Sampling Distribution of the Sample Proportion — Quiz
Question 9 • Math 4
A professional tennis player has a serve-return rate of p is equal to 0 point 7 1 A random sample of 55 serve returns is selected. Which of the following is the correct calculation and interpretation of the standard deviation of the sampling distribution of hat p ?
Answer
A
sigma sub hat p is equal to 0 point 0 0 4 . In SRSs of size 55 , the sample proportion of this tennis player’s serve-return rate typically varies 0 point 0 0 4 from the true proportion p is equal to 0 point 7 1 .
B
sigma sub hat p is equal to 0 point 1 1 4 . In SRSs of size 55 , the sample proportion of this tennis player’s serve-return rate typically varies 0 point 1 1 4 from the true proportion p is equal to 0 point 7 1 .
C
sigma sub hat p is equal to 0 point 0 6 1 . In SRSs of size 55 , the sample proportion of this tennis player’s serve-return rate typically varies 0 point 0 6 1 from the true proportion p is equal to 0 point 7 1 .
D
sigma sub hat p is equal to 0 point 0 1 3 . In SRSs of size 55 , the sample proportion of this tennis player’s serve-return rate typically varies 0 point 0 1 3 from the true proportion p is equal to 0 point 7 1 .
Explanation
The standard deviation of the sampling distribution of a sample proportion is σ̂p = √[p(1 − p)/n]. Substituting p = 0.71 and n = 55 gives √[(0.71)(0.29)/55] ≈ 0.061. So the sample proportion typically varies about 0.061 from the true proportion of 0.71; this is choice C.
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