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A professional tennis player has a serve-return rate of [tex]$p=0.71$[/tex]. A random sample of 55 serve returns is selected. Which of the following is the mean of the sampling distribution of [tex]\hat{p}[/tex]?

A. [tex]\mu_{\hat{p}} = p = 0.71[/tex]

B. [tex]u_{\hat{p}} = n p = 55(0.71) = 39.05[/tex]

C. [tex]\mu_{\widehat{p}} = 1 - p = 1 - 0.71 = 0.29[/tex]

D. [tex]\mu_{\hat{p}} = n(1 - p) = 55(1 - 0.71) = 15.95[/tex]

Sagot :

GinnyAnsweridn
To find the mean of the sampling distribution of [tex]\(\hat{p}\)[/tex], it is important to understand the properties of the sampling distribution of a sample proportion.

Given:
- [tex]\( p \)[/tex] (the population proportion) = 0.71
- [tex]\( n \)[/tex] (the sample size) = 55

The mean of the sampling distribution of [tex]\(\hat{p}\)[/tex] is actually just the population proportion [tex]\( p \)[/tex]. This is because [tex]\(\hat{p}\)[/tex] is an unbiased estimator of [tex]\( p \)[/tex]. The symbol [tex]\(\hat{p}\)[/tex] typically denotes the sample proportion, and because we are dealing with a proportion, the mean of the sampling distribution (expected value) is just the population proportion [tex]\( p \)[/tex].

Therefore, the correct interpretation for the mean of the sampling distribution of [tex]\(\hat{p}\)[/tex] is:

[tex]\[ \mu_{\hat{p}} = p = 0.71 \][/tex]

So, the correct answer is:

[tex]\[ \mu_{\hat{p}} = p = 0.71 \][/tex]
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