Chapter 11

34. A researcher wishes to investigate the difference between the mean scores on a standardized test for students who were exposed to two different methods of teaching. How large a sample should the researcher take (equal sample size for each method) to be 99 percent certain of knowing the difference of the average scores to be within 3 points if the standard deviations for the populations are 5 and 8? (a) 66 (b) 38 (c) 27 (d) 54

A

37. When will it be reasonable to construct a confidence interval for a parameter if the values for the entire population are known? (a) Never (b) When the population size is greater than 30 (c) When the population size is less than 30 (d) When only lower confidence levels are used

A

9. The confidence interval for the population proportion p can be computed from p_hat +or - z_α/2*p_hat*(1 −p_hat)/n, where σis the population standard deviation.

F

25. In a religious survey of southerners, it was found that 164 out of 200 believed in angels. The 90 percent confidence interval for the true proportion of southerners who believe in angels (a) is 0.7753 to 0.8647. (b) is 0.7499 to 0.8901. (c) is 0.8188 to 0.8212. (d) cannot be computed since the assumptions that are required to compute this confidence interval are violated.

A

32. In a religious survey of southerners, it was found that 82 out of 100 believed in angels. If we wanted to construct a 99 percent confidence interval for the true proportion of southerners who believe in angels, what would be the margin of error? (a) 0.0991 (b) 0.9191 (c) 0.0381 (d) 0.7209

A

23. The length of time it takes a car salesperson to close a deal on a car sale is assumed to be normally distributed. A random sample of 100 such times was selected and yielded a mean of 3 h and variance of 30 min. The 98 percent confidence interval for the mean length of time it takes a car salesperson to sell a car is (a) 2.8835 to 3.1165 h. (b) 2.8176 to 3.1824 h. (c) 2.8352 to 3.1648 h. (d) 2.8710 to 3.1290 h.

C

28. The 95 percent confidence interval for the proportion of drunk drivers who are female is to be constructed and must be accurate to within 0.08. A preliminary sample provides an initial estimate of pˆ= 0.09. The smallest sample size that will provide the desired accuracy with 95 percent confidence is (a) 26. (b) 77. (c) 50. (d) 151.

C

30. In a random sample of 100 observations, pˆ= 0.1. The 84 percent confidence interval for p is (a) 0.1 +or- 0.578. (b) 0.1 +or- 0.282. (c) 0.1 +or- 0.0423. (d) 0.1 +or- 0.001.

C

13. What value of the population proportion p will maximize p(1 −p)? (a) 0.50 (b) 0.25 (c) 0.75 (d) 0.05

A

21. The most common confidence levels and the corresponding z values are listed below. Which corresponding z value is incorrect? (a) 99 percent, z value = 1.280 (b) 95 percent, z value = 1.960 (c) 98 percent, z value = 2.330 (d) 90 percent, z value = 1.645

A

3. As the sample size increases, the confidence interval for the population mean will (a) decrease. (b) increase. (c) stay the same. (d) decrease and then increase.

A

1. The best point estimate for the population mean µ is the sample mean x

T

10. A 90 percent confidence interval for a population mean implies that there is a 0.90 probability that the population mean will be contained in the confidence interval.

F

13. When repeated samples are selected from a population, the point estimate for a given parameter will always be the same value.

F

17. In order to determine the sample size when determining the population proportion, it is necessary to know the level of confidence, the margin of error, and an estimate of the population mean.

F

19. Based on the Central Limit Theorem for the difference of two population proportions, we can assume, for large enough sample sizes, that the sampling distribution for the difference between two sample proportions is exactly normally distributed.

F

21. When computing large-sample confidence intervals for the difference between two population means, it is necessary to know the variances for the two populations.

F

3. If the length of a confidence interval is very large, then the corresponding prediction is very meaningful.

F

4. The z score corresponding to a 98 percent confidence level is 1.96.

F

5. The confidence interval for the population mean µ can be computed from x =z_α/2*σ/n , where σis the population standard deviation and n is the sample size.

F

7. For a fixed confidence level, when the sample size decreases, the length of the confidence interval for a population mean decreases.

