URGENT

38. Recall that a confidence interval is too small if the number being estimated is larger than every number in the confidence interval. Similarly, a confidence interval is too large if the number being estimated is smaller than every number in the confidence interval. Each of four researchers selects a random sample from the same population. Each researcher calculates a confidence interval for the median of the population. The intervals are below. [14, 31], [20, 29], [10, 23], and [25, 35].

(a) Nature announces, "Two of the intervals are correct and two are too large." Given this information, what is the narrowest interval that is known to contain the median? (Hint: The answer is not any of the four confidence intervals.)

(b) Nature announces, "Two of the intervals are correct, one interval is too small and one interval is too large." Given this information, what is the narrowest interval that is known to contain the median? (Hint: The answer is not any of the four confidence intervals.)

(c) It is possible all of these intervals are incorrect. For example, if = 100 then every interval is incorrect. But what is the maximum number of these intervals that can be correct? What values of will give this maximum number of correct intervals? (Hint: The answer is not any of the four confidence intervals.)

39. Homer performs three simulation studies. His population is skewed to the right. For one study he has his computer generate 10,000 random samples of size n = 10 from the population. For each random sample, the computer calculates the Gosset 95% confidence interval for and checks to see whether the interval is correct. His second study is like his first, but n = 100. Finally, his third study is like the first, but n = 200. In one of his studies, Homer obtains 9,504 correct intervals; in another he obtains 9,478 correct intervals; and in the remaining study he obtains 8,688 correct intervals. Based on what we learned in class, match each sample size to its number of correct intervals. Explain your answer.

40. Independent random samples are selected from two populations. Below are the sorted data from the first population.

362 373 399 428 476 481 545 564 585 589 590 600 671 694 723 724 904

Hint: The mean and standard deviation of these numbers are 571.1 and 144.7. Below are the sorted data from the second population. 387 530 544 547 646 766 786 864

Hint: The mean and standard deviation of these numbers are 633.8 and 160.8.

(a) Calculate Gosset's 90% confidence interval for the mean of the first population.

(b) Calculate a confidence interval for the median of the second population. Select your confidence level and report it with your answer.

(c) Suppose that we now learn that the two samples came from the same population. Thus, the two samples can be combined into one random sample from the one population. Use this combined sample to obtain the 95% confidence interval for the median of the population.

41. Independent random samples are selected from two populations. Below are the sorted data from the first population.

53.2 54.2 54.7 55.3 55.9 56.0 56.3 57.0 58.2 58.5 58.7 61.0 62.5 62.8 64.4 66.3 67.0 69.0

Hint: The mean and standard deviation of these numbers are 59.50 and 4.80. Below are the sorted data from the second population. 49.2 53.8 56.9 57.8 58.1 58.4 62.0 65.4 69.4 Hint: The mean and standard deviation of these numbers are 59.00 and 6.00.

(a) Calculate Gosset's 90% confidence interval for the mean of the first population.

(b) Calculate a confidence interval for the median of the second population. Select your confidence level and report it with your answer.

(c) Suppose that we now learn that the two samples came from the same population. Thus, the two samples can be combined into one random sample from the one population. Use this combined sample to obtain the 95% confidence interval for the median of the population.

42. Independent random samples are selected from two populations. Below are selected summary statistics. Pop.

Mean Stand. Dev. Sample size 1 62.00 10.00 17 2 54.00 6.00 10

(a) Construct the 95% confidence interval for X Y . (

b) Obtain the P-value for the alternative X 6= Y . Show your work.

43. Independent random samples are selected from two populations. Below are selected summary statistics. Pop. Mean Stand. Dev. Sample size 1 73.00 10.00 14 2 62.50 6.00 8 (a) Construct the 95% confidence interval for X Y . (b) Obtain the P-value for the alternative X 6= Y