Chapter 8 Review Exercises

483

section 8.1

For Exercises 1 and 2, answer the following questions:

  1. Calculate .
  2. Find for a confidence interval for with 95% confidence.
  3. Compute and interpret , the margin of error for a confidence interval with 95% confidence.
  4. Construct and interpret a 95% confidence interval for .

Question 8.302

1. A sample of with sample mean is drawn from a normal population in which .

8.99.1

(a) 5 (b) (c) . We can estimate the population mean to within 9.8 with 95% confidence.

(d) (90.2, 109.8). We are 95% confident that the population mean lies between 90.2 and 109.8.

Question 8.303

2. A sample of with sample mean is drawn from a population in which .

Question 8.304

3. The Mozart Effect. A random sample of 45 children showed a mean increase of 7 IQ points after listening to a Mozart piano sonata for about 10 minutes. The distribution of such increases is unknown, but the standard deviation is assumed to be 2 IQ points.

  1. Find the point estimate of the increase in IQ points for all children after they have listened to Mozart.
  2. Calculate
  3. Find for a confidence interval with 90% confidence.
  4. Compute and interpret the margin of error for a confidence interval with 90% confidence.
  5. Construct and interpret a 90% confidence interval for the mean increase in IQ points for all children after listening to a Mozart piano sonata for about 10 minutes.

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(a) 7 points (b) 0.2981 point (c) 1.645 (d) 0.4904 point. We can estimate to within 0.4904 point with 90% confidence. (e) (6.5096, 7.4904). We are 90% confident that the true mean increase in IQ points for all children after listening to a Mozart piano sonata for about 10 minutes lies between 6.5106 points and 7.4904 points.

Suppose we are estimating . For Exercises 4–6, find the required sample size.

Question 8.305

4. , confidence level 95%, margin of error 5

Question 8.306

5. , confidence level 95%, margin of error 5

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139

Question 8.307

6. , confidence level 95%, margin of error 5

Question 8.308

7. Clinical Psychology. A clinical psychologist wants to estimate the population mean number of episodes her patients have suffered in the past year. Assume that the standard deviation is 10 episodes. How many patients will she have to examine if she wants her estimate to be within two episodes with 90% confidence?

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68

section 8.2

For Exercises 8–10, construct the indicated confidence interval if appropriate. If it is not appropriate, explain why not.

Question 8.309

8. Confidence level 90%, , , , non-normal population

Question 8.310

9. Confidence level 90%, , , , normal population

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(40.3, 43.7)

Question 8.311

10. Confidence level 90%, , , , non-normal population

Question 8.312

biomass

11. Biomass Power Plants. The table25 contains a random sample of eight biomass-fueled power plants and the amount of energy generated (capacity) in megawatts (MW) in 2014. Construct and interpret a 95% confidence interval for the mean capacity.

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 8.35: Source: biomassmagazine.com/plants/listplants/biomass/US.

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(18.91, 59.61)

section 8.3

For Exercises 12 and 13, follow steps (a)–(d).

  1. Find .
  2. Determine whether the conditions are met.
  3. Calculate and interpret the margin of error,
  4. Construct a confidence interval for with the indicated confidence level, and sketch the confidence interval on the number line.

Question 8.313

12. Confidence level 95%, ,

Question 8.314

13. Confidence level 95%, ,

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(a) (b) is and is (c) . We can estimate the population proportion to within 0.0087 with 95% confidence. (d) (0.0013, 0.0187). We are 95% confident that the population proportion lies between 0.0013 and 0.0187.

Question 8.315

14. Ecstasy and Emergency Room Visits. According to the National Institute on Drug Abuse (www.drugabuse.gov), 77% of the emergency room patients who mentioned MDMA (Ecstasy) as a factor in their admission were age 25 and under. Assume that the sample size is 200.

  1. Calculate and interpret the margin of error for confidence level 95%.
  2. Construct and interpret a 95% confidence interval for the population proportion of all emergency room patients mentioning MDMA (Ecstasy) as a factor in their admission who are age 25 and under.

484

For Exercises 15–17, we are estimating and we know the value of . Find the required sample size.

Question 8.316

15. Confidence level 99%, margin of error 0.05,

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239

Question 8.317

16. Confidence level 95%, margin of error 0.05,

Question 8.318

17. Confidence level 95%, margin of error 0.05,

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2

For Exercises 18–20, we are estimating and we do not know the value of . Find the required sample size.

Question 8.319

18. Confidence level 95%, margin of error 0.06

Question 8.320

19. Confidence level 95%, margin of error 0.05

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385

Question 8.321

20. Confidence level 95%, margin of error 0.04

section 8.4

For Exercises 21–24, a random sample is drawn from a normal population. The sample of size has a sample variance of . Construct the specified confidence interval.

Question 8.322

21. 90% confidence interval for the population variance

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(224.00, 366.63)

Question 8.323

22. 95% confidence interval for the population variance

Question 8.324

23. 90% confidence interval for the population standard deviation

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(14.97, 19.15)

Question 8.325

24. 95% confidence interval for the population standard deviation

Question 8.326

unionmember

25. Union Membership. The table contains the total union membership for seven randomly selected states. Construct and interpret a 95% confidence interval for . Assume the data are normally distributed.

State Union membership
(1000s)
Florida 397
Indiana 334
Maryland 342
Massachusetts 414
Minnesota 395
Texas 476
Wisconsin 386
Table 8.36: Source: U.S. Bureau of Labor Statistics, 2010.

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Lower bound = 30.537, upper bound = 104.367. We are 95% confident that σ, the population standard deviation of total union membership per state, lies between 30.537 and 104.367 thousand.