Section 8.4 Exercises

CLARIFYING THE CONCEPTS

Question 8.263

1. To construct a confidence interval for or , what must be true about the population? (p. 476)

8.4.1

The population must be normal.

Question 8.264

2. Explain the difference between and . (p. 476)

Question 8.265

3. Explain why we need to find two different critical values to construct the confidence intervals in this section. Why can't we just use the “point estimate ± margin of error” method we used earlier in this chapter? (p. 474)

8.4.3

To use this method, the distribution has to be symmetric and the curve is not symmetric.

Question 8.266

4. Provide an example from the real world where it would be important to estimate the variability of a data set. (p. 474)

Determine whether each proposition in Exercises 5–8 is true or false. If it is false, restate the proposition correctly.

Question 8.267

5. The curve is symmetric. (p. 474)

8.4.5

False. The not symmetric. It is right-skewed.

Question 8.268

6. The value of the random variable is never negative. (p. 474)

Question 8.269

7. The curve is right-skewed. (p. 474)

8.4.7

True

Question 8.270

8. The total area under the curve equals 1. (p. 474)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 9–16 Example 23 Finding the critical
values
Exercises 17–32 Example 24 Confidence intervals for
the population variance
and the population
standard deviation

For Exercises 9–14, find the critical values and for the given confidence level and sample size.

Question 8.271

9. Confidence level 90%,

8.4.9

and .

[Using Minitab: and ]

Question 8.272

10. Confidence level 95%,

Question 8.273

11. Confidence level 99%,

8.4.11

and .

[Using Minitab: and ]

Question 8.274

12. Confidence level 95%,

Question 8.275

13. Confidence level 95%,

8.4.13

and .

Question 8.276

14. Confidence level 95%,

Question 8.277

15. Consider the critical values you calculated in Exercises 9–11. Describe what happens to the critical values for a given sample size as the confidence level increases.

8.4.15

decreases and increases.

Question 8.278

16. Consider the critical values you calculated in Exercises 12–14. Describe what happens to the critical values for a given confidence level as the sample size increases.

In Exercises 17–22, a random sample is drawn from a normal population. The sample of size has a sample variance of . Construct the specified confidence interval.

Question 8.279

17. 90% confidence interval for the population variance

8.4.17

(21.87, 35.80) [Using Minitab: (20.1, 32.1)]

Question 8.280

18. 95% confidence interval for the population variance

Question 8.281

19. 99% confidence interval for the population variance

8.4.19

(19.29, 41.81) [Using Minitab: (17.8, 37.2)]

Question 8.282

20. 90% confidence interval for the population standard deviation

Question 8.283

21. 95% confidence interval for the population standard deviation

8.4.21

(4.58, 6.14) [Using Minitab: (4.39, 5.81)]

Question 8.284

22. 99% confidence interval for the population standard deviation

Question 8.285

23. Consider the confidence intervals you constructed in Exercises 17–19. Describe what happens to the lower bound and upper bound of a confidence interval for as the confidence level increases but the sample size stays the same.

8.4.23

Lower bound decreases while the upper bound increases.

Question 8.286

24. Consider the confidence intervals you constructed in Exercises 20–22. Describe what happens to the lower bound and upper bound of a confidence interval for as the confidence level increases but the sample size stays the same.

In Exercises 25–30, a random sample is drawn from a normal population. The sample variance is . Construct the specified confidence interval.

481

Question 8.287

25. 95% confidence interval for the population variance for a sample of size

8.4.25

(15.86, 45.18)

Question 8.288

26. 95% confidence interval for the population variance for a sample of size

Question 8.289

27. 95% confidence interval for the population variance for a sample of size

8.4.27

(20.64, 50.14) [Using Minitab: (17.4, 38.8)]

Question 8.290

28. 95% confidence interval for the population standard deviation for a sample of size

Question 8.291

29. 95% confidence interval for the population standard deviation for a sample of size

8.4.29

(4.56, 7.62) [Using Minitab: (4.10, 6.42)]

Question 8.292

30. 95% confidence interval for the population standard deviation for a sample of size

Question 8.293

31. Consider the confidence intervals you constructed in Exercises 25–27. Describe what happens to the lower bound and upper bound of a confidence interval for as the sample size increases but the confidence level stays the same.

8.4.31

Lower bound increases while the upper bound decreases.

Question 8.294

32. Consider the confidence intervals you constructed in Exercises 28–30. Describe what happens to the lower bound and upper bound of a confidence interval for as the sample size increases but the confidence level stays the same.

