Survey results show that 52.3% of men and 58.7% of women have gym memberships. Is the difference in proportions significant? Using a calculated test statistic value of z = -1.52 and P-value = 0.1286, choose the most correct conclusion (use α = 0.05).

Question 6 options:

Reject H0 since the P-value < 0.05 – a significantly greater proportion of women than men have gym memberships

Fail to reject H0 since P-value > 0.05 – there appears to be no significant difference in the proportions of men and women who have gym memberships

Fail to reject H0 since P-value < 0.05 – there appears to be a significant difference in the proportions of men and women who have gym memberships

Reject H0 since P-value > 0.05 – there appears to be a significant difference in the proportions of men and women who have gym memberships

Reject Ha since P-value > 0.05 – there appears to be no significant difference in the proportions of men and women who have gym memberships

To determine the correct conclusion, let's analyze the given information:

- **Significance Level (\(\alpha\))**: 0.05
- **Test Statistic**: \(z = -1.52\)
- **P-value**: 0.1286

### Hypothesis Setup:
- **Null Hypothesis (\(H_0\))**: There is no difference in the proportions of men and women who have gym memberships.
- **Alternative Hypothesis (\(H_1\))**: There is a difference in the proportions of men and women who have gym memberships.

### Decision Rule:
- If the P-value is less than \(\alpha\) (0.05), we reject the null hypothesis.
- If the P-value is greater than \(\alpha\) (0.05), we fail to reject the null hypothesis.

### Conclusion:
- Given that the P-value (0.1286) is greater than the significance level (\(\alpha = 0.05\)), we **fail to reject** the null hypothesis.
- This indicates that there is **no significant difference** in the proportions of men and women who have gym memberships based on the data provided.

### Most Correct Conclusion:
**Fail to reject H0 since P-value > 0.05 – there appears to be no significant difference in the proportions of men and women who have gym memberships.**

Would you like any further details or explanations?

---

**Related Questions:**
1. What is the importance of the significance level in hypothesis testing?
2. How would the conclusion change if \(\alpha\) was 0.10 instead of 0.05?
3. What does the z-value of -1.52 tell us about the difference in proportions?
4. If a different test statistic was used, how could the result differ?
5. How does the sample size affect the outcome of a hypothesis test?

**Tip:** Always compare the P-value directly to the significance level (\(\alpha\)) to determine whether to reject or fail to reject the null hypothesis.

Understand the significance of differences in gym membership proportions between men and women through hypothesis testing. Analyze a scenario with calculated test statistic and P-value, determining the conclusion with α = 0.05.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation