Question 1206945
**1. Set up the Hypotheses**

* **Null Hypothesis (H₀):** σ² ≥ 0.15² 
    * The population variance of the soda can volumes is greater than or equal to 0.15² (0.0225).
* **Alternative Hypothesis (Hₐ):** σ² < 0.15² 
    * The population variance of the soda can volumes is less than 0.15² (0.0225).

**2. Calculate the Test Statistic**

* The test statistic for this hypothesis test is the chi-square statistic:
   χ² = (n - 1) * (s² / σ₀²) 
   where:
       * n is the sample size (22)
       * s is the sample standard deviation (0.09)
       * σ₀ is the hypothesized population standard deviation (0.15)

* Calculate the test statistic:
   χ² = (22 - 1) * (0.09² / 0.15²) 
   χ² = 21 * (0.0081 / 0.0225) 
   χ² = 7.56

**3. Determine the P-value**

* The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. 
* Since this is a left-tailed test (Hₐ: σ² < 0.15²), we need to find the area to the left of the calculated chi-square value in the chi-square distribution with (n - 1) degrees of freedom.

* **Using a chi-square table or statistical software:**
    * Find the p-value associated with χ² = 7.56 and degrees of freedom (df) = n - 1 = 21. 
    * The p-value will be greater than 0.025 (as the test statistic falls in the right tail of the chi-square distribution).

**Therefore:**

* **A) Test Statistic:** χ² = 7.56
* **B) P-value:** p-value > 0.025

**Conclusion**

Since the p-value is greater than the significance level (0.025), we **fail to reject the null hypothesis**. There is not enough evidence to support the claim that the population standard deviation of the soda can volumes is less than 0.15 oz.