We are performing a hypothesis test for the population mean $\mu$, with the null hypothesis $H_0: \mu = 115$ and the alternative hypothesis $H_A: \mu \neq 115$. This is a two-tailed test since the alternative hypothesis specifies that $\mu$ is not equal to 115.

Given information:

• Sample mean ($\overline{x}$) = 108
• Population standard deviation ($\sigma$) = 25
• Sample size ($n$) = 42
• Significance level ($\alpha$) = 0.01
• $H_0: \mu = 115$
• $H_A: \mu \neq 115$

a. State the decision rule in terms of the critical values of the test statistic.

Since we know the population standard deviation, we will use a z-test (normal distribution) to calculate the test statistic. For a two-tailed z-test, the decision rule is based on the critical values of the standard normal distribution corresponding to the significance level $\alpha$.

Step 1: Find the critical values

Since this is a two-tailed test, we split the significance level $\alpha = 0.01$ into two equal parts (for the two tails), so each tail has a probability of $\frac{\alpha}{2} = 0.005$.

From the z-table, the critical z-values for $\alpha/2 = 0.005$ are:

• $z_{\text{critical}} = \pm 2.576$

Step 2: Decision rule

The decision rule is:

• Reject the null hypothesis if the calculated value of the test statistic (z) is not contained within the interval $[-2.576, 2.576]$.
• Otherwise, do not reject the null hypothesis.

Summary:

• Reject $H_0$ if $z$ is not contained within the interval $[-2.576, 2.576]$.
• Do not reject $H_0$ if $z$ is contained within this interval.

Would you like to proceed with the next parts of the question? If yes, we can continue to calculate the test statistic and make a decision based on this rule.

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Follow-up questions:

1. What is the formula to compute the z-test statistic in a hypothesis test for the mean?
2. How would the decision rule change if this were a one-tailed test?
3. What are the implications of changing the significance level $\alpha$ to 0.05 in a two-tailed test?
4. How does the sample size affect the width of the confidence interval in hypothesis testing?
5. Can you explain how to interpret a z-score in the context of this hypothesis test?

Tip:

Always check whether you're dealing with a one-tailed or two-tailed test, as it directly affects the decision rule and critical values.