Question

test the claim about the population mean, μ, at the given level of significance using the given sample statistics. claim: μ = 50; α = 0.02; σ = 3.73. sample statistics: \\( \overline { x } = 48.6 \\), n = 77 o e. \\( h _ { 0 } : \mu > 50 \\) \\( h _ { a } : \mu = 50 \\) o f. \\( h _ { 0 } : \mu \
eq 50 \\) \\( h _ { a } : \mu = 50 \\) calculate the standardized test statistic. the standardized test statistic is - 3.29. (round to two decimal places as needed.) determine the critical value(s). select the correct choice below and fill in the answer box to complete your choice. (round to two decimal places as needed.) o a. the critical value is. o b. the critical values are ± 2.33. determine the outcome and conclusion of the test. choose the correct answer below. o a. reject \\( h _ { 0 } \\). at the 2% significance level, there is enough evidence to support the claim. o b. fail to reject \\( h _ { 0 } \\). at the 2% significance level, there is not enough evidence to support the claim. o c. reject \\( h _ { 0 } \\). at the 2% significance level, there is enough evidence to reject the claim. o d. fail to reject \\( h _ { 0 } \\). at the 2% significance level, there is not enough evidence to reject the claim.

Explanation:

Step1: Determine the null and alternative hypotheses

The claim is \(\mu = 50\). So, \(H_{0}:\mu = 50\) and \(H_{a}:\mu
eq50\) (two - tailed test).

Step2: Analyze the critical values

For a two - tailed test with \(\alpha=0.02\), the critical values are \(z_{\alpha/2}=\pm z_{0.01}\). From the standard normal table, \(z_{0.01} = 2.33\), so the critical values are \(\pm2.33\).

Step3: Compare the test statistic and critical values

The test statistic \(z=- 3.29\). Since \(-3.29<-2.33\) (i.e., the test statistic falls in the rejection region).

Answer:

C. Reject \(H_{0}\). At the \(2\%\) significance level, there is enough evidence to reject the claim.