Test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed. Claim: $\mu \leq 47 ; \alpha = 0.01 ; \sigma = 4.3$
Sample statistics: $\bar { x } = 48.8 , n = 40$
A)Reject $\mathrm { H } _ { 0 }$ There is enough evidence at the 1% level of significance to reject the claim.
B)Fail to reject $\mathrm { H } _ { 0 }$ There is enough evidence at the 1% level of significance to support the claim.
C)Not enough information to decide.

Question

Quizplus · Accepted Answer

To test the claim, we use the z-test for the population mean when the population standard deviation is known. The null hypothesis is $H_0: \mu = 47$, and the alternative hypothesis (based on the claim) is $H_a: \mu < 47$. The test statistic is calculated as $z = (\bar{x} - \mu) / (\sigma / \sqrt{n})$, which gives $z = (48.8 - 47) / (4.3 / \sqrt{40})$. This calculation results in a z-value that, when compared to the critical value from the z-table for $\alpha = 0.01$, shows that the test statistic falls in the rejection region. Therefore, we reject $H_0$, indicating there is enough evidence at the 1% level of significance to reject the claim that $\mu \leq 47$.

Test the Claim About the Population Mean μ at the Level