Section 13.2 Exercises

CLARIFYING THE CONCEPTS

Question 13.79

1. Explain the difference between the confidence interval and the prediction interval we learned about in this section. (pp. 738, 739)

13.2.1

The confidence interval is for the mean value of for a given and the prediction interval is for a randomly selected value of for a given .

Question 13.80

2. True or false: For a given value of , the confidence interval is always wider than the prediction interval. (p. 740)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 3–8 Example 8 Constructing and interpreting
a confidence interval for the
mean value of for a given
Exercises 9–14 Example 9 Constructing and interpreting
a prediction interval for a
randomly selected value of
for a given

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For Exercises 3–8, use the data provided to construct a 95% confidence interval for the mean value of for the given value of . The regression equation is provided; assume the regression assumptions are met.

Question 13.81

3.

1 15
2 20
3 20
4 25
5 25

13.2.3

Lower bound: 18.7500 (Minitab: 18.7497); Upper bound: 23.5000 (Minitab: 23.2500)

Question 13.82

4.

0 10
5 20
10 45
15 50
20 75

Question 13.83

5.

−5 0
−4 8
−3 8
−2 16
−1 16

13.2.5

Lower bound: 1.1909 (Minitab: 1.19027); Upper bound: 10.0091 (Minitab: 10.0097)

Question 13.84

6.

−3 −5
−1 −15
1 −20
3 −25
5 −30

Question 13.85

7.

10 100
20 95
30 85
40 85
50 80

13.2.7

Lower bound: 92.636 (Minitab: 92.6351); Upper bound: 105.364 (Minitab: 105.365)

Question 13.86

8.

0 11
20 11
40 16
60 21
80 26

For Exercises 9–14, use the data from the indicated exercise to construct a 95% prediction interval for a randomly chosen value of for the given value of . Assume the regression assumptions are met.

Question 13.87

9. The data in Exercise 3

13.2.9

Lower bound: 15.4886 (Minitab: 15.4878); Upper bound: 26.5114 (Minitab: 26.5122)

Question 13.88

10. The data in Exercise 4

Question 13.89

11. The data in Exercise 5

13.2.11

Lower bound: −3.5783 (Minitab: −3.57958); Upper bound: 14.7783 (Minitab: 14.7796)

Question 13.90

12. The data in Exercise 6

Question 13.91

13. The data in Exercise 7

13.2.13

Lower bound: 88.6076 (Minitab: 88.6062); Upper bound: 109.3924 (Minitab: 109.394)

Question 13.92

14. The data in Exercise 8

APPLYING THE TECHNIQUES

For Exercises 15–24, use the data and the regression equations that you calculated in the Section 13.1 exercises.

Question 13.93

15. Volume and Weight. For the data from Exercise 39 in Section 13.1, do the following:

  1. Predict the weight of a package that has a volume of 4 cubic meters.
  2. Construct a 95% confidence interval for the population mean weight of all packages that have a volume of 4 cubic meters.

13.2.15

(a) 10.4 kilograms (b) Lower bound: 7.1550 kilograms (Minitab: 7.15453) Upper bound: 13.6450 kilograms (Minitab: 13.6455)

Question 13.94

16. Family Size and pets. For the data from Exercise 40 in Section 13.1, do the following:

  1. Predict the population mean number of pets for a family of size 5.
  2. Construct a 95% confidence interval for the population mean number of pets for all families of size 5.

Question 13.95

17. World Temperatures. For the data from Exercise 41 in Section 13.1, do the following:

  1. Predict the population mean high temperature for a city with a low temperature of 30 degrees.
  2. Construct a 99% confidence interval for the population mean high temperature for all cities with a low temperature of 30 degrees.

13.2.17

(a) 43.3443 degrees (b) Lower bound: 34.4488 degrees (Minitab: 34.4486); Upper bound: 52.2398 degrees (Minitab: 52.2400)

Question 13.96

18. Video game Sales. For the data from Exercise 42 in Section 13.1, do the following:

  1. Predict the population mean total sales for a game that has been on the top 30 list for 20 weeks.
  2. Construct a 99% confidence interval for the population mean total sales for all games that have been on the top 30 list for 20 weeks.

Question 13.97

19. Volume and Weight. Refer to the data from Exercise 15.

  1. Construct a 95% prediction interval for the weight of a randomly selected package that has a volume of 4 cubic meters.
  2. Compare the intervals from Exercise 15(b) and Exercise 19(a). Which interval is wider, and why?

13.2.19

(a) Lower bound: 5.1009 kilograms (Minitab: 5.1009); Upper bound: 15.6991 kilograms (Minitab: 15.6998) (b) The interval in (a) is wider. Individual values are more variable than their mean. The interval in Exercise 15(b) is a 95% confidence interval for the mean value of for the given value cubic meters and the interval in (a) is a prediction interval for a randomly chosen value of for the given value cubic meters.

Question 13.98

20. Family Size and pets. Refer to the data from Exercise 16.

  1. Construct a 95% prediction interval for the number of pets for a randomly selected family of size 5.
  2. Compare the intervals from Exercise 16(b) and Exercise 20(a). Which interval is wider, and why?

Question 13.99

21. World Temperatures. Refer to the data from Exercise 17.

  1. Construct a 90% prediction interval for the high temperature for a randomly chosen city with a low temperature of 30 degrees.

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  2. Suppose we were asked to provide a prediction interval for the high temperature for a randomly chosen city with a low temperature of zero degrees. Explain what the danger would be in constructing such a prediction interval.

13.2.21

(a) Lower bound: 22.1568 degrees (Minitab: 22.1564); Upper bound: 64.5317 degrees (Minitab: 64.5322) (b) The range of the low temperatures in the data set is from 7°C to 70°C, inclusive. Therefore, a low temperature of 0°C is outside of the range of our data set, and any predictions or estimates using the regression equation for 0°C represent extrapolation. Extrapolation should be avoided if possible because the relationship between the variables may no longer be linear outside the range of .

Question 13.100

22. Video game Sales. Refer to the data from Exercise 18.

  1. Construct a 99% prediction interval for the total sales for a randomly chosen video game that has been on the top list for 20 weeks.
  2. Suppose we were asked to provide a prediction interval for the total sales for a randomly chosen video game that has been on the top list for 104 weeks. Explain what the danger would be in constructing such a prediction interval.

Question 13.101

pulseandtemp

23. Working with large Data Sets. Open the Pulse and Temp data set. Select the females data only. Use technology to do the following:

  1. Construct a 95% confidence interval for the population mean body temperature for all females with a heart rate of 72.
  2. Construct a 95% prediction interval for the body temperature for a randomly selected female with a heart rate of 72.

13.2.23

(a) (98.153, 98.521) (b) (96.891, 99.783)

Question 13.102

nutrition

24. Working with large Data Sets. Open the Nutrition data set. Use technology to do the following:

  1. Construct a 95% confidence interval for the population mean number of calories per gram for all foods with the amount of fat per gram of 0.5.
  2. Construct a 95% prediction interval for the number of calories per gram for a randomly selected food with the amount of fat per gram of 0.5.