Section 7.4 The Central Limit Theorem for Proportions 319 of heads is p = 0.5. There are 32 possible samples. Table 7.2 lists them and presents the sample proportion ̂p of heads for each. Table 7.2 The 32 Possible Samples of Size 5 and Their Sample Proportions of Heads Sample ̂p Sample ̂p Sample ̂p Sample ̂p TTTTT 0.0 THTTT 0.2 HTTTT 0.2 HHTTT 0.4 TTTTH 0.2 THTTH 0.4 HTTTH 0.4 HHTTH 0.6 TTTHT 0.2 THTHT 0.4 HTTHT 0.4 HHTHT 0.6 TTTHH 0.4 THTHH 0.6 HTTHH 0.6 HHTHH 0.8 TTHTT 0.2 THHTT 0.4 HTHTT 0.4 HHHTT 0.6 TTHTH 0.4 THHTH 0.6 HTHTH 0.6 HHHTH 0.8 TTHHT 0.4 THHHT 0.6 HTHHT 0.6 HHHHT 0.8 TTHHH 0.6 THHHH 0.8 HTHHH 0.8 HHHHH 1.0 The columns labeled ‘‘ ̂p’’ contain the values of the sample proportion for each of the 32 possible samples. Some of these values appear more than once, because several samples have the same proportion. The mean of the sampling distribution is the average of these 32 values. The standard deviation of the sampling distribution is the population standard deviation of these 32 values, which can be computed by the method presented in Section 3.2. The mean and standard deviation are Mean: ��̂p = 0.5 Standard deviation: ��̂p = 0.2236 The values of ��̂p and ��̂p are related to the values of the population proportion p = 0.5 and the sample size n = 5. Specifically, ��̂p = 0.5 = p The mean of the sample proportion is equal to the population proportion. The relationship among ��̂p, p, and n is less obvious. However, note that ��̂p = 0.2236 = √ 0.5(1 − 0.5) 5 = √ p(1 − p) n These relationships hold in general. SUMMARY Let ̂p be the sample proportion of a simple random sample of size n, drawn from a population with population proportion p. The mean and standard deviation of the sampling distribution of ̂p are ��̂p = p ��̂p = √ p(1 − p) n EXAMPLE 7.22 Find the mean and standard deviation of a sampling distribution The soft-drink cups at a certain fast-food restaurant have tickets attached to them. Customers peel off the tickets to see whether they win a prize. The proportion of tickets that are winners is p = 0.25. A total of n = 70 people purchase soft drinks between noon and 1:00 P.M. on a certain day. Let ̂p be the proportion that win a prize. Find the mean and standard deviation of ̂p.
navidi_monk_elementary_statistics_2e_ch7-9
To see the actual publication please follow the link above