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1. Measurements are collected from 100 subjects from each of two sources. The data yield the following frequency histograms. The number above each rectangle is its height. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Source 1 1 1 2 3 4 5 7 8 9 10 11 2 3 4 5 7 8 9 10 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Source 2 1 5 13 21 16 8 1 4 5 6 8 Each sample has the same mean, 10.00. In order to answer (b) and (c) below, refer to the empirical rule for interpreting s, taking into account the shape of the histogram. Do not try to calculate s because you do not have enough information to do so. In addition, you will receive no credit for simply identifying the correct s; you must provide an explanation.

(a) What is the most precise correct statement that you can make about the numerical value of the median of the data from source 2? Do not explain your answer. Hint: Here is a correct statement: The median is between 0 and 20. This statement is not precise enough to receive any credit.

(b) Among the possibilities 1.50, 2.00 and 2.50, which is the numerical value of s for the data from source 1? Explain your answer.

(c) Among the possibilities 1.00, 1.50 and 2.00, which is the numerical value of s for the data from source 2? Explain your answer.

2. The mean and median of Al's n = 3 observations both equal 10. The mean and median of Bev's n = 5 observations both equal 18.

(a) Carol combines Al's and Bev's data into one collection of n = 8 observations. Can the mean of Carol's data be calculated from the information given? If you think not, just say that. If you think it can, then calculate Carol's mean.

(b) Refer to part (a). Demonstrate, by an explicit example, that there is not enough information to determine Carol's median. Hint: Find two sets of data sets that satisfy Al's and Bev's conditions, yet, when combined, give different medians.

3. A sample of size 40 yields the following sorted data. Note that I have x-ed out x(39) (the second largest number). This fact will NOT prevent you from answering the questions below. 14.1 46.0 49.3 53.0 54.2 54.7 54.7 54.7 54.8 55.4 57.6 58.2 58.3 58.7 58.9 60.8 60.9 61.0 61.1 63.0 64.3 65.6 66.3 66.6 67.0 67.9 70.1 70.3 72.1 72.4 72.9 73.5 74.2 75.3 75.4 75.9 76.5 77.0 x 88.9

(a) Calculate range, IQR, and median of these data.

(b) Given that the mean of these data is 63.50 (exactly) and the standard deviation is 12.33, what proportion of the data lie within one standard deviation of the mean?

(c) How does your answer to (b) compare to the empirical rule approximation?

(d) Ralph decides to delete the smallest observation, 14.1, from these data. Thus, Ralph has a data set with n = 39. Calculate the range, IQR, and median of Ralph's new data set.

(e) Refer to (d). Calculate the mean of Ralph's new data set.

6. Consider a balanced study with six subjects, identified as A, B, C, D, E and G. In the actual study, Subjects A, B and C are assigned to the first treatment, and the other subjects are assigned to the second treatment. There are exactly four successes, obtained by A, D, E and G. This information is needed for parts (a)-(c) below.

(a) Compute the observed value of the test statistic.

(b) Assume that the Skeptic is correct. Determine the observed value of the test statistic for the assignment that places C, D and E on the first treatment, and the remaining subjects on the second treatment.

(c) We have obtained the sampling distribution of the test statistic on the assumption that the Skeptic is correct. It also is possible to obtain a sampling distribution of the test statistic if the Skeptic is wrong provided we specify exactly how the Skeptic is in error. These new sampling distributions are used in the study of statistical power which is briefly described in Chapter 7 of the text. Assume that the Skeptic is correct about subjects C, D and E, but incorrect about subjects A, B and G. For the assignment that puts D, E and G on the first treatment, and the other subjects on the second treatment, determine the response for each of the six subjects.

7. Consider an unbalanced study with six subjects, identified as A, B, C, D, E and G. In the actual study, Subjects A and B are assigned to the first treatment, and the other subjects are assigned to the second treatment. There are exactly two successes, obtained by A and C. This information is needed for parts (a)-(c) below.

(a) Compute the observed value of the test statistic.

(b) Assume that the Skeptic is correct. Determine the observed value of the test statistic for the assignment that places D and E on the first treatment, and the remaining subjects on the second treatment.

(c) We have obtained the sampling distribution of the test statistic on the assumption that the Skeptic is correct. It also is possible to obtain a sampling distribution of the test statistic if the Skeptic is wrong provided we specify exactly how the Skeptic is in error. These new sampling distributions are used in the study of statistical power which is briefly described in Chapter 7 of the text. Assume that the Skeptic is correct about subjects A and G, but incorrect about subjects B, C, D and E. For the assignment that puts D and G on the first treatment, and the other subjects on the second treatment, determine the response for each of the six subjects.