Question

test the claim about the population mean μ at the level of significance α. assume the population is normally distributed. claim: μ>27; α=0.05; σ=1.2 sample statistics: x̄=27.3, n=50 a. reject h₀. there is enough evidence at the 5% level of significance to support the claim. b. fail to reject h₀. there is not enough evidence at the 5% level of significance to support the claim. c. there is not enough information to decide.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:\mu\leq27\) and the alternative hypothesis \(H_1:\mu > 27\) (the claim).

Step2: Calculate the test - statistic

The formula for the \(z\) - test statistic is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
Given \(\bar{x} = 27.3\), \(\mu = 27\), \(\sigma=1.2\), and \(n = 50\).
Substitute the values into the formula:

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Step3: Find the critical value

For a right - tailed test with \(\alpha = 0.05\), the critical value \(z_{\alpha}\) is \(z_{0.05}=1.645\) (from the standard normal distribution table).

Step4: Make a decision

Since the test statistic \(z = 1.77>z_{\alpha}=1.645\), 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.