According to a study, the proportion of people who are satisfied with the way things are going in their lives is 0.84 . Suppose that a random sample of 100 people is obtained. Complete parts (a) through (e) below.

Click here to view the standard normal distribution table (page 1).
Click here to view the standard normal distribution table (page 2).
(c) Describe the sampling distribution of $\hat{p}$, the proportion of people who are satisfied with the way things are going in their life. Be sure to verify the model requirements. Since the sample size is no more than $5 \%$ of the population size and $n p(1-p)=\square \geq 10$, the distribution of $\hat{p}$ is approximately normal with $\mu_{\hat{p}}=\square$ and $\sigma_{\hat{p}}=\square$.
(Round to three decimal places as needed.)

Step 1 ：The problem is asking for the sampling distribution of the proportion of people who are satisfied with the way things are going in their life. The distribution of this proportion is approximately normal because the sample size is no more than 5% of the population size and the product of the sample size and the probability of success and failure is greater than or equal to 10.

Step 2 ：The mean of the sampling distribution of the proportion, denoted as \(\mu_{\hat{p}}\), is equal to the population proportion, which is given as 0.84.

Step 3 ：The standard deviation of the sampling distribution of the proportion, denoted as \(\sigma_{\hat{p}}\), is calculated as the square root of the product of the population proportion and its complement (1 - population proportion) divided by the sample size.

Step 4 ：Let's calculate \(\sigma_{\hat{p}}\).

Step 5 ：\(p = 0.84\)

Step 6 ：\(n = 100\)

Step 7 ：\(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\)

Step 8 ：\(\sigma_{\hat{p}} = \sqrt{\frac{0.84(1-0.84)}{100}}\)

Step 9 ：\(\sigma_{\hat{p}} = 0.036660605559646724\)

Step 10 ：Rounding \(\sigma_{\hat{p}}\) to three decimal places, we get \(\sigma_{\hat{p}} = 0.037\)

Step 11 ：Final Answer: The distribution of \(\hat{p}\) is approximately normal with \(\mu_{\hat{p}}=0.84\) and \(\sigma_{\hat{p}}=0.037\)

Step 12 ：\(\boxed{\mu_{\hat{p}}=0.84, \sigma_{\hat{p}}=0.037}\)