Question

for the given data, (a) find the test statistic, (b) find the standardized test statistic, (c) decide whether the standardized test statistic is in the rejection region, and (d) decide whether you should reject or fail to reject the null hypothesis. the samples are random and independent.
claim: $\mu_1 < \mu_2$, $\alpha = 0.01$. sample statistics: $\bar{x}_1 = 1235$, $n_1 = 35$, $\bar{x}_2 = 1195$, and $n_2 = 55$. population parameters: $\sigma_1 = 75$ and $\sigma_2 = 100$.

(a) the test statistic for $\mu_1 - \mu_2$ is 40.

(b) the standardized test statistic for $\mu_1 - \mu_2$ is 2.16.
(round to two decimal places as needed.)

(c) is the standardized test statistic in the rejection region?

• no
• yes

(d) should you reject or fail to reject the null hypothesis?
$h_0: \mu_1 \geq \mu_2$; $h_a: \mu_1 < \mu_2$ $h_0$. at the 1% significance level, there is evidence to support the claim.

Explanation:

Step1: Determine the critical value

For a left - tailed test with \(\alpha = 0.01\) and the population standard deviations known (z - test), the critical value \(z_{\alpha}\) is found from the standard normal distribution. The critical value for \(\alpha=0.01\) (left - tailed) is \(z=- 2.33\) (since \(P(Z < - 2.33)\approx0.01\)).

Step2: Compare standardized test statistic with critical value

The standardized test statistic is \(z = 2.16\). For a left - tailed test, the rejection region is \(z < - 2.33\). Since \(2.16>-2.33\), the standardized test statistic is not in the rejection region.

Step3: Make a decision about the null hypothesis

Since the standardized test statistic is not in the rejection region, we fail to reject the null hypothesis \(H_0:\mu_1=\mu_2\) (the original null hypothesis should be \(H_0:\mu_1=\mu_2\) and \(H_a:\mu_1 < \mu_2\) for the claim \(\mu_1<\mu_2\)). At the \(1\%\) significance level, since we fail to reject \(H_0\), there is not enough evidence to support the claim.

Answer:

(a) The test statistic for \(\mu_1-\mu_2\) is \(\boldsymbol{40}\).
(b) The standardized test statistic for \(\mu_1 - \mu_2\) is \(\boldsymbol{2.16}\).
(c) The standardized test statistic is not in the rejection region, so the answer is \(\boldsymbol{No}\).
(d) We fail to reject \(H_0\). At the \(1\%\) significance level, there is not enough evidence to support the claim. The first dropdown (for the decision on \(H_0\)) should be "Fail to reject", and the second dropdown (for evidence) should be "not enough".