Questions: Nikita is a government official looking to find evidence on whether the mean taxable income for an individual taxpayer in the region dropped since the previous year. She surveyed 32 individual taxpayers in the region and found the taxable income of each individual. Instead of using the standard deviation from the survey, Nikita decided to use the census data for the region to assume that the population standard deviation of income is 26,744. The mean taxable income in the region was 62,712 for the previous year. Nikita conducts a one-mean hypothesis at the 10% significance level, to test whether this year's mean taxable income for an individual taxpayer in the region is less than the mean taxable income from the previous year. (a) Which answer choice shows the correct null and alternative hypotheses for this test? Select the correct answer below: H0: μ ≥ 62,712; Ha: μ<62,712, which is a left-tailed test. H0: μ ≥ 26,744; Ha: μ<26,744, which is a left-tailed test. H0: μ ≤ 62,712; Ha: μ>62,712, which is a right-tailed test. H0: μ ≤ 26,744; Ha: μ>26,744, which is a right-tailed test.

Nikita is a government official looking to find evidence on whether the mean taxable income for an individual taxpayer in the region dropped since the previous year. She surveyed 32 individual taxpayers in the region and found the taxable income of each individual. Instead of using the standard deviation from the survey, Nikita decided to use the census data for the region to assume that the population standard deviation of income is 26,744. The mean taxable income in the region was 62,712 for the previous year. Nikita conducts a one-mean hypothesis at the 10% significance level, to test whether this year's mean taxable income for an individual taxpayer in the region is less than the mean taxable income from the previous year. (a) Which answer choice shows the correct null and alternative hypotheses for this test? Select the correct answer below: H0: μ ≥ 62,712; Ha: μ<62,712, which is a left-tailed test. H0: μ ≥ 26,744; Ha: μ<26,744, which is a left-tailed test. H0: μ ≤ 62,712; Ha: μ>62,712, which is a right-tailed test. H0: μ ≤ 26,744; Ha: μ>26,744, which is a right-tailed test.
Transcript text: Nikita is a government official looking to find evidence on whether the mean taxable income for an individual taxpayer in the region dropped since the previous year. She surveyed 32 individual taxpayers in the region and found the taxable income of each individual. Instead of using the standard deviation from the survey, Nikita decided to use the census data for the region to assume that the population standard deviation of income is $\$ 26,744$. The mean taxable income in the region was $\$ 62,712$ for the previous year. Nikita conducts a one-mean hypothesis at the $10 \%$ significance level, to test whether this year's mean taxable income for an individual taxpayer in the region is less than the mean taxable income from the previous year. (a) Which answer choice shows the correct null and alternative hypotheses for this test? Select the correct answer below: $H_{0}: \mu \geq \$ 62,712 ; H_{a}: \mu<\$ 62,712$, which is a left-tailed test. $H_{0}: \mu \geq \$ 26,744 ; H_{a}: \mu<\$ 26,744$, which is a left-tailed test. $H_{0}: \mu \leq \$ 62,712 ; H_{a}: \mu>\$ 62,712$, which is a right-tailed test. $H_{0}: \mu \leq \$ 26,744 ; H_{a}: \mu>\$ 26,744$, which is a right-tailed test.
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Solution

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Solution Steps

Step 1: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{26744}{\sqrt{32}} \approx 4727.7159 \]

Step 2: Calculate the Test Statistic

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

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{60000 - 62712}{4727.7159} \approx -0.5736 \]

Step 3: Calculate the P-value

For a left-tailed test, the P-value is given by:

\[ P = T(z) \approx 0.2831 \]

Step 4: State the Hypotheses

The null and alternative hypotheses for this test are:

\[ H_0: \mu \geq 62712 \quad \text{(the mean taxable income has not dropped)} \] \[ H_a: \mu < 62712 \quad \text{(the mean taxable income has dropped)} \]

Step 5: Conclusion

At a significance level of \( \alpha = 0.10 \), we compare the P-value to \( \alpha \):

\[ P \approx 0.2831 > 0.10 \]

Since the P-value is greater than the significance level, we fail to reject the null hypothesis \( H_0 \).

Final Answer

The correct hypotheses are: \[ \boxed{H_0: \mu \geq 62712; \, H_a: \mu < 62712} \]

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