APPLYING THE CONCEPTS

Question 8.295

biomass

33. Biomass Power Plants. Power plants around the country are retooling in order to consume biomass instead of, or in addition to, coal. The table21 contains a random sample of eight such power plants and the amount of energy generated in megawatts (MW) in 2014.

image
Normal probability plot of energy capacity in megawatts.
Company Location Capacity
(MW)
Hoge Lumber Co. New Knoxville,
Ohio		 3.7
Evergreen Clean Energy Eagle, CO 12.0
GreenHunter Energy Grapevine, TX 18.5
Covanta Energy
Corporation		 Niagara Falls,
NY		 30.0
Northwest Energy Systems
Co.		 Warm Springs,
OR		 37.0
Riverstone Holdings Kenansville, NC 44.1
Lee County Solid Waste
Authority		 Ft. Myers, FL 57.0
Energy Investor Funds Detroit, MI 68.0
Dominion Virginia Power Hurt, VA 83.0
Table 8.30: Source: biomassmagazine.com/plants/listplants/biomass/US.
  1. Check whether the normality condition is met.
  2. Find the critical values and for a 95% confidence interval for .
  3. Construct and interpret a 95% confidence interval for the population variance of the MW of energy generated.
  4. Construct and interpret a 95% confidence interval for the population standard deviation of the MW of energy generated.

8.4.33

(a) Acceptable normality. (b) and

(c) (319.75, 2571.91). We are 95% confident that the population variance lies between 319.75 megawatts squared and 2571.91 megawatts squared.

(d) (17.88, 50.71). We are 95% confident that the population standard deviation lies between 17.88 megawatts and 50.71 megawatts.

Question 8.296

carbon

34. Carbon Emissions. The following table represents the carbon emissions (in millions of tons) from consumption of fossil fuels for a random sample of five nations.22

Nation Emissions
Brazil 361
Germany 844
Mexico 398
Great Britain 577
Canada 631
Table 8.31: Source: U.S. Energy Information Administration.
  1. Find the critical values and for a 95% confidence interval for .
  2. Construct and interpret a 95% confidence interval for the population variance of carbon emissions.

Question 8.297

35. Biomass Power Plants. Refer to Exercise 33.

  1. What are the units you used to interpret your confidence interval in (b)?
  2. What are the units you used to interpret your confidence interval in (c)?
  3. Which units are more easily understood by most people?

8.4.35

(a) Megawatts squared (b) Megawatts (c) Megawatts

Question 8.298

36. Carbon Emissions. Refer to Exercise 34.

  1. What are the units you used to interpret your confidence interval in (b)?
  2. Do you think that those units would be easily understood by most people?
  3. What would the units be for a confidence interval for the population standard deviation ?
  4. Construct and interpret a 95% confidence interval for .

Question 8.299

cerealcalories

37. Calories in Breakfast Cereals. A random sample of six well-known breakfast cereals yielded the following calorie data. Can we construct a confidence interval for the variance of the number of calories? Why or why not?

Cereal Calories
Apple Jacks 110
Cocoa Puffs 110
Mueslix 160
Cheerios 110
Corn Flakes 100
Shredded Wheat 80

8.4.37

No; not normally distributed

Question 8.300

deepwaterclean

38. Deepwater Horizon Cleanup Costs. The following table represents the amount of money disbursed by BP to a random sample of six Florida counties for cleanup of the Deepwater Horizon oil spill, in millions of dollars.23 The normality of the data was confirmed in the Section 8.1 exercises. Construct and interpret a 90% confidence interval for .

482

County Cleanup costs
($ millions)
Broward 0.85
Escambia 0.70
Franklin 0.50
Pinellas 1.15
Santa Rosa 0.50
Walton 1.35
Table 8.33: Source: Florida Department of Financial Services, 2011.

Question 8.301

wiisales

39. Wii Game Sales. The following table represents the number of units sold in the United States for the week ending March 26, 2011, for a random sample of eight Wii games.24 The normality of the data was confirmed in the Section 8.1 exercises. Construct and interpret a 99% confidence interval for .

Game Units
(1000s)		 Game Units
(1000s)
Wii Sports Resort 65 Zumba Fitness 56
Super Mario All Stars 40 Wii Fit Plus 36
Just Dance 2 74 Michael Jackson 42
New Super Mario
Brothers		 16 Lego Star Wars 110
Table 8.34: Source: www.vgchartz.com, April 1, 2011.

8.4.39

(16.86, 76.33). We are 99% confident that the population standard deviation lies between 16.86 thousand units and 76.33 thousand units.