Questions: Question An economist is studying salaries for high technology companies and wants to test the claim that the average salary for high tech employees is less than 70,000. The economist selects a sample of 35 random employees from various high tech companies and records their salaries. Based on past studies, the economist determines that the population standard deviation is 8,500. The economist conducts a one-mean hypothesis test at the 5% significance level to test the claim that the average salary for high tech employees is less than 70,000. The setup for the null and alternative hypothesis is given as: H0: μ ≥ 70,000; Ha: μ<70,000, which is a left-tailed test. The sample data for 35 salaries is shown in the dataset below. Use Excel to test the claim that the average salary for high tech employees is less than 70,000, where α=0.05. Calculate the test statistic, z, and the p-value, rounding to three decimal places.

Question An economist is studying salaries for high technology companies and wants to test the claim that the average salary for high tech employees is less than 70,000. The economist selects a sample of 35 random employees from various high tech companies and records their salaries. Based on past studies, the economist determines that the population standard deviation is 8,500. The economist conducts a one-mean hypothesis test at the 5% significance level to test the claim that the average salary for high tech employees is less than 70,000. The setup for the null and alternative hypothesis is given as: H0: μ ≥ 70,000; Ha: μ<70,000, which is a left-tailed test. The sample data for 35 salaries is shown in the dataset below. Use Excel to test the claim that the average salary for high tech employees is less than 70,000, where α=0.05. Calculate the test statistic, z, and the p-value, rounding to three decimal places.
Transcript text: Question An economist is studying salaries for high technology companies and wants to test the claim that the average salary for high tech employees is less than $\$ 70,000$. The economist selects a sample of 35 random employees from various high tech companies and records their salaries. Based on past studies, the economist determines that the population standard deviation is $\$ 8,500$. The economist conducts a one-mean hypothesis test at the $5 \%$ significance level to test the claim that the average salary for high tech employees is less than $\$ 70,000$. The setup for the null and alternative hypothesis is given as: $H_{0}: \mu \geq 70,000 ; H_{a}: \mu<70,000$, which is a left-tailed test. The sample data for 35 salaries is shown in the dataset below. Use Excel to test the claim that the average salary for high tech employees is less than $\$ 70,000$, wh $\alpha=0.05$. Calculate the test statistic, $z$, and the $p$-value, rounding to three decimal places.
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Solution

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Solution Steps

Step 1: Calculate the Sample Mean

The sample mean \( \bar{x} \) is calculated as follows:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{2429000}{35} = 69400.0 \]

Step 2: Calculate the Standard Error

The standard error \( SE \) is computed using the population standard deviation \( \sigma \) and the sample size \( n \):

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{8500}{\sqrt{35}} \approx 1436.762 \]

Step 3: Calculate the Test Statistic

The test statistic \( Z \) for the left-tailed test is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{69400.0 - 70000}{1436.762} \approx -0.418 \]

Step 4: Calculate the P-value

For a left-tailed test, the P-value is determined using the Z-score:

\[ P = T(z) \approx 0.338 \]

Final Answer

The test statistic is approximately \( Z \approx -0.418 \) and the P-value is approximately \( P \approx 0.338 \).

Thus, the final results are:

\[ \boxed{Z \approx -0.418, \, P \approx 0.338} \]

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