2. Note: Normal probability plots indicate that the normality assumption is not violated. Test whether the population standard deviations of the two manufacturers' fuel efficiency differ, using .
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