Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics.
Claim: $\mu=30 ; \alpha=0.05 ; \sigma=3.43$. Sample statistics: $\bar{x}=28.5, n=71$
Identify the null and alternative hypotheses. Choose the correct answer below.
A. $\mathrm{H}_{0}: \mu> 30$
B. $\mathrm{H}_{0}: \mu=30$
$\mathrm{H}_{\mathrm{a}}: \mu=30$
$\mathrm{H}_{\mathrm{a}}: \mu< 30$
C. $\mathrm{H}_{0}: \mu=30$
D. $H_{0}: \mu=30$
$H_{a}: \mu> 30$
$H_{a}: \mu \neq 30$
E. $H_{0}: \mu< 30$
F. $H_{0}: \mu \neq 30$
$\mathrm{H}_{\mathrm{a}}: \mu=30$
$\mathrm{H}_{\mathrm{a}}: \mu=30$
Answer
Final Answer: \(\boxed{B. \mathrm{H}_{0}: \mu=30, \mathrm{H}_{\mathrm{a}}: \mu<30}\)
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis is always a statement of no effect or no difference. In this case, the claim is that the population mean is 30. So, the null hypothesis would be that the population mean is equal to 30. The alternative hypothesis is the opposite of the null hypothesis. Since the sample mean is less than the claimed population mean, the alternative hypothesis would be that the population mean is less than 30.
Step 2 ：Final Answer: \(\boxed{B. \mathrm{H}_{0}: \mu=30, \mathrm{H}_{\mathrm{a}}: \mu<30}\)
