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Coefficient of skewness
2(8.5275)3 (b) From Problem 5.41, m4 Then 199.3759 m2 s2 a4 8.5275 m4 s4 199.3759 (8.5275)2 Coefficient of kurtosis
From (a) we see that the distribution is moderately skewed to the left. From (b) we see that it is slightly less peaked than the normal distribution (which has coefficient of kurtosis 3). Miscellaneous problems 5.43. (a) Show how to select 30 random samples of 4 students each (with replacement) from the table of heights on page 161 by using random numbers, (b) Find the mean and standard deviation of the sampling distribution of means in (a). (c) Compare the results of (b) with theoretical values, explaining any discrepancies. (a) Use two digits to number each of the 100 students: 00, 01, 02, . . . , 99 (see Table 513). Therefore, the 5 students with heights 60 62 inches are numbered 00 04, the 18 students with heights 63 65 inches are numbered 05 22, etc. Each student number is called a sampling number. CHAPTER 5 Sampling Theory
Table 513 Height (inches) 60 62 63 65 66 68 69 71 72 74 Frequency 5 18 42 27 8 Sampling Number 00 04 05 22 23 64 65 91 92 99 We now draw sampling numbers from the random number table (Appendix H). From the first line we find the sequence 51, 77, 27, 46, 40, etc., which we take as random sampling numbers, each of which yields the height of a particular student. Therefore, 51 corresponds to a student having height 66 68 inches, which we take as 67 inches (the class mark). Similarly 77, 27, 46 yield heights of 70, 67, 67 inches, respectively. By this process we obtain Table 514, which shows the sampling numbers drawn, the corresponding heights, and the mean height for each of 30 samples. It should be mentioned that although we have entered the random number table on the first line, we could have started anywhere and chosen any specified pattern. Table 514 Sampling Numbers Drawn 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 51, 77, 27, 46 40, 42, 33, 12 90, 44, 46, 62 16, 28, 98, 93 58, 20, 41, 86 19, 64, 08, 70 56, 24, 03, 32 34, 91, 83, 58 70, 65, 68, 21 96, 02, 13, 87 76, 10, 51, 08 63, 97, 45, 39 05, 81, 45, 93 96, 01, 73, 52 07, 82, 54, 24 Corresponding Heights 67, 70, 67, 67 67, 67, 67, 64 70, 67, 67, 67 64, 67, 73, 73 67, 64, 67, 70 64, 67, 64, 70 67, 67, 61, 67 67, 70, 70, 67 70, 70, 70, 64 73, 61, 64, 70 70, 64, 67, 64 67, 73, 67, 67 64, 70, 67, 73 73, 61, 70, 67 64, 70, 67, 67 Mean Height 67.75 66.25 67.75 69.25 67.00 66.25 65.50 68.50 68.50 67.00 66.25 68.50 68.50 67.75 67.00 Sampling Numbers Drawn 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 11, 64, 55, 58 70, 56, 97, 43 74, 28, 93, 50 79, 42, 71, 30 58, 60, 21, 33 75, 79, 74, 54 06, 31, 04, 18 67, 07, 12, 97 31, 71, 69, 88 11, 64, 21, 87 03, 58, 57, 93 53, 81, 93, 88 23, 22, 96, 79 98, 56, 59, 36 08, 15, 08, 84 Corresponding Heights 64, 67, 67, 67 70, 67, 73, 67 70, 67, 73, 67 70, 67, 70, 67 67, 67, 64, 67 70, 70, 70, 67 64, 67, 61, 64 70, 64, 64, 73 67, 70, 70, 70 64, 67, 64, 70 61, 67, 67, 73 67, 70, 73, 70 67, 64, 73, 70 73, 67, 67, 67 64, 64, 64, 70 Mean Height 66.25 69.25 69.25 68.50 66.25 69.25 64.00 67.75 69.25 66.25 67.00 70.00 68.50 68.50 65.50 (b) Table 515 gives the frequency distribution of sample mean heights obtained in (a). This is a sampling distribution of means. The mean and the standard deviation are obtained as usual by the coding methods already described. Mean a cu # a c a fu n c2u2 # u2 # 67.00 c (0.75)(23) 30 67.58 inches

