Part 1: Hypothesis Identification

We are testing whether the return rate of surveys is less than 20%. This indicates that the null hypothesis assumes the return rate is 20%, and the alternative hypothesis suggests that the return rate is less than 20%.

• Null Hypothesis ($H_0$): The proportion of returned surveys is 20% (i.e., $p = 0.20$).
• Alternative Hypothesis ($H_1$): The proportion of returned surveys is less than 20% (i.e., $p < 0.20$).

Thus, the correct hypothesis pair is:

E. $H_0: p = 0.20$, $H_1: p < 0.20$

Part 2: Calculating the Test Statistic

We will use the normal approximation to the binomial distribution to compute the test statistic.

Step 1: Identify the values

• Sample size ($n$) = 1290 (the number of surveys returned)
• Sample proportion ($\hat{p}$) = $\frac{1290}{6962} = 0.185$ (the proportion of returned surveys)
• Hypothesized proportion ($p_0$) = 0.20

Step 2: Compute the standard error of the proportion

The standard error ($SE$) for the sample proportion is calculated using the formula: $SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$ Substitute $p_0 = 0.20$ and $n = 6962$: $SE = \sqrt{\frac{0.20(1 - 0.20)}{6962}} = \sqrt{\frac{0.20(0.80)}{6962}} = \sqrt{\frac{0.16}{6962}} \approx \sqrt{0.00002295} \approx 0.00479$

Step 3: Calculate the test statistic (z)

The z-test statistic is calculated by the formula: $z = \frac{\hat{p} - p_0}{SE}$ Substitute $\hat{p} = 0.185$, $p_0 = 0.20$, and $SE = 0.00479$: $z = \frac{0.185 - 0.20}{0.00479} = \frac{-0.015}{0.00479} \approx -3.14$

So, the test statistic is z ≈ -3.14.

Summary of Results:

• Null Hypothesis ($H_0$): $p = 0.20$
• Alternative Hypothesis ($H_1$): $p < 0.20$
• Test Statistic (z): -3.14

Would you like to continue with calculating the p-value or testing the hypothesis further? Let me know if you'd like additional steps or clarifications.

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Related Questions:

1. How do you compute the p-value for a given z-score?
2. What is the interpretation of the p-value in hypothesis testing?
3. How would the test change if the significance level was 0.05 instead of 0.01?
4. What does the normal approximation to the binomial distribution imply in this context?
5. How do you interpret the results if the z-score were greater than the critical value?

Tip: In hypothesis testing, always ensure your sample size is large enough to use the normal approximation, and check if both $np_0 \geq 10$ and $n(1 - p_0) \geq 10$.