Question 1169146
Here's how to solve this hypothesis test for the standard deviation:

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

* Null Hypothesis (H₀): σ = 0.15 oz (The population standard deviation is 0.15 oz)
* Alternative Hypothesis (H₁): σ < 0.15 oz (The population standard deviation is less than 0.15 oz)
* Significance level (α) = 0.05

**2. Gather the Data:**

* Sample size (n) = 19
* Sample mean (x̄) = 12.19 oz (This is actually irrelevant for the chi-square test for standard deviation)
* Sample standard deviation (s) = 0.08 oz
* Population standard deviation (σ₀) = 0.15 oz (from the null hypothesis)

**3. Calculate the Chi-Square Test Statistic (χ²):**

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

    χ² = (n - 1) * s² / σ₀²

* Plug in the values:

    χ² = (19 - 1) * (0.08)² / (0.15)²
    χ² = 18 * 0.0064 / 0.0225
    χ² = 18 * (64/225)
    χ² = 18 * 0.284444...
    χ² = 5.12

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

* Degrees of freedom (df) = n - 1 = 19 - 1 = 18

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

* Since this is a left-tailed test (H₁: σ < 0.15), we need to find the critical chi-square value from a chi-square distribution table with 18 degrees of freedom and an alpha level of 0.05.
* Using a chi-square table or calculator, the critical chi-square value is approximately 9.390.

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

* Our calculated chi-square test statistic (5.12) is less than the critical chi-square value (9.390).

**7. Make a Decision:**

* Because our calculated chi-square value (5.12) falls within the critical region (it is less than the critical value), we reject the null hypothesis.

**8. Conclusion:**

* There is significant evidence at the α = 0.05 level to conclude that the population standard deviation of the soda drink volumes is less than 0.15 oz.

**Therefore, x to the second power is 5.12**