According to a study conducted by a statistical organization, the proportion of people who are satisfied with the way things are going in their lives is 0.74 . Suppose that a randorn sample of 100 people is obtained Complete parts (a) through (e) below. (b) Explain why the sample proportion, $\hat{p}$, is a random variable. What is the source of the variability? A. The sample proportion $\hat{p}$ is a random variable because the value of $\hat{p}$ represents a random person included in the sample. The variability is due to the fact that different people feel differently regarding their satisfaction B. The sample proportion $\hat{p}$ is a random variable because the value of $\hat{p}$ represents a random person included in the sample. The variability is due to the fact that people may not be responding to the question truthfully. C. The sample proportion $\hat{p}$ is a random variable because the value of $\hat{p}$ varies from sample to sample. The variability is due to the fact that people may not be responding to the question truthfully. D. The sample proportion $\hat{p}$ is a random variable because the value of $\hat{p}$ varies from sample to sample The variability is due to the fact that different people feel differently regarding their satisfaction. (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 $\square$, $\quad \nabla$ ith $\mu_{\hat{p}}=\square$ and $\sigma_{\hat{p}}=\square$ (Round to three decimal places as needed)
Solution
Step 1 ：The sample proportion, denoted as $\hat{p}$, is a random variable because its value varies from sample to sample. The source of this variability is the differing levels of satisfaction among individuals.
Step 2 ：We can calculate the standard deviation of the sampling distribution using the formula $\sqrt{p(1-p)/n}$, where $p$ is the population proportion and $n$ is the sample size. In this case, $p = 0.74$ and $n = 100$.
Step 3 ：The standard deviation of the sampling distribution, $\sigma_{\hat{p}}$, is approximately 0.044.
Step 4 ：The model requirements for the sampling distribution of $\hat{p}$ are that the sample size is no more than 5% of the population size and that $np(1-p) \geq 10$. Since we don't know the population size, we can't check the first condition. However, the second condition is satisfied.
Step 5 ：The sampling distribution of $\hat{p}$ is approximately normal because the sample size is large enough (n = 100) and the condition $np(1-p) \geq 10$ is satisfied. The mean of the distribution, $\mu_{\hat{p}}$, is equal to the population proportion, p = 0.74. The standard deviation of the distribution, $\sigma_{\hat{p}}$, is approximately 0.044.
Step 6 ：Final Answer: The sample proportion $\hat{p}$ is a random variable because the value of $\hat{p}$ varies from sample to sample. The variability is due to the fact that different people feel differently regarding their satisfaction. The sampling distribution of $\hat{p}$ is approximately normal with $\mu_{\hat{p}} = 0.74$ and $\sigma_{\hat{p}} \approx 0.044$. The model requirements are satisfied because the sample size is large enough and $np(1-p) \geq 10$.
