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): σ=2\sigma = 2
  • Alternative Hypothesis (Hₐ): σ2\sigma \neq 2

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 df=n1df = n - 1, where nn is the sample size.
    • Here, n=15n = 15, so df=151=14df = 15 - 1 = 14. This is consistent with the information given.

3. Test Statistic (Chi-Square Value)

  • The formula for the chi-square test statistic for variance is: χ2=(n1)s2σ02\chi^2 = \frac{(n - 1) s^2}{\sigma_0^2} where s2s^2 is the sample variance, n1n - 1 is the degrees of freedom, and σ02\sigma_0^2 is the hypothesized population variance.
  • Given: χ2=5.2094\chi^2 = 5.2094, which aligns with the test statistic provided.

4. Significance Level (α)

  • Given: α=0.05\alpha = 0.05, which is the standard 5% significance level.

5. P-Value

  • P-value: 0.0346 is provided, which is less than the significance level α=0.05\alpha = 0.05.
    • Since p<αp < \alpha, we would reject the null hypothesis.

6. Decision

  • Based on the p-value comparison:
    • Decision: Reject H0H_0, 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:

  1. How would the results change if the significance level was 1% instead of 5%?
  2. Can you use a one-tailed test for standard deviation? Why or why not?
  3. What happens if the sample size changes to 30? How does that affect the chi-square distribution?
  4. What are the conditions that justify using the chi-square test for variance?
  5. 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