F

6. If the 98 percent confidence limits for the population mean µ are 73 and 80, which of the following could be the 95 percent confidence limits? (a) 73 and 81 (b) 72 and 79 (c) 72 and 81 (d) 74 and 79

D

9. Which of the following confidence intervals will be the widest? (a) 90 percent (b) 95 percent (c) 80 percent (d) 98 percent

D

38. A researcher wants to determine the difference between the proportions of males and females who do volunteer work. If a margin of error of 0.02 is acceptable at the 90 percent confidence level, what is the maximum sample size that should be taken? Assume equal sample sizes are selected for the two sample proportions. (a) 3,383 (b) 2,048 (c) 6,787 (d) 8,295

A

8. Interval estimates of a parameter provide information on (a) how close an estimate of the parameter is to the parameter. (b) what proportion of the estimates of the parameter are contained in the interval. (c) exactly what values the parameter can assume. (d) the z score.

A

11. When determining the sample size in constructing confidence intervals for the population mean µ, for a fixed maximum error of estimate and level of confidence, the sample size will (a) increase when the population standard deviation is decreased. (b) increase when the population standard deviation is increased. (c) decrease when the population standard deviation is increased. (d) decrease and then increase when the population standard deviation is increased.

B

14. Suppose that a sample of size 100 is selected from a population with unknown variance. If this information is used in constructing a confidence interval for the population mean, which of the following statements is true? (a) The sample must have a normal distribution. (b) The population is assumed to have a normal distribution. (c) Only 95 percent confidence intervals may be computed. (d) The sample standard deviation cannot be used to estimate the population standard deviation because the sample size is large.

B

35. In 1973, the Graduate Division at the University of California, Berkeley, did an observational study on sex bias in admissions to the graduate school. It was found that in a particular major, out of 800 male applicants, 65 percent were admitted, and out of 120 female applicants, 85 percent were admitted. Establish a 95 percent confidence interval estimate of the difference in the proportions of females and males for this particular major. (a) 0.2 +or- 0.09 (b) 0.2 +or- 0.07 (c) 0.2 +or- 0.11 (d) 0.2 +or- 0.12

B

4. If we change the confidence level from 98 percent to 95 percent when constructing a confidence interval for the population mean,we can expect the size of the interval to (a) increase. (b) decrease. (c) stay the same. (d) do none of the above.

B

1. If we are constructing a 98 percent confidence interval for the population mean, the confidence level will be (a) 2 percent. (b) 2.29. (c) 98 percent. (d) 2.39.

C

16. A 95 percent confidence interval for the population proportion is to be constructed and must be accurate to within 0.1 unit. The largest sample size n that provides the desired accuracy with 95 percent confidence (a) cannot be determined. (b) is 73. (c) is 97. (d) is 100.

C

2. The z value corresponding to a 97 percent confidence interval is (a) 1.88. (b) 2.17. (c) 1.96. (d) 3 percent.

B

15. A 95 percent confidence interval for the mean of a population is to be constructed and must be accurate to within 0.3 unit. A preliminary sample standard deviation is 2.9. The smallest sample size n that provides the desired accuracy with 95 percent confidence is (a) 253. (b) 359. (c) 400. (d) 380.

B

17. In a survey about a murder case that was widely reported by the TV networks, 201 out of 300 persons surveyed said that they believed that the accused was guilty. The 95 percent confidence interval for the proportion of people who did not believe that the accused was guilty is (a) 0.617 to 0.723. (b) 0.277 to 0.383. (c) 0.285 to 0.375. (d) 0.625 to 0.715.

B

19. Suppose the heights of the population of basketball players at a certain college are normally distributed with a standard deviation of 2 ft. If a sample of heights of size 16 is randomly selected from this population with a mean of 6.2 ft, the 90 percent confidence interval for the mean height of these basketball players is (a) 4.555 to 7.845 ft. (b) 5.378 to 7.022 ft. (c) 4.447 to 7.953 ft. (d) 5.324 to 7.077 ft.

B

22. The heights (in inches) of the students on a campus are assumed to have a normal distribution with a standard deviation of 4 in. A random sample of 49 students was taken and yielded a mean of 68 in. The 95 percent percent confidence interval for the population mean µ is (a) 67.06 to 68.94 in. (b) 66.88 to 69.12 in. (c) 63.42 to 72.48 in. (d) 64.24 to 71.76 in.

B

26. In a random sample of 150 drunk drivers, 91 percent were males. The 99 percent confidence interval for the proportion of drunk drivers who are male is (a) 0.8716 to 0.9484. (b) 0.8498 to 0.9702. (c) 0.8641 to 0.9559. (d) 0.8555 to 0.9645.

