8.15. For the population in Table 8.1, and for samples with N = 8, what proportion of the sample means will be 8.5 or less? 8.16. For the population in Table 8.1, and for samples with / = 16, what proportion of the sample means will be 8.5 or less? 10 or greater? TABLE 8.1 20 scores used as a population; u = 9, o = 2 11 13 10 10 10 12 9 10 5 9 10 One method of getting a random sample is to write each number in the population on a slip of paper, put the 20 slips in a box, jumble them around, and draw out 8 slips. The numbers on the slips are a random sample. This method works fine if the slips are all the same size, they are jumbled thoroughly, and the population has only a few members. If the population is large, this method is tedious. A second (usually easier) method of selecting a random sample is to use a table of random numbers, such as Table B in Appendix C. To use the table, you must first assign an identifying number to each of the scores in the population. My version is Table 8.2. The population scores do not have to be arranged in any order. Each score in the population is identified by a two-digit number from 01 to 20. Now turn to Appendix C, Table B. Pick an intersection of a row and a column. Any haphazard method will work; close your eyes and put your finger on a spot. Suppose you found yourself at row 80, columns 15-19 ([lage 388). Find that place. Reading horizontally, the digits are 82279. You need only two digits to identify any member of your population, so you might as well use the first two (columns 15 and 16), which give you 8 and 2 (82). Unfortunately, 82 is larger than any of the identifying numbers, so it doesn't match a score in the population, but at least you are started. From this point you can read two-digit numbers in any direction-up, down, or sideways-but the decision should be made before you look at the numbers. If you decide to read down, you find 04. The identifying number 04 corresponds to a score of 13 in Table 8.2, so 13 becomes the first number in the sample. The next identifying number is 34, which again does not correspond to a population score. Indeed, the next ten numbers are too large. The next usable ID number is 16, which places an 8 in the sample. Continuing the search, you reach the bottom of the table. At this point you can go in any direction; I moved to the right and started back up the two outside columns (18 and 19). The first number, 83, was too large, but the next identifying number, 06, corresponded to an 8, which went into the sample. 8.23. A social worker conducted an 8-week assertiveness training workshop. Afterward, the 14 clients took the Door Manifest Assertiveness Test, which has a national mean of 24.0. Use the data that follow to construct a 95 percent confidence interval about the sample mean. Write an interpretation about the effectiveness of the workshop. 24 25 31 25 33 29 21 22 23 32 27 29 30 27 8.25. This problem will give you practice in constructing confidence intervals and in discovering how to reduce their size (without sacrificing confidence). Smaller confidence intervals tell you the value of / more precisely. a. How wide is the 95 percent confidence interval about a sample mean when s = 4 and N = 4? b. How much smaller is the 95 percent confidence interval about a sample mean when A is increased fourfold to 16 (s remains 4)? c. Compared to your answer in part a, how much smaller is the 95 percent confidence interval about a sample mean when / remains 4 and s is reduced to 2