We are conducting a hypothesis test for the population mean $\mu$ using a two-tailed test with the following hypotheses:

• Null Hypothesis ($H_0$): $\mu = 125$
• Alternative Hypothesis ($H_A$): $\mu \neq 125$

Given Information:

• Sample mean ($\bar{x}$): 118
• Population standard deviation ($\sigma$): 26
• Sample size ($n$): 46
• Significance level ($\alpha$): 0.05

Step 1: State the test statistic formula

We use the z-test because the population standard deviation $\sigma$ is known. The formula for the z-test statistic is:

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x} = 118$
• $\mu = 125$
• $\sigma = 26$
• $n = 46$

Step 2: Determine the critical value(s)

Since this is a two-tailed test at $\alpha = 0.05$, we need to find the critical z-values for a two-tailed test with $\alpha/2 = 0.025$ in each tail.

Using a z-table or calculator, the critical z-values for $\alpha = 0.05$ are: $z_{\text{critical}} = \pm 1.96$

Step 3: Decision Rule

The decision rule is based on whether the calculated z-statistic falls within or outside the critical region.

• Reject the null hypothesis if the calculated z-statistic is not contained within the critical values, i.e., if: $z < -1.96 \quad \text{or} \quad z > 1.96$

• Do not reject the null hypothesis if the calculated z-statistic is within the range $-1.96 \leq z \leq 1.96$.

Thus, the decision rule is:

• Reject $H_0$ if $z$ is not contained within the interval $[-1.96, 1.96]$.
• Otherwise, do not reject $H_0$.

Would you like to proceed with calculating the test statistic and concluding the hypothesis test?

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Here are 5 related questions:

1. How do you calculate a z-statistic when the sample size changes?
2. What is the meaning of a two-tailed test compared to a one-tailed test?
3. How does the significance level $\alpha$ affect the critical values?
4. Why do we use a z-test when the population standard deviation is known?
5. What happens if the sample mean is exactly equal to the population mean?

Tip: Always check whether the test is one-tailed or two-tailed before determining critical values.