Questions: A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1033 adult Americans in which 524 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the alpha=0.1 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 H1: p=0.50 E. H0: p>0.50 H1: p=0.50 F. H0: p=0.50 H1: p<0.50 Find the test statistic for this hypothesis test. z= (Round to two decimal places as needed.)

A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1033 adult Americans in which 524 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the alpha=0.1 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 H1: p=0.50

E. H0: p>0.50 H1: p=0.50

F. H0: p=0.50 H1: p<0.50

Find the test statistic for this hypothesis test.
z= (Round to two decimal places as needed.)

Transcript text: A headline read, "More Than Half of Americans Say Federal Taxes Too High." The headline was based on a random sample of 1033 adult Americans in which 524 stated the amount of federal tax they have to pay is too high. Is this an accurate headline? Assume the $\boldsymbol{\alpha}=0.1$ level of significance. Identify the null and alternative hypotheses for this test. A. $\mathrm{H}_{0}: p=0.50$ B. $H_{0}: p \neq 0.50$ C. $H_{0}: p=0.50$ $H_{1}: p \neq 0.50$ $H_{1}: p=0.50$ $\mathrm{H}_{1}: p>0.50$ D. $H_{0}: p<0.50$ $H_{1}: p=0.50$ \[ \text { E. } \begin{array}{l} H_{0}: p>0.50 \\ H_{1}: p=0.50 \end{array} \] \[ \text { F. } \begin{aligned} H_{0}: p=0.50 \\ H_{1}: p<0.50 \end{aligned} \] Find the test statistic for this hypothesis test. $z=$ $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Sample Proportion Calculation

The sample proportion ($\hat{p}$) of Americans who believe federal taxes are too high is calculated as follows:

\[ \hat{p} = \frac{524}{1033} \approx 0.5073 \]

Step 2: Hypotheses Formulation

The null and alternative hypotheses for this test are defined as:

Null Hypothesis ($H_0$): $p = 0.50$
Alternative Hypothesis ($H_1$): $p > 0.50$

Step 3: Test Statistic Calculation

The test statistic ($Z$) is calculated using the formula:

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

Substituting the values:

\[ Z = \frac{0.5073 - 0.50}{\sqrt{\frac{0.50(1 - 0.50)}{1033}}} \approx 0.4667 \]

Step 4: P-value Calculation

The p-value associated with the test statistic is found to be:

\[ \text{P-value} \approx 0.3204 \]

Step 5: Critical Region Determination

At a significance level of $\alpha = 0.1$, the critical value for a one-tailed test is:

\[ Z_{critical} \approx 1.2816 \]

Step 6: Conclusion

Since the calculated test statistic $Z \approx 0.4667$ is less than the critical value $Z_{critical} \approx 1.2816$, we fail to reject the null hypothesis. The p-value $0.3204$ is greater than $\alpha = 0.1$, further supporting this conclusion.