Let's analyze each part of the problem based on the provided information and image.

Problem Analysis

1. Hypotheses:

   • Null Hypothesis $H_0$: $\mu = 7$
   • Alternative Hypothesis $H_1$: $\mu < 7$

   This is a one-tailed test since we are testing if the mean is less than 7.

2. Critical Value:

   • The critical value at a 0.05 significance level is given as $-1.645$ (for a one-tailed test).

3. Test Statistic:

   • The calculated test statistic is $-2.631$.

Solution Steps

Part (a)

• Since this is a one-tailed test (lower tail), we look at the left side of the distribution for the rejection region.
• Rejection Region: Values less than $-1.645$ fall into the rejection region.
• Test Statistic: $-2.631$, which is indeed less than $-1.645$.

This confirms that the test statistic $-2.631$ lies in the rejection region.

Part (b)

• Based on the information above, we select:
  • The value of the test statistic lies in the rejection region.

Part (c)

• Since the test statistic is in the rejection region, at a 0.05 significance level, we reject the null hypothesis.
• Therefore, we select:
  • The null hypothesis should be rejected.

Summary of Answers

• (a): Indicate on the graph that the rejection region is to the left of $-1.645$, and the test statistic is $-2.631$.
• (b): Select "The value of the test statistic lies in the rejection region."
• (c): Select "The null hypothesis should be rejected."

Would you like a more detailed explanation of each step or have any questions?

Related Questions

1. What would happen if the test statistic was exactly equal to the critical value?
2. How would the decision change if this was a two-tailed test?
3. What is the interpretation of a one-tailed versus a two-tailed test in hypothesis testing?
4. How does changing the significance level affect the critical value?
5. Why do we reject the null hypothesis when the test statistic falls into the rejection region?

Tip

Remember that in hypothesis testing, the rejection region depends on both the significance level and whether the test is one-tailed or two-tailed.