Questions: A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1221 adult Americans in which 632 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the α=0.05 level of significance. Identify the null and alternative hypotheses for this test. A. H0: p=0.50 B. H0: p<0.50 C. H0: p ≠ 0.50 H1: p ≠ 0.50 H1: p=0.50 H1: p=0.50 D. H0: p=0.50 E. H0: p=0.50 F. H0: p>0.50 H1: p>0.50 H1: p<0.50 H1: p=0.50

A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1221 adult Americans in which 632 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the α=0.05 level of significance. Identify the null and alternative hypotheses for this test. A. H0: p=0.50 B. H0: p<0.50 C. H0: p ≠ 0.50 H1: p ≠ 0.50 H1: p=0.50 H1: p=0.50 D. H0: p=0.50 E. H0: p=0.50 F. H0: p>0.50 H1: p>0.50 H1: p<0.50 H1: p=0.50
Transcript text: A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1221 adult Americans in which 632 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the $\alpha=0.05$ level of significance. Identify the null and alternative hypotheses for this test. A. $H_{0}: p=0.50$ B. $H_{0}: p<0.50$ C. $H_{0}: p \neq 0.50$ $H_{1}: p \neq 0.50$ $H_{1}: p=0.50$ $H_{1}: p=0.50$ D. $H_{0}: p=0.50$ E. $H_{0}: p=0.50$ F. $H_{0}: p>0.50$ $H_{1}: p>0.50$ $H_{1}: p<0.50$ $H_{1}: p=0.50$
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Solution

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Solution Steps

Step 1: Define the Hypotheses

We are testing whether more than half of Americans believe that federal taxes are too high. The hypotheses are defined as follows:

  • Null Hypothesis: \( H_0: p = 0.50 \)
  • Alternative Hypothesis: \( H_1: p > 0.50 \)
Step 2: Calculate the Test Statistic

The test statistic for the sample proportion is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Where:

  • \( \hat{p} = \frac{632}{1221} \approx 0.5180 \)
  • \( p_0 = 0.50 \)
  • \( n = 1221 \)

Substituting the values, we find:

\[ Z = \frac{0.5180 - 0.50}{\sqrt{\frac{0.50(1 - 0.50)}{1221}}} \approx 1.2306 \]

Step 3: Calculate the P-value

The P-value associated with the test statistic \( Z = 1.2306 \) is calculated to be:

\[ \text{P-value} = 0.1092 \]

Step 4: Determine the Critical Region

For a significance level of \( \alpha = 0.05 \) in a one-tailed test, the critical value is:

\[ Z_{critical} = 1.6449 \]

Step 5: Make a Decision

We compare the P-value with the significance level:

  • Since \( \text{P-value} = 0.1092 > \alpha = 0.05 \), we fail to reject the null hypothesis.
Step 6: Conclusion

Based on the results, we conclude that there is not enough evidence to support the claim that more than half of Americans believe that federal taxes are too high. Therefore, the headline is not accurate.

Final Answer

\(\boxed{\text{The headline is not accurate.}}\)

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