Solution:

We are testing the claim that the return rate is less than $20\%$ using the P-value method. Here's the step-by-step solution:

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Step 1: Identify the hypotheses

• Null hypothesis ($H_0$): $p \geq 0.20$ (The return rate is at least 20%)
• Alternative hypothesis ($H_1$): $p < 0.20$ (The return rate is less than 20%)

This is a left-tailed test.

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Step 2: Identify the significance level

$\alpha = 0.01$

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Step 3: Compute the test statistic

• Sample size ($n$): $69876987$
• Number of returned surveys ($x$): $12911291$
• Sample proportion ($\hat{p}$): $\hat{p} = \frac{x}{n} = \frac{12911291}{69876987} \approx 0.1848$

The test statistic $z$ for a proportion is calculated as: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Where:

• $p_0 = 0.20$ (hypothesized proportion)
• $\hat{p} = 0.1848$
• $n = 69876987$

Substitute the values: $z = \frac{0.1848 - 0.20}{\sqrt{\frac{0.20 \cdot (1 - 0.20)}{69876987}}}$

First, calculate the standard error: $\text{SE} = \sqrt{\frac{0.20 \cdot 0.80}{69876987}} = \sqrt{\frac{0.16}{69876987}} \approx 0.000478$

Now, compute $z$: $z = \frac{0.1848 - 0.20}{0.000478} = \frac{-0.0152}{0.000478} \approx -31.8$

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Step 4: Find the P-value

Using a z-table or normal distribution calculator, find the area to the left of $z = -31.8$.

The P-value is approximately: $P \approx 0 \, (\text{extremely small, essentially zero})$

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Step 5: Compare P-value with $\alpha$

$P = 0 < \alpha = 0.01$

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Step 6: Decision and conclusion

Since $P < \alpha$, we reject the null hypothesis.

Conclusion: At the $0.01$ significance level, there is sufficient evidence to support the claim that the return rate is less than $20\%$.

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Do you need further clarification or detailed steps?
Here are 5 related questions for exploration:

1. How does the sample size influence the outcome of hypothesis testing?
2. What are the assumptions for using the normal approximation to the binomial distribution?
3. How would the test change if the significance level were $0.05$ instead of $0.01$?
4. What is the importance of using the P-value method versus the critical value method?
5. How would the conclusion change if the alternative hypothesis were $p \neq 0.20$?

Tip: Always verify that the sample size is large enough for the normal approximation to be valid ($n \cdot p_0 \geq 5$ and $n \cdot (1 - p_0) \geq 5$).