To address this problem, we are testing whether the proportion of U.S. adults who support an internet sales tax is less than 45% using a hypothesis test.

Step 1: Define the hypotheses

• Null Hypothesis ($H_0$): $p = 0.45$
  (The proportion of U.S. adults who support the internet sales tax is 45%.)

• Alternative Hypothesis ($H_1$): $p < 0.45$
  (The proportion of U.S. adults who support the internet sales tax is less than 45%.)

This is a left-tailed test because we are testing if the proportion is less than 45%.

Step 2: Calculate the test statistic

We will use the sample proportion ($\hat{p}$) and perform a z-test for a proportion.

Given:

• Sample size ($n$) = 1031
• Number of successes = 443
  So, $\hat{p} = \frac{443}{1031} \approx 0.4297$
• Hypothesized proportion ($p_0$) = 0.45
• Significance level ($\alpha$) = 0.05

Formula for the z-test statistic:

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

Substitute the values: $z = \frac{0.4297 - 0.45}{\sqrt{\frac{0.45 (1 - 0.45)}{1031}}}$ $z = \frac{-0.0203}{\sqrt{\frac{0.45 \cdot 0.55}{1031}}}$ $z = \frac{-0.0203}{\sqrt{0.00023955}}$ $z \approx \frac{-0.0203}{0.01548} \approx -1.31$

Step 3: Find the p-value

Using a z-table or calculator, find the p-value for $z = -1.31$:

• For $z = -1.31$, the p-value is approximately 0.0951.

Step 4: Conclusion

Compare the p-value with $\alpha = 0.05$:

• Since $0.0951 > 0.05$, we fail to reject the null hypothesis.

Conclusion: There is insufficient evidence to conclude that less than 45% of U.S. adults support an internet sales tax.

Would you like detailed steps for calculating the z-value or finding the p-value? Let me know!

Here are some related questions to expand this concept:

1. What assumptions must be satisfied for this test to be valid?
2. How would the conclusion change if the p-value were smaller than $\alpha$?
3. How does the choice of a one-tailed vs. two-tailed test affect the result?
4. What is the confidence interval for the population proportion based on this data?
5. How can this test be replicated using a TI-84 Plus calculator?

Tip: Always verify that sample size is large enough for normal approximation when performing hypothesis tests for proportions.