Question

test the claim about the population mean μ at the level of significance α. assume the population is no
claim: μ ≠ 21; α = 0.05; σ = 2.7
sample statistics: x̄ = 20.1, n = 36

a. reject h₀. there is enough evidence at the 5% level of significance to support the claim.
b. fall to reject h₀. there is not enough evidence at the 5% level of significance to support the c
c. there is not enough information to decide.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:\mu = 21\) and the alternative hypothesis \(H_1:\mu
eq21\) (since the claim is \(\mu
eq21\)).

Step2: Calculate the test - statistic

The formula for the z - test statistic is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
Substitute \(\bar{x} = 20.1\), \(\mu = 21\), \(\sigma=2.7\), and \(n = 36\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Find the critical values

For a two - tailed test with \(\alpha=0.05\), the critical values are \(z_{\alpha/2}=\pm z_{0.025}\).
From the standard normal table, \(z_{0.025}=1.96\). So the critical values are \(z=-1.96\) and \(z = 1.96\).

Step4: Make a decision

Since the test statistic \(z=-2\lt - 1.96\), we reject the null hypothesis \(H_0\).

Answer:

A. Reject \(H_0\). There is enough evidence at the \(5\%\) level of significance to support the claim.