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



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):** \( \sigma = 2 \)
- **Alternative Hypothesis (Hₐ):** \( \sigma \neq 2 \)

### 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 \( df = n - 1 \), where \( n \) is the sample size.
    - Here, \( n = 15 \), so \( df = 15 - 1 = 14 \). This is consistent with the information given.

### 3. Test Statistic (Chi-Square Value)
- The formula for the chi-square test statistic for variance is:
  \[
  \chi^2 = \frac{(n - 1) s^2}{\sigma_0^2}
  \]
  where \( s^2 \) is the sample variance, \( n - 1 \) is the degrees of freedom, and \( \sigma_0^2 \) is the hypothesized population variance.
- **Given:** \( \chi^2 = 5.2094 \), which aligns with the test statistic provided.

### 4. Significance Level (α)
- **Given:** \( \alpha = 0.05 \), which is the standard 5% significance level.

### 5. P-Value
- **P-value:** 0.0346 is provided, which is less than the significance level \( \alpha = 0.05 \).
    - Since \( p < \alpha \), we would reject the null hypothesis.

### 6. Decision
- Based on the p-value comparison:
  - **Decision:** Reject \( H_0 \), 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.

Test if the population standard deviation is 2, based on a sample of size 15. Use a 5% significance level. Ha: σ ≠ 2, df = 14, Chi-square distribution with df = 14, χ² = 5.2094, p-value = 0.0346, α = 0.05, Decision: Reject H₀, H₀: σ = 2

Explore a hypothesis test for population standard deviation using the chi-square distribution. This problem tests if the population standard deviation is 2 based on a sample of size 15 at a 5% significance level, with a chi-square value of 5.2094 and p-value of 0.0346, leading to a rejection of the null hypothesis.

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