To determine the test statistic and analyze whether the survey provides convincing evidence that more than 42% of U.S. adults favor an internet sales tax, we can follow these steps:

Step 1: State the hypotheses

• Null Hypothesis ($H_0$): $p = 0.42$ (The proportion of U.S. adults who favor the tax is 42%).
• Alternative Hypothesis ($H_a$): $p > 0.42$ (The proportion of U.S. adults who favor the tax is more than 42%).

This is a one-tailed test since we are testing if the proportion is greater than 42%.

Step 2: Identify the sample proportion

The sample proportion ($\hat{p}$) is: $\hat{p} = \frac{x}{n} = \frac{472}{1025} \approx 0.4605$

Step 3: Compute the test statistic

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

• $\hat{p} = 0.4605$ (sample proportion),
• $p_0 = 0.42$ (null hypothesis proportion),
• $n = 1025$ (sample size).

Substitute the values into the formula: $z = \frac{0.4605 - 0.42}{\sqrt{\frac{0.42 \cdot (1 - 0.42)}{1025}}}$ First, calculate the denominator: $\sqrt{\frac{0.42 \cdot 0.58}{1025}} = \sqrt{\frac{0.2436}{1025}} = \sqrt{0.0002376} \approx 0.0154$ Now compute the numerator and the test statistic: $z = \frac{0.4605 - 0.42}{0.0154} = \frac{0.0405}{0.0154} \approx 2.63$

Step 4: Interpret the test statistic

The test statistic is: $z \approx 2.63$

At a significance level of $\alpha = 0.01$, compare the p-value to determine if we reject $H_0$.

Step 5: Compute the P-value

Using standard normal tables or a z-score calculator, the p-value for $z = 2.63$ in a one-tailed test is approximately: $P(z > 2.63) \approx 0.0043$

Since the p-value ($0.0043$) is less than $\alpha = 0.01$, we reject $H_0$.

Conclusion

There is convincing evidence at the 0.01 level of significance to conclude that more than 42% of U.S. adults favor an internet sales tax.

Would you like a breakdown of these steps or further clarification?

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1. What does the test statistic represent in hypothesis testing?
2. Why do we use the P-value method in this test?
3. How is the sample proportion ($\hat{p}$) calculated?
4. What are the steps to interpret a z-test in hypothesis testing?
5. Can you explain why we chose a one-tailed test?

Tip: Always ensure the sample size is sufficiently large for proportions to approximate a normal distribution, as done here!