This problem involves conducting a hypothesis test for the standard deviation of a normally distributed population. Specifically, we will use a chi-square test for standard deviation to determine if the standard deviation of the piston diameters has significantly decreased after recalibration.

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Step 1: Define the hypotheses

• Null Hypothesis ($H_0$): The population standard deviation remains the same, $\sigma = 0.004$.
• Alternative Hypothesis ($H_a$): The population standard deviation has decreased, $\sigma < 0.004$.

This is a left-tailed test because we are testing if the standard deviation has decreased.

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Step 2: Test statistic

The test statistic for a chi-square test for standard deviation is given by:

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

Where:

• $n = 28$ (sample size)
• $s = 0.0027$ (sample standard deviation)
• $\sigma_0 = 0.004$ (hypothesized population standard deviation)

First, calculate the degrees of freedom ($df$):

$df = n - 1 = 28 - 1 = 27$

Next, calculate $s^2$ and $\sigma_0^2$:

$s^2 = (0.0027)^2 = 0.00000729,\ \ \sigma_0^2 = (0.004)^2 = 0.000016$

Now, compute $\chi^2$:

$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2} = \frac{(28-1)(0.00000729)}{0.000016} = \frac{27 \times 0.00000729}{0.000016} \approx 12.29$

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Step 3: Critical value

We need the critical value for a left-tailed test at $\alpha = 0.05$ with $df = 27$. Using a chi-square distribution table or calculator:

$\chi^2_{\text{critical}} = \chi^2_{0.05, 27} = 16.151$

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Step 4: Decision rule

• If $\chi^2 < \chi^2_{\text{critical}}$, reject $H_0$.
• If $\chi^2 \geq \chi^2_{\text{critical}}$, fail to reject $H_0$.

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Step 5: Conclusion

The computed test statistic is $\chi^2 = 12.29$, and the critical value is $\chi^2_{\text{critical}} = 16.151$. Since $12.29 < 16.151$, we reject the null hypothesis.

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Final Conclusion

At the $\alpha = 0.05$ level of significance, there is significant evidence to conclude that the standard deviation of the piston diameters has decreased after recalibration.

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the chi-square distribution different from the normal distribution?
2. What are the implications of rejecting the null hypothesis in this case?
3. How would the test change if this were a two-tailed test instead of a left-tailed test?
4. What role does the sample size ($n$) play in this type of hypothesis testing?
5. How would increasing the significance level ($\alpha$) affect the test's outcome?

Tip: Always ensure the population is approximately normal before using the chi-square test for standard deviation.