Questions: In a simple random sample of size 58, there were 36 individuals in the category of interest. Part 1 of 4 (a) Compute the sample proportion p̂. Round the answer to at least three decimal places. Part 2 of 4 (b) Are the assumptions for a hypothesis test satisfied? Explain. Yes, the number of individuals in each category is greater than 10. Part 3 of 4 (c) It is desired to test H0: p=0.6 versus H1: p>0.6. Compute the test statistic z. Round the answer to at least two decimal places. Part 4 of 4 (d) Find the critical value(s) at the 0.1 level. Round the answer(s) to at least three decimal places. If there is more than one critical value, separate them with commas.

Solution Steps

The sample proportion \( \hat{p} \) is calculated as follows:

\[ \hat{p} = \frac{\text{number of successes}}{\text{sample size}} = \frac{36}{58} \approx 0.621 \]

To satisfy the assumptions for the hypothesis test, we check if both \( n \hat{p} \) and \( n (1 - \hat{p}) \) are greater than 10:

\[ n \hat{p} = 58 \times 0.621 \approx 36.078 \quad (\text{greater than } 10) \] \[ n (1 - \hat{p}) = 58 \times (1 - 0.621) \approx 21.922 \quad (\text{greater than } 10) \]

Since both conditions are satisfied, the assumptions for the hypothesis test are confirmed to be true.

We are testing the null hypothesis \( H_0: p = 0.6 \) against the alternative hypothesis \( H_1: p > 0.6 \). The test statistic \( z \) is calculated using the formula:

\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

\[ z = \frac{0.621 - 0.6}{\sqrt{\frac{0.6(1 - 0.6)}{58}}} \approx 0.32 \]

Final Answer