CHAPTER 5 Sampling Theory

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5.100. Compute the mean for the data in Table 5-21.

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Table 5-21

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Class 10 under 15 15 under 20 20 under 25 25 under 30 30 under 35 35 under 40 40 under 45 TOTAL Frequency 3 7 16 12 9 5 2 54

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5.101. Find the standard deviation of the numbers: (a) 3, 6, 2, 1, 7, 5; (b) 3.2, 4.6, 2.8, 5.2, 4.4; (c) 0, 0, 0, 0, 0, 1, 1, 1. 5.102. (a) By adding 5 to each of the numbers in the set 3, 6, 2, 1, 7, 5, we obtain the set 8, 11, 7, 6, 12, 10. Show that the two sets have the same standard deviation but different means. How are the means related (b) By multiplying each of the numbers 3, 6, 2, 1, 7, 5 by 2 and then adding 5, we obtain the set 11, 17, 9, 7, 19, 15. What is the relationship between the standard deviations and between the means for the two sets (c) What properties of the mean and standard deviation are illustrated by the particular sets of numbers in

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(a) and (b)

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5.103. Find the standard deviation of the set of numbers in the arithmetic progression 4, 10, 16, 22, . . . , 154. 5.104. Find the standard deviations for the distributions of: (a) Problem 5-97, (b) Problem 5.98. 5.105. Find (a) the mean, (b) the standard deviation for the distribution of Problem 5.30, explaining the significance of the results obtained. 5.106. (a) Find the standard deviation s of the rivet diameters in Problem 5.99 (b) What percentage of rivet diameters lie in (x s), (x 2s), (x 3s) (c) Compare the percentages in (b) with those that would theoretically be # # # expected if the distribution were normal, and account for any observed differences. 5.107. (a) Find the mean and standard deviation for the data of Problem 5.28. (b) Construct a frequency distribution for the data, and find the standard deviation. (c) Compare the result of (b) with that of (a). 5.108. Work Problem 5.107 for the data of Problem 5.87. 5.109. (a) Of a total of n numbers, the fraction p are ones while the fraction q 1 p are zeros. Prove that the standard deviation of the set of numbers is !pq. (b) Apply the result of (a) to Problem 5.101(c). 5.110. Find the (a) first, (b) second, (c) third, (d) fourth moment about the origin for the set of numbers 4, 7, 5, 9, 8, 3, 6. 5.111. Find the (a) first, (b) second, (c) third, (d) fourth moment about the mean for the set of numbers in Problem 5.110.

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CHAPTER 5 Sampling Theory

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5.112. Find the (a) first, (b) second, (c) third, (d) fourth moment about the number 7 for the set of numbers in Problem 5.110. 5.113. Using the results of Problems 5.110 and 5.111, verify the following relations between the moments: (a) m2 mr2 mr2, (b) m3 mr3 3mr1 mr2 2mr13, (c) m4 mr4 4mr1 mr3 6mr12mr2 3mr14. 1 5.114. Find the first four moments about the mean of the set of numbers in the arithmetic progression 2, 5, 8, 11, 14, 17. 5.115. If the first moment about the number 2 is equal to 5, what is the mean 5.116. If the first four moments of a set of numbers about the number 3 are equal to 2, 10, 25, and 50, determine the corresponding moments (a) about the mean, (b) about the number 5, (c) about zero. 5.117. Find the first four moments about the mean of the numbers 0, 0, 0, 1, 1, 1, 1, 1. 5.118. (a) Prove that m5 mr5 5mr1 mr4 10mr12mr3 10mr13mr2 4mr15. (b) Derive a similar formula for m6. 1 p are zeros. Find (a) m1, (b) m2,

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5.119. Of a total of n numbers, the fraction p are ones while the fraction q (c) m3, (d) m4 for the set of numbers. Compare with Problem 5.117.

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5.120. Calculate the first four moments about the mean for the distribution of Table 5-22.

Table 5-22