Questions: Test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed. Claim: μ<4915 ; α=0.05 Sample statistics: x̄=5017, s=5613, n=56 What are the null and alternative hypotheses? H₀: μ=4915 Hₐ: μ<4915 (Type integers or decimals. Do not round.)

Solution Steps

We are testing the claim about the population mean $\mu$. The null and alternative hypotheses are defined as follows:

$$H_0: \mu = 4915$$ $$H_a: \mu < 4915$$

The standard error $SE$ is calculated using the formula:

$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5613}{\sqrt{56}} \approx 750.0687$$

The test statistic $Z_{test}$ is calculated using the formula:

$$Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{5017 - 4915}{750.0687} \approx 0.136$$

For a left-tailed test, the P-value is determined as follows:

$$P = T(z) \approx 0.5541$$

At a significance level of $\alpha = 0.05$, we compare the P-value with $\alpha$:

$$P \approx 0.5541 > 0.05$$

Since the P-value is greater than the significance level, we fail to reject the null hypothesis $H_0$.

Final Answer