: The Gallup Poll asked 1019 . adults whether they believed that people should pay sales tax on items purchased over the internet. Of these, 428 said they supported such a tax. Does the survey provide convincing evidence that more than 39%  of U.S. adults favor an internet sales tax? Use the a=.05 level of significance and the P-value method with the TI-84 Plus calculator.

To determine if the survey provides convincing evidence that more than 39% of U.S. adults favor an internet sales tax, we will conduct a hypothesis test for a proportion using the given data. We'll use the \( \alpha = 0.05 \) level of significance and the P-value method. Here are the steps:

1. **Define the hypotheses:**

   - Null hypothesis (\(H_0\)): \( p = 0.39 \)
   - Alternative hypothesis (\(H_1\)): \( p > 0.39 \)

2. **Calculate the sample proportion (\(\hat{p}\)):**

   \[
   \hat{p} = \frac{x}{n} = \frac{428}{1019} \approx 0.4200
   \]

3. **Calculate the standard error (SE) of the sample proportion:**

   \[
   SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.39 (1 - 0.39)}{1019}} \approx 0.0154
   \]

4. **Calculate the test statistic (z):**

   \[
   z = \frac{\hat{p} - p_0}{SE} = \frac{0.4200 - 0.39}{0.0154} \approx 1.948
   \]

5. **Determine the P-value:**

   Using the TI-84 Plus calculator, we can find the P-value for the z-test. 

   - Press [2ND] then [VARS] to access the DISTR menu.
   - Select "normalcdf(".
   - Enter the lower bound as 1.948, the upper bound as a very large number (e.g., 9999), the mean as 0, and the standard deviation as 1.
   
   So, the input will be `normalcdf(1.948, 9999, 0, 1)`. 

   The calculator will return the P-value. For our calculations:

   \[
   \text{P-value} \approx 0.0257
   \]

6. **Compare the P-value with \( \alpha \):**

   \[
   \text{P-value} = 0.0257 < 0.05 = \alpha
   \]

Since the P-value is less than the significance level \( \alpha \), we reject the null hypothesis.

7. **Conclusion:**

   There is sufficient evidence at the \( \alpha = 0.05 \) level of significance to conclude that more than 39% of U.S. adults favor an internet sales tax.

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Would you like more details or have any questions?

Here are 8 related questions for further exploration:

1. How would the conclusion change if the level of significance was \( \alpha = 0.01 \)?
2. What if the sample size was smaller, say 500 adults? How would that affect the test?
3. How does changing the hypothesized proportion (\( p_0 \)) to 0.45 affect the test?
4. What are Type I and Type II errors in the context of this hypothesis test?
5. How would you calculate the confidence interval for the sample proportion?
6. How can you use the TI-84 Plus to conduct a two-proportion z-test?
7. What assumptions must be satisfied for this hypothesis test to be valid?
8. How does the choice of the level of significance (\( \alpha \)) influence the hypothesis test results?

**Tip:** Always verify that the assumptions of the hypothesis test are met before interpreting the results.

Explore a hypothesis test to determine if more than 39% of U.S. adults support an internet sales tax based on a survey. Step-by-step solution using significance level and P-value method with TI-84 Plus calculator.

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