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Question 1193541: 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 significance level of 0.05.
Answer by ElectricPavlov(122)   (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 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.**
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