Question

an employment information service claims the mean annual salary for senior level product engineers is $95,000. the annual salaries (in dollars) for a random sample of 16 senior level product engineers are shown in the table to the right. at \LXI0, test the claim that the mean salary is $95,000. complete parts (a) through (e) below. assume the population is normally distributed. (c) find the standardized test statistic, \\( t \\). the standardized test statistic is \\( t = - 0.93 \\) (round to two decimal places as needed) (d) decide whether to reject or fail to reject the null hypothesis. fail to reject \\( h _ { 0 } \\) because the standardized test statistic is not in the rejection region. (e) interpret the decision in the context of the original claim. there evidence at the \\( \\% \\) level of significance to the claim that the mean annual salary for senior level product engineers is $ (type integers or decimals. do not round.)

Explanation:

Step1: Recall the decision rule for hypothesis testing

If the standardized test statistic \(t\) is not in the rejection region, we fail to reject the null hypothesis \(H_0\). Here, \(t=- 0.93\). For a two - tailed test with \(\alpha = 0.01\) and \(n = 16\) (so degrees of freedom \(df=n - 1=15\)), the critical values are \(t_{\alpha/2}=\pm 2.947\) (from the t - distribution table). Since \(-2.947<-0.93 < 2.947\), the test statistic is not in the rejection region.

Step2: Interpret the result in the context of the claim

When we fail to reject the null hypothesis \(H_0:\mu = 95000\), it means that there is not enough evidence. The level of significance is \(\alpha=1\%\) (or \(1\) in percentage form). The claim was about the mean annual salary being \(\$95000\).

Answer:

There is not enough evidence at the \(1\%\) level of significance to reject the claim that the mean annual salary for senior - level product engineers is \(\$95000\).