Question

Test the claim about the population​ mean, µ​, at the given level of significance using the given sample statistics. Claim: µ = 30​; α = 0.01​; σ = 3.78. Sample​ statistics: xbar = 29.2​, n = 57. Identify the null and alternative hypotheses.

Answer 1

A z-test is used to test the claim about the population mean at a significance level of
`0.01`. The null hypothesis is that the population mean is
`30` and the alternative is that it is not
`30`. If the p-value is less than
`0.01`, the null hypothesis is rejected.

Step-by-step explanation:

To test the claim about the population mean, μ, we begin by identifying the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is a statement of no effect or no difference, and in this case, it is
`H0: μ = 30`. The alternative hypothesis is what you want to test for, and it is either
`μ < 30, μ > 30, or μ ≠ 30` depending on the nature of the claim. Since the claim is that
`μ = 30`, and no direction (greater than or less than) is specified, it suggests a two-tailed test where
`Ha: μ ≠ 30`. Given the level of significance
`α = 0.01`, the population standard deviation σ = 3.78, the sample mean
`xbar = 29.2`, and the sample size
`n = 57,` we can perform a z-test to determine if there is enough evidence to reject the null hypothesis.

To conduct the z-test, we calculate the z-score using the sample statistics and the known value of the population standard deviation. The corresponding p-value is then compared to the significance level α. If the p-value is less than α, we reject the null hypothesis. If it's higher, we do not reject the null hypothesis. This helps us determine if the sample provides enough evidence to support the claim that the population mean is or is not 30.