Math Problem Statement
is this right Question: Test if the population standard deviation is 2, based on a sample of size 15. Use a 5% significance level. b. Ha: σ ≠ 2 c. df = 14 d. Chi-square distribution with df = 14 e. χ² = 5.2094 f. p-value = 0.0346 h. i. α = 0.05 Decision: Reject H₀ H₀: σ = 2
Solution
Let's go over the information step-by-step to verify if the question setup and solution are correct. Here's a breakdown of the hypothesis test:
1. Hypotheses
- Null Hypothesis (H₀):
- Alternative Hypothesis (Hₐ):
2. Distribution and Degrees of Freedom
- This test uses the chi-square distribution because we're testing the population standard deviation (variance).
- Degrees of Freedom (df): The formula for degrees of freedom in a chi-square test for variance is , where is the sample size.
- Here, , so . This is consistent with the information given.
3. Test Statistic (Chi-Square Value)
- The formula for the chi-square test statistic for variance is: where is the sample variance, is the degrees of freedom, and is the hypothesized population variance.
- Given: , which aligns with the test statistic provided.
4. Significance Level (α)
- Given: , which is the standard 5% significance level.
5. P-Value
- P-value: 0.0346 is provided, which is less than the significance level .
- Since , we would reject the null hypothesis.
6. Decision
- Based on the p-value comparison:
- Decision: Reject , which is consistent with the final conclusion.
Final Verification
- Everything matches up with the requirements of the hypothesis test for standard deviation:
- Correct hypotheses, degrees of freedom, test distribution, significance level, chi-square value, and decision-making based on p-value.
Conclusion
Yes, the solution appears correct as presented.
Would you like further details on any part of this process or an explanation of the chi-square distribution's use in variance testing?
Here are five related questions to expand on this topic:
- How would the results change if the significance level was 1% instead of 5%?
- Can you use a one-tailed test for standard deviation? Why or why not?
- What happens if the sample size changes to 30? How does that affect the chi-square distribution?
- What are the conditions that justify using the chi-square test for variance?
- How do you interpret the p-value in hypothesis testing for population standard deviation?
Tip: Always check the p-value against the significance level to ensure the correct decision in hypothesis testing.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Chi-Square Distribution
Population Standard Deviation
Formulas
Chi-square test statistic for variance: χ² = (n - 1) * s² / σ₀²
Theorems
Chi-Square Distribution in Variance Testing
Suitable Grade Level
College/University Level
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