To analyze this hypothesis test, let’s address each part of the problem based on the provided image:

Key Details:

• Hypotheses:

  • Null hypothesis ($H_0$): $\mu = 5$
  • Alternative hypothesis ($H_1$): $\mu \neq 5$ (Two-tailed test)

• Significance level ($\alpha$): 0.10

• Test statistic: $1.873$

• p-value: $0.061$

Steps to Complete:

Step 1: Select one-tailed or two-tailed

The problem involves testing whether $\mu \neq 5$, which corresponds to a two-tailed test.

Step 2: Enter the test statistic

The test statistic, rounded to three decimal places, is 1.873.

Step 3: Shade the area represented by the p-value

Since this is a two-tailed test, the shaded regions are the two extreme ends of the normal curve, corresponding to a total area (p-value) of 0.061. Each tail contains half of this probability ($0.0305$ in each tail).

Step 4: Enter the p-value

The p-value is 0.061.

Decision Based on p-value:

• If $p \leq \alpha$ ($0.061 \leq 0.10$), reject the null hypothesis.
• Since $p = 0.061$ is less than 0.10, we reject $H_0$ at the 0.10 level of significance, suggesting there is enough evidence to support that the population mean is different from 5.

Would you like more clarification on any step? Here are 5 related questions for further exploration:

1. How is the p-value calculated for two-tailed tests using the test statistic?
2. What does rejecting the null hypothesis mean in practical terms?
3. How does the significance level affect hypothesis testing decisions?
4. Could the conclusion change if $\alpha = 0.05$?
5. What are Type I and Type II errors in the context of this test?

Tip: Always check whether the test is one-tailed or two-tailed before shading or interpreting the p-value.