From the image provided, we are working on a hypothesis test with the following details:

Hypotheses:

• Null hypothesis $H_0: \mu = 7$
• Alternative hypothesis $H_1: \mu > 7$

Information provided:

• Test statistic $z = 0.871$ (rounded to 3 decimal places)
• $p$-value = 0.192 (rounded to 3 decimal places)
• Level of significance $\alpha = 0.10$

Steps to complete the hypothesis test:

1. Determine one-tailed or two-tailed test:

   • Since $H_1: \mu > 7$, this is a one-tailed test.

2. Input the test statistic:

   • Test statistic $z = 0.871$.

3. Shade the area representing the $p$-value:

   • For $z = 0.871$, the area to the right represents the $p$-value (since this is a one-tailed test).

4. Input the $p$-value:

   • $p = 0.192$.

Decision Rule:

• If $p \leq \alpha$, reject $H_0$.
• If $p > \alpha$, fail to reject $H_0$.

Here, $p = 0.192$ is greater than $\alpha = 0.10$. Hence, we fail to reject $H_0$. This means there is not enough evidence to support the claim that $\mu > 7$ at the 0.10 level of significance.

Would you like further explanation or assistance with any specific step?

5 Questions for Expansion:

1. What is the relationship between $p$-value and significance level $\alpha$?
2. How is the test statistic $z$ calculated in hypothesis testing?
3. What are the assumptions required for conducting a $z$-test for the mean?
4. How would the decision change if $\alpha$ were 0.05 instead of 0.10?
5. Can you explain why this test is one-tailed rather than two-tailed?

Tip:

Always visualize the problem on a normal distribution curve to better understand the area represented by the $p$-value, especially for one-tailed and two-tailed tests.