Questions: A test of H0: μ=53 versus H1: μ ≠ 53 is performed using a significance level of α=0.05. The value of the test statistic is z=-1.97. Determine whether to reject H0.

A test of H0: μ=53 versus H1: μ ≠ 53 is performed using a significance level of α=0.05. The value of the test statistic is z=-1.97.

Determine whether to reject H0.

Transcript text: A test of $H_{0}: \mu=53$ versus $H_{1}: \mu \neq 53$ is performed using a significance level of $\alpha=0.05$. The value of the test statistic is $z=-1.97$. Determine whether to reject $H_{0}$.

Solution

Solution Steps

Step 1: Determine the Critical Z-Values

For a two-tailed hypothesis test at a significance level of $ \alpha = 0.05 $, the critical z-values can be calculated using the formula:

\[ Z = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \]

Substituting $ \alpha = 0.05 $:

\[ Z = \Phi^{-1}(1 - 0.025) = \Phi^{-1}(0.975) \]

The critical z-values are found to be:

\[ \text{Critical z-values: } \pm 1.96 \]

Step 2: Compare the Test Statistic with Critical Values

The test statistic given is $ z = -1.97 $. We compare this value with the critical z-values:

Critical region for rejection of $ H_0 $: $ z < -1.96 $ or $ z > 1.96 $

Since $ -1.97 < -1.96 $, the test statistic falls within the critical region.

Step 3: Make a Decision

Since the test statistic $ z = -1.97 $ is in the critical region, we reject the null hypothesis $ H_0: \mu = 53 $ at the $ \alpha = 0.05 $ level.

Final Answer

We reject the null hypothesis $ H_0 $ at the $ \alpha = 0.05 $ level.

$\boxed{\text{Reject } H_0}$

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