Question 1193541
**1. Set up Hypotheses**

* **Null Hypothesis (H0):** The variance of the drug weight in the sample is equal to the specified population variance. 
    * σ² = 0.36 kg² 

* **Alternative Hypothesis (H1):** The variance of the drug weight in the sample is different from the specified population variance.
    * σ² ≠ 0.36 kg²

**2. Test Statistic**

* We will use the chi-square test statistic for this hypothesis test:

   χ² = (n - 1) * s² / σ² 

   where:
     * n is the sample size (15)
     * s² is the sample variance (0.05 kg²)
     * σ² is the population variance (0.36 kg²)

**3. Calculate Test Statistic**

χ² = (15 - 1) * 0.05 / 0.36 
χ² = 14 * 0.05 / 0.36
χ² ≈ 1.94

**4. Determine Critical Values**

* **Degrees of Freedom:** df = n - 1 = 15 - 1 = 14
* **Significance Level:** α = 0.05
* **Two-tailed test:** We need to find the critical values for both tails of the chi-square distribution.

* **Find critical values using a chi-square table or statistical software:**
    * Lower critical value (χ²_lower) ≈ 5.629
    * Upper critical value (χ²_upper) ≈ 26.119

**5. Decision Rule**

* If the calculated chi-square statistic (χ²) falls within the critical region (below χ²_lower or above χ²_upper), we reject the null hypothesis.
* If the calculated chi-square statistic falls within the acceptance region (between χ²_lower and χ²_upper), we fail to reject the null hypothesis.

**6. Make a Decision**

* Our calculated χ² (1.94) is less than the lower critical value (5.629). 

* **Conclusion:** We reject the null hypothesis. 

**Interpretation**

The evidence suggests that the variance of the drug weight in the sample is significantly different from the specified population variance at the 0.05 significance level. 

**Therefore, the quality engineer's claim that the variance of the drug weight does not differ significantly from the specified variance is not supported by the sample data.**