Questions: Question 5 12.5 Points A random sample is to be selected from a population that has a proportion of successes p=0.75. Determine the mean of the sample proportion p̂ when the sample size n=400. Add your answer Integer, decimal, or E notation allowed Question 6 15 Points Regarding question 5, what is the standard deviation of the sample proportion when the sample size n=400? Round your answer to five decimal places. Add your answer Integer, decimal, or E notation allowed Question 7 15 Points Regarding question 5, what is the probability that the sample proportion will be between 0.7 and 0.8 when the sample size n =400? Round your answer to four decimal places. Add your answer Just output the content of the question, DO NOT output additional information or explanations.

Solution Steps

The mean of the sample proportion \( \widehat{p} \) is equal to the population proportion \( p \). Given that \( p = 0.75 \), we have:

\[ \text{Mean of the sample proportion} = \widehat{p} = p = 0.75 \]

The standard deviation of the sample proportion \( \widehat{p} \) can be calculated using the formula:

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

Substituting the values \( p = 0.75 \) and \( n = 400 \):

\[ \sigma_{\widehat{p}} = \sqrt{\frac{0.75 \times (1 - 0.75)}{400}} = \sqrt{\frac{0.75 \times 0.25}{400}} = \sqrt{\frac{0.1875}{400}} = \sqrt{0.00046875} \approx 0.02165 \]

To find the probability that the sample proportion \( \widehat{p} \) falls between 0.7 and 0.8, we first convert these values to Z-scores using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

Where \( \mu = 0.75 \) and \( \sigma = 0.02165 \).

Calculating the Z-scores:

For \( X = 0.7 \):

\[ Z_{start} = \frac{0.7 - 0.75}{0.02165} \approx -2.3094 \]

For \( X = 0.8 \):

\[ Z_{end} = \frac{0.8 - 0.75}{0.02165} \approx 2.3094 \]

Using the standard normal distribution, we find:

\[ P(0.7 < \widehat{p} < 0.8) = \Phi(Z_{end}) - \Phi(Z_{start}) \approx \Phi(2.3094) - \Phi(-2.3094) \approx 0.9791 \]

Final Answer