Let's step through the solution:
a. Expected value of [tex]\(\bar{p}\)[/tex]:
The expected value of the sample proportion [tex]\(\bar{p}\)[/tex] is simply the population proportion [tex]\(p\)[/tex]. For a given population proportion [tex]\(p = 0.40\)[/tex]:
[tex]\[ E(\bar{p}) = p = 0.40 \][/tex]
Thus, the expected value of [tex]\(\bar{p}\)[/tex] is [tex]\(0.40\)[/tex].
b. Standard error of [tex]\(\bar{p}\)[/tex]:
The standard error of the sample proportion [tex]\(\bar{p}\)[/tex] measures the variability of [tex]\(\bar{p}\)[/tex] around the population proportion [tex]\(p\)[/tex]. It can be found using the formula:
[tex]\[ \sigma_{\bar{p}} = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
Given:
- [tex]\(p = 0.40\)[/tex]
- [tex]\(n = 100\)[/tex]
From the numerical result, the standard error is:
[tex]\[ \sigma_{\bar{p}} = 0.05 \][/tex]
Thus, the standard error of [tex]\(\bar{p}\)[/tex] is [tex]\(0.05\)[/tex].
c. Sampling distribution of [tex]\(\bar{p}\)[/tex]:
The sampling distribution of [tex]\(\bar{p}\)[/tex] is normally distributed with parameters given by the expected value (mean) and the standard error. It can be described by:
[tex]\[
\begin{array}{l}
\sigma_{\bar{p}} = 0.05 \\
E(\bar{p}) = 0.40
\end{array}
\][/tex]
d. What does the sampling distribution of [tex]\(\bar{p}\)[/tex] show?
The sampling distribution of [tex]\(\bar{p}\)[/tex] provides a probability distribution of the sample proportion around the population proportion [tex]\(p\)[/tex]. Given that the sample size is sufficiently large (n = 100), the Central Limit Theorem assures us that the distribution of [tex]\(\bar{p}\)[/tex] will be approximately normal.
- The mean of this distribution (expected value of [tex]\(\bar{p}\)[/tex]) is equal to the population proportion [tex]\(p = 0.40\)[/tex].
- The standard error (standard deviation of [tex]\(\bar{p}\)[/tex]) measures the extent to which the sample proportion [tex]\(\bar{p}\)[/tex] is expected to deviate from the true population proportion [tex]\(p\)[/tex].
In summary, this sampling distribution tells us that if we repeatedly took samples of size 100 from the population, the sample proportions [tex]\(\bar{p}\)[/tex] would on average be 0.40, with most of the sample proportions falling within roughly [tex]\(0.40 \pm 0.05\)[/tex].
