Question 1169147
Here's how to solve this problem step-by-step:

**1. Define the Hypotheses and Significance Level:**

* Null Hypothesis (H₀): σ = 0.004 inch (The standard deviation has not decreased)
* Alternative Hypothesis (H₁): σ < 0.004 inch (The standard deviation has decreased)
* Significance level (α) = 0.01

**2. Gather the Data:**

* Sample size (n) = 21
* Sample standard deviation (s) = 0.0035 inch
* Population standard deviation (σ₀) = 0.004 inch

**3. Calculate the Chi-Square Test Statistic:**

* The formula for the chi-square test statistic is:

    χ² = (n - 1) * s² / σ₀²

* Plug in the values:

    χ² = (21 - 1) * (0.0035)² / (0.004)²
    χ² = 20 * (0.00001225) / (0.000016)
    χ² = 20 * (1225 / 1600)
    χ² = 20 * 0.765625
    χ² = 15.3125

**4. Determine the Degrees of Freedom:**

* Degrees of freedom (df) = n - 1 = 21 - 1 = 20

**5. Find the Critical Chi-Square Value:**

* Since this is a left-tailed test (H₁: σ < 0.004), we need to find the critical chi-square value from a chi-square distribution table with 20 degrees of freedom and an alpha level of 0.01.
* Using a chi-square table or calculator, the critical chi-square value is approximately 8.260.

**6. Compare the Test Statistic to the Critical Value:**

* Our calculated chi-square test statistic (15.3125) is greater than the critical chi-square value (8.260).

**7. Make a Decision:**

* Because our calculated chi-square value (15.3125) is *not* less than the critical value (8.260), we fail to reject the null hypothesis.

**8. Conclusion:**

* There is *not* significant evidence at the α = 0.01 level to conclude that the standard deviation of the piston diameters has decreased after recalibrating the production machine.

**Therefore, x to the second power is 15.3125**