Question

Describe the sampling distribution of p(hat). Assume the size of the population is 30,000.
n=800, p=0.6
a) Determine the mean of the sampling distribution
b) Dtermine the standard deviation of the sampling distribution

Answer 1

Answer:

a) $\mathbf{\mu_ \hat p = 0.6}$

b) $\mathbf{\sigma_p =0.01732}$

Step-by-step explanation:

Given that:

population mean $\mu$ = 30,000

sample size n = 800

population proportion p = 0.6

a)

The mean of the the sampling distribution is equal to the population proportion.

$\mu_ \hat p = p$

$\mathbf{\mu_ \hat p = 0.6}$

b)

The standard deviation of the sampling distribution can be estimated by using the formula:

$\sigma_p = \sqrt{\dfrac{p(1-p)}{n}}$

$\sigma_p = \sqrt{\dfrac{0.6(1-0.6)}{800}}$

$\sigma_p = \sqrt{\dfrac{0.6(0.4)}{800}}$

$\sigma_p = \sqrt{\dfrac{0.24}{800}}$

$\sigma_p = \sqrt{3 \times 10^{-4}}$

$\mathbf{\sigma_p =0.01732}$

Answer 2

The mean of a distribution is the average of the distribution.

• The mean is 480
• The standard deviation is 13.86

The given parameters are:

$\mathbf{n = 800}$

$\mathbf{p = 0.6}$

(a) The mean

This is calculated as:

$\mathbf{\bar x = np}$

So, we have:

$\mathbf{\bar x = 800 \times 0.6}$

$\mathbf{\bar x = 480}$

Hence, the mean is 480

(b) The standard deviation

This is calculated as:

$\mathbf{\sigma = \sqrt{np(1 - p)}}$

So, we have:

$\mathbf{\sigma = \sqrt{480 \times (1 - 0.6)}}$

$\mathbf{\sigma = \sqrt{480 \times 0.4}}$

$\mathbf{\sigma = \sqrt{192}}$

Take square roots

$\mathbf{\sigma = 13.86}$

Hence, the standard deviation is 13.86

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