SOLUTION: The weight of a drug produced by a pharmaceutical company follows normal distribution. The specified variances of the weight of the drug of this population is 0.36KG. The quality e

Question 1193379: The weight of a drug produced by a pharmaceutical company follows normal distribution. The specified variances of the weight of the drug of this population is 0.36KG. The quality engineer of the firm claims that the variance of the weight of the drug does not differ significantly from the specified variance of the weight of the drug of the population. So , the purchase manager of a hospital who places order for the drug with the pharmaceutical company has selected a random sample of 15 drugs. The variance of the weight of the sample is found to be 0.05KG. Verify the intuition of the quality engineer of the pharmaceutical company at a significant level of 0.05.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**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 not equal to the specified population variance.
* σ² ≠ 0.36 kg²
**2. Determine Test Statistic**
* We will use the **chi-square test statistic** for testing the variance of a normally distributed population.
* The test statistic is calculated as:
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**
* **Significance Level (α):** 0.05
* **Degrees of Freedom (df):** n - 1 = 15 - 1 = 14
* **Critical Values:**
* Find the critical values from the chi-square distribution table for α/2 = 0.025 and 1 - α/2 = 0.975 with 14 degrees of freedom.
* Using a statistical software or table, we find:
* Lower Critical Value (χ²_lower) ≈ 5.629
* Upper Critical Value (χ²_upper) ≈ 26.119
**5. Decision Rule**
* **Reject H0** if the calculated chi-square statistic (χ²) falls outside the critical region (i.e., below χ²_lower or above χ²_upper).
* **Fail to reject H0** if the calculated chi-square statistic (χ²) falls within the critical region.
**6. Make a Decision**
* Our calculated χ² value (1.94) is less than the lower critical value (5.629).
* Therefore, we **reject the null hypothesis (H0)**.
**Conclusion**
* There is sufficient evidence at the 0.05 significance level to conclude that the variance of the drug weight in the sample **differs significantly** from the specified population variance.
**Interpretation**
* 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.**
* This suggests that the drug manufacturing process might be experiencing more variability in drug weight than expected.
**Note:**
* This analysis assumes that the sample is randomly selected and that the drug weights are normally distributed.
* If these assumptions are not met, the results of the test may not be valid.
I hope this comprehensive explanation is helpful! Let me know if you have any further questions.

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