Question

test the claim about the population mean μ at the level of significance α. assume the population is normally distributed.
claim: μ = 1,110; α = 0.01; σ = 82
sample statistics: (overline{x}=1,080), n = 35

a. fail to reject (h_0). there is not enough evidence at the 1% level of significance to reject the claim.
b. reject (h_0). there is enough evidence at the 1% level of significance to reject the claim.
c. there is not enough information to decide.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0:\mu = 1110\) (the claim), and the alternative hypothesis \(H_1:\mu
eq1110\) (two - tailed test).

Step2: Calculate the test statistic

The formula for the \(z\) - test statistic is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
Substitute \(\bar{x} = 1080\), \(\mu = 1110\), \(\sigma = 82\), and \(n = 35\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Find the critical values

For a two - tailed test with \(\alpha=0.01\), the critical values are \(z_{\alpha/2}=\pm z_{0.005}\). From the standard normal table, \(z_{0.005}=\pm2.576\).

Step4: Make a decision

Since \(- 2.576<-2.16 < 2.576\) (the test statistic \(z=-2.16\) does not fall in the rejection region), we fail to reject the null hypothesis \(H_0\).

Answer:

A. Fail to reject \(H_0\). There is not enough evidence at the \(1\%\) level of significance to reject the claim.