Questions: There is a very popular lottery in which a ticket is called a scratcher. In this lottery, 41% of the scratchers are winning ones. Suppose that we will take a random sample of 7 scratchers. Let p̂ represent the proportion of winning scratchers from the sample. Consider the sampling distribution of the sample proportion p̂. Complete the following. Carry your intermediate computations to four or more decimal places. Write your answers with two decimal places, rounding if needed. (a) Find μp (the mean of the sampling distribution of the sample proportion). μp̂= (b) Find σp̂ (the standard deviation of the sampling distribution of the sample proportion). σp̂=

There is a very popular lottery in which a ticket is called a scratcher. In this lottery, 41% of the scratchers are winning ones. Suppose that we will take a random sample of 7 scratchers. Let p̂ represent the proportion of winning scratchers from the sample. Consider the sampling distribution of the sample proportion p̂.

Complete the following. Carry your intermediate computations to four or more decimal places. Write your answers with two decimal places, rounding if needed. (a) Find μp (the mean of the sampling distribution of the sample proportion). μp̂= (b) Find σp̂ (the standard deviation of the sampling distribution of the sample proportion). σp̂=

Transcript text: There is a very popular lottery in which a ticket is called a scratcher. In this lottery, $41 \%$ of the scratchers are winning ones. Suppose that we will take a random sample of 7 scratchers. Let $\hat{p}$ represent the proportion of winning scratchers from the sample. Consider the sampling distribution of the sample proportion $\widehat{p}$. Complete the following. Carry your intermediate computations to four or more decimal places. Write your answers with two decimal places, rounding if needed. (a) Find $\mu_{p}$ '(the mean of the sampling distribution of the sample proportion). \[ \mu_{\hat{p}}= \] (b) Find $\sigma_{\hat{p}}$ (the standard deviation of the sampling distribution of the sample proportion). \[ \sigma_{\hat{p}}= \]

Solution

Solution Steps

Step 1: Mean of the Sampling Distribution

The mean of the sampling distribution of the sample proportion $ \mu_{\hat{p}} $ is equal to the population proportion. Given that $ p = 0.41 $, we have:

\[ \mu_{\hat{p}} = p = 0.41 \]

Step 2: Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of the sample proportion $ \sigma_{\hat{p}} $ is calculated using the formula:

\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} \]

Substituting the values $ p = 0.41 $ and $ n = 7 $:

\[ \sigma_{\hat{p}} = \sqrt{\frac{0.41 \times (1 - 0.41)}{7}} = \sqrt{\frac{0.41 \times 0.59}{7}} \approx 0.19 \]