To determine if there is significant evidence that the standard deviation has decreased, we can perform a hypothesis test for the variance using the Chi-square distribution.

Step 1: Set Up Hypotheses

• Null Hypothesis ($H_0$): The standard deviation has not decreased (i.e., $\sigma = 0.006$ inch).
• Alternative Hypothesis ($H_a$): The standard deviation has decreased (i.e., $\sigma < 0.006$ inch).

Step 2: Calculate the Test Statistic

The test statistic for the variance is calculated using the formula:

$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$

Where:

• $n$ = sample size = 27
• $s$ = sample standard deviation = 0.0052 inch
• $\sigma_0$ = population standard deviation under the null hypothesis = 0.006 inch

First, we need to compute $s^2$ and $\sigma_0^2$:

• $s^2 = (0.0052)^2 = 0.00002704$
• $\sigma_0^2 = (0.006)^2 = 0.000036$

Now, substituting the values into the Chi-square formula:

$\chi^2 = \frac{(27-1)(0.00002704)}{0.000036} = \frac{26 \times 0.00002704}{0.000036}$

Calculating this:

$\chi^2 = \frac{0.00070304}{0.000036} \approx 19.5$

Step 3: Determine the Critical Value

For a one-tailed test at the $\alpha = 0.05$ level of significance with $n-1 = 26$ degrees of freedom, we can find the critical value from the Chi-square distribution table.

Using a Chi-square table or calculator, we find:

• Critical value at $df = 26$ and $\alpha = 0.05$ is approximately $14. Margins$.

Step 4: Conclusion

• Since our calculated $\chi^2$ (approximately 19.5) is greater than the critical value (approximately 14. Margins), we do not reject the null hypothesis.

Summary

• Test statistic: $\chi^2 \approx 19.5$
• Decision: There is insufficient evidence at the $\alpha = 0.05$ level to conclude that the standard deviation has decreased.

Would you like any more details or have further questions? Here are some related topics you might consider exploring:

1. What are the assumptions of the Chi-square test?
2. How to interpret the p-value in this context?
3. What would happen if the sample size were larger?
4. How can we improve the reliability of the test results?
5. What other statistical tests could be used to analyze this data?

Tip: Always check the assumptions of normality and independence before conducting hypothesis tests for variance.