Question

is the difference between the mean annual salaries of statisticians in region 1 and region 2 more than $6000? to decide, you select a random sample of statisticians from each region. the results of each survey are shown to the right. at α = 0.10, what should you conclude?
region 1: ( \bar{x}_1 = $67,800 ), ( sigma_1 = $8950 ), ( n_1 = 43 )
region 2: ( \bar{x}_2 = $58,000 ), ( sigma_2 = $9125 ), ( n_2 = 40 )
click here to view page 1 of the standard normal distribution table.
click here to view page 2 of the standard normal distribution table.
calculate the standardized test statistic for ( mu_1 - mu_2 ).
( z = 1.91 ) (round to two decimal places as needed.)
choose the correct answer below.
a. reject ( h_0 ). at the 10% significance level, there is insufficient evidence to support the claim that the difference between the mean annual salaries is more than $6000.
b. fail to reject ( h_0 ). at the 10% significance level, there is insufficient evidence to support the claim that the difference between the mean annual salaries is more than $6000.
c. reject ( h_0 ). at the 10% significance level, there is sufficient evidence to support the claim that the difference between the mean annual salaries is more than $6000.
d. fail to reject ( h_0 ). at the 10% significance level, there is sufficient evidence to support the claim that the difference between the mean annual salaries is more than $6000.

Explanation:

Step1: Identify Hypotheses

The claim is that the difference between the mean annual salaries (\(\mu_1 - \mu_2\)) is more than \$6000. So, the null hypothesis \(H_0: \mu_1 - \mu_2 \leq 6000\) and the alternative hypothesis \(H_a: \mu_1 - \mu_2 > 6000\) (right - tailed test).

Step2: Find Critical Value

For a right - tailed test with \(\alpha = 0.10\), we look up the critical value from the standard normal distribution table. The critical value \(z_{\alpha}\) for \(\alpha=0.10\) (right - tailed) is \(z_{0.10}=1.28\) (from the standard normal table, the \(z\) - score that separates the upper 10% of the standard normal distribution from the rest).

Step3: Compare Test Statistic and Critical Value

The test statistic \(z = 1.91\) (given). Since \(1.91>1.28\) (the test statistic is in the rejection region), we reject the null hypothesis \(H_0\). When we reject \(H_0\), it means that at the 10% significance level, there is sufficient evidence to support the claim that the difference between the mean annual salaries is more than \$6000.

Answer:

C. Reject \(H_0\). At the 10% significance level, there is sufficient evidence to support the claim that the difference between the mean annual salaries is more than \$6000.