Question

test the claim about the difference between two population means $\mu_1$ and $\mu_2$ at the level of significance $\alpha$. assume the samples are and independent, and the populations are normally distributed.
claim: $\mu_1=\mu_2$; $\alpha = 0.01$
population parameters: $\sigma_1 = 3.3$, $\sigma_2 = 1.5$
sample statistics: $\bar{x}_1 = 16$, $n_1 = 28$, $\bar{x}_2 = 18$, $n_2 = 30$
determine the p-value.
p-value = 0.003 (round to three decimal places as needed.)
what is the proper decision?
a. fail to reject $h_0$. there is enough evidence at the 1% level of significance to reject the claim.
b. fail to reject $h_0$. there is not enough evidence at the 1% level of significance to reject the claim.
c. reject $h_0$. there is enough evidence at the 1% level of significance to reject the claim.
d. reject $h_0$. there is not enough evidence at the 1% level of significance to reject the claim.

Explanation:

Step1: State the hypotheses

The claim is \(\mu_1=\mu_2\), so the null hypothesis \(H_0:\mu_1 - \mu_2=0\) and the alternative hypothesis \(H_1:\mu_1-\mu_2
eq0\) (two - tailed test).

Step2: Calculate the test statistic \(z\)

The formula for the test statistic \(z\) for two independent samples with known population variances is \(z=\frac{(\overline{x}_1-\overline{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}\).
Substitute \(\overline{x}_1 = 16\), \(\overline{x}_2=18\), \(\mu_1-\mu_2 = 0\), \(\sigma_1 = 3.3\), \(\sigma_2=1.5\), \(n_1 = 28\), \(n_2=30\) into the formula:

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Step3: Find the P - value

For a two - tailed test, the P - value is \(P = 2P(Z\lt|z|)\).
Since \(z=-2.94\), \(P(Z\lt - 2.94)\) can be found using the standard normal table. \(P(Z\lt - 2.94)=0.0016\)
So \(P - value=2\times0.0016 = 0.0032\approx0.003\) (rounded to three decimal places)

Step4: Make a decision

The level of significance \(\alpha = 0.01\).
Since \(P - value=0.003\lt\alpha = 0.01\), we reject the null hypothesis \(H_0\).

Answer:

C. Reject \(H_0\). There is enough evidence at the \(1\%\) level of significance to reject the claim.