Question

the sampling distribution of the sample proportion describes how the sample proportion will vary from sample to sample. the standard deviation of the sampling distribution of the sample proportion gives us a way to measure sample - to - sample variability.
consider sampling from a binomial model with $p = 0.44$ and sample size $n$. find the mean and standard deviation of the sampling distribution for the sample proportion, $hat{p}$.
a. for a random sample of size $n = 100$:
the sample proportion has a mean of $p=$ and a standard deviation of
$sd=sqrt{(\frac{pcdot(1 - p)}{n})}=$ (round to 4 decimal places)
b. for a random sample of size $n = 500$:
the sample proportion has a mean of $p=$ and a standard deviation of $sd=$ (round to 4 decimal places)
c. as the sample size increases, the standard deviation of the sample proportion (decreases increases).

Explanation:

Step1: Recall mean formula

The mean of the sampling - distribution of the sample proportion $\hat{p}$ is $\mu_{\hat{p}}=p$. Given $p = 0.44$, the mean remains $p$ for all sample sizes.

Step2: Calculate standard deviation for $n = 100$

We use the formula $SD=\sqrt{\frac{p(1 - p)}{n}}$. Substitute $p = 0.44$, $1-p=1 - 0.44 = 0.56$ and $n = 100$ into the formula:

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Step3: Calculate standard deviation for $n = 500$

Substitute $p = 0.44$, $1 - p=0.56$ and $n = 500$ into the formula $SD=\sqrt{\frac{p(1 - p)}{n}}$:

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Step4: Analyze the effect of sample - size on standard deviation

As the denominator $n$ in the formula $SD=\sqrt{\frac{p(1 - p)}{n}}$ increases, the value of $SD$ decreases since $p(1 - p)$ is constant.

Answer:

a. Mean: $0.44$, Standard deviation: $0.0496$
b. Mean: $0.44$, Standard deviation: $0.0222$
c. decreases