B

29. If the population proportion is being estimated, the sample size needed in order to be 90 percent confident that the estimate is within 0.05 of the true proportion is (a) 20. (b) 271. (c) 196. (d) 400.

B

33. A statistician wishes to investigate the difference between the proportions of males and the proportion of females who believe in aliens. How large a sample should be taken (equal sample size for each group) to be 95 percent certain of knowing the difference to within 0.02? (a) 3,383 (b) 4,802 (c) 2,048 (d) 6,787

B

7. A 90 percent confidence interval for a population mean indicates that (a) we are 90 percent confident that the interval will contain all possible sample means with the same sample size taken from the given population. (b) we are 90 percent confident that the population mean will be the same as the sample mean used in constructing the interval. (c) we are 90 percent confident that the population mean will fall within the interval. (d) none of the above is true.

C

10. The best point estimate for the population variance is (a) a statistic. (b) the sample standard deviation. (c) the sample mean. (d) the sample variance.

D

12. When computing the sample size to help construct confidence intervals for the population proportion, for a fixed margin of error of estimate and level of confidence, the sample size will be maximum when (a) p = 0.25. (b) (1 −p) = 0.25. (c) p(1 −p) = 0.5. (d) p = 0.5.

D

18. In constructing a confidence interval for the population mean µ, if the level of confidence is changed from 98 percent to 90 percent, the standard deviation of the mean will (a) be equal to 90 percent of the original standard deviation of the mean. (b) increase. (c) decrease. (d) remain the same.

D

20. A 99 percent confidence interval is to be constructed for a population mean from a random sample of size 22. If the population standard deviation is known, the table value to be used in the computation is (a) 2.518. (b) 2.330. (c) 2.831. (d) 2.580.

D

24. In a recent study, it was found that 11 out of every 100 Pap smears sampled were misdiagnosed by a certain lab. If a sample of 100 is taken, the 99 percent confidence interval for the proportion of misdiagnosed Pap smears is (a) 0.1075 to 0.1125. (b) 0.0371 to 0.1829. (c) 0.1077 to 0.1123. (d) 0.0293 to 0.1908.

D

27. The heights (in inches) of the students on a campus are assumed to have a normal distribution with a variance of 25 in. Suppose that we want to construct a 95 percent confidence interval for the population mean µ and have it accurate to within 0.5 in. The minimum sample size required is (a) 9,604. (b) 269. (c) 98. (d) 385.

D

31. When computing the sample size needed to estimate the population proportion p, which of the following is not necessary? (a) The required confidence level (b) The margin of error (c) An estimate of p (d) An estimate of the population variance

D

36. Two brands of similar tires were tested, and their lifetimes, in miles, were compared. The data are given below. Find the 95 percent confidence interval for the true difference in the means. Assume that the lifetimes are normally distributed. BRAND A BRAND B x_1= 41,000 x_ 2= 39,600 s_1= 3,000 s_2= 2,600 n1= 100 n2= 100 (a) 1,400 1,022.5 (b) 1,400 653 (c) 1,400 508.1 (e) 1,400 778.1

D

5. Generally, lower confidence levels will yield (a) smaller standard deviations for the sampling distribution. (b) larger margins of error. (c) broader confidence intervals. (d) narrower confidence intervals.

D

14. The larger the level of confidence, the shorter the confidence interval.

F

11. A 90 percent confidence interval for a population parameter means that if a large number of confidence intervals were constructed from repeated samples, then on average, 90 percent of these intervals would contain the true parameter.

T

12. The point estimate of a population parameter is always at the center of the confidence interval for the parameter.

T

15. The best point estimate for a population parameter is its corresponding sample statistic.

T

16. In order to determine the sample size when considering confidence intervals for the population mean µ, it is necessary to know the level of confidence, the margin of error, and either an estimate of the population standard deviation or the standard deviation itself.

T

18. The maximum error of estimate gives a measure of accuracy when computing the sample size required to make inferences.

T

2. As the length of a confidence interval increases, the degree of confidence in its actually containing the population parameter being estimated also increases.

T

20. In computing equal sample sizes when considering confidence intervals for the difference between two population proportions, the sample size will increase when the margin of error is decreased and the significance level is held fixed.

T

6. For a fixed confidence level, when the sample size increases, the length of the confidence interval for a population mean decreases.

T

8. The distribution of sample proportions is approximately normal provided that the sample size n≥30.

T