The question you're working on seems to be related to hypothesis testing for proportions. Let's break it down step-by-step:

Given:

• 7% of 300 working college students are employed as teachers or teaching assistants.
• The education researcher claims that at most 5% of working college students are employed as teachers or teaching assistants.

Part (a): Hypotheses

• Null Hypothesis (H₀): The proportion of working college students employed as teachers or teaching assistants is equal to 5%, or $p = 0.05$.
• Alternative Hypothesis (H₁): The proportion of working college students employed as teachers or teaching assistants is greater than 5%, or $p > 0.05$.

Thus, the correct choice for the hypotheses would be:

• H₀: p = 0.05
• H₁: p > 0.05

Part (b): Test Statistic You are asked to compute the test statistic using the formula for a z-test for proportions:

$z = \frac{p̂ - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Where:

• $p̂ = \frac{7\% \times 300}{300} = 0.07$ (observed proportion)
• $p_0 = 0.05$ (claimed population proportion)
• $n = 300$ (sample size)

Substituting the values into the formula, we can calculate the value for z. Let me do the calculation for you.

Part (c): Rejection Regions The problem asks about the rejection region based on the z-value and your significance level (alpha = 0.10). This would correspond to a one-tailed test since you are testing if the proportion is greater than 5%. A z-table or critical value can be used to find this region.

Let's first calculate the z-test statistic. math

Given values

p_hat = 0.07 # sample proportion p_0 = 0.05 # claimed proportion n = 300 # sample size

Standard error formula

se = math.sqrt((p_0 * (1 - p_0)) / n)

z-test statistic formula

z = (p_hat - p_0) / se zThe calculated z-test statistic is approximately 1.59.

Now, for part (c), the critical value for a one-tailed z-test with a significance level of 0.10 (alpha = 0.10) corresponds to a z-value of about 1.28.

Since the z-test statistic (1.59) is greater than the critical value (1.28), we can reject the null hypothesis and conclude that there is evidence to support the claim that the proportion of working college students employed as teachers or teaching assistants is greater than 5%.

Rejection Region:

• The rejection region for this one-tailed test is z > 1.28.

Would you like further clarification on any of the steps or additional details on hypothesis testing?

Here are some related questions you could explore:

1. How do we interpret a p-value in hypothesis testing?
2. What is the difference between a one-tailed and two-tailed test?
3. What other assumptions should be checked for the z-test for proportions?
4. How do you calculate a confidence interval for the population proportion?
5. How does changing the sample size affect the results of a hypothesis test?

Tip: Always check if your sample size is large enough to use the z-test by ensuring that both $n \cdot p_0$ and $n \cdot (1 - p_0)$ are greater than 5.