MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Analysis of Proportions of Defective Components from Three Suppliers
**Background:**
Benson Manufacturing is considering ordering electronic components from three different suppliers. The suppliers may differ in terms of quality, specifically in the proportion of defective components. To evaluate the proportion of defective components from the suppliers, Benson requested a sample shipment of 500 components from each supplier. The data on the number of defective components and the number of good components from each supplier is as follows:
| Component | Supplier A | Supplier B | Supplier C |
|------------|------------|------------|------------|
| Good | 483 | 478 | 458 |
| Defective | 17 | 22 | 42 |
**Objectives:**
1. Formulate the hypotheses to test if the proportion of defective components is equal among the three suppliers.
2. Conduct a hypothesis test using a 0.05 level of significance.
3. Use multiple comparison tests to determine if one supplier can be eliminated based on poor quality.
#### (a) Formulate the Hypotheses:
- **Null Hypothesis \( H_0 \):** All population proportions of defective components are equal.
\( p_A = p_B = p_C \)
- **Alternative Hypothesis \( H_3 \):** Not all population proportions of defective components are equal.
#### (b) Conduct the Hypothesis Test:
Using a 0.05 level of significance:
1. **Test Statistic Calculation:**
\[
\chi^2 = 13.703
\]
2. **p-value:**
\[
p\text{-value} = 0.0011
\]
Conclusion:
- Since the p-value (0.0011) is less than 0.05, we reject the null hypothesis \( H_0 \).
- We conclude that the suppliers do not provide equal proportions of defective components.
#### (c) Multiple Comparison Test:
To determine if there is an overall best supplier or if one supplier can be eliminated due to poor quality, we perform multiple comparisons using a 0.05 level of significance.
| Comparison | \( |\hat{p}_i - \hat{p}_j| \) | \( CV_{ij} \) | Significant \( \text{Diff} > CV_{ij} \) |
|------------|-------------------------------|--------------|-----------------------------------------|
| A vs. B |](https://content.bartleby.com/qna-images/question/4959d4f7-c515-4b84-8a14-00389cef5562/b3b893d5-8434-44c0-9b12-06c26160f017/o8nfk9_processed.jpeg)
Transcribed Image Text:### Analysis of Proportions of Defective Components from Three Suppliers
**Background:**
Benson Manufacturing is considering ordering electronic components from three different suppliers. The suppliers may differ in terms of quality, specifically in the proportion of defective components. To evaluate the proportion of defective components from the suppliers, Benson requested a sample shipment of 500 components from each supplier. The data on the number of defective components and the number of good components from each supplier is as follows:
| Component | Supplier A | Supplier B | Supplier C |
|------------|------------|------------|------------|
| Good | 483 | 478 | 458 |
| Defective | 17 | 22 | 42 |
**Objectives:**
1. Formulate the hypotheses to test if the proportion of defective components is equal among the three suppliers.
2. Conduct a hypothesis test using a 0.05 level of significance.
3. Use multiple comparison tests to determine if one supplier can be eliminated based on poor quality.
#### (a) Formulate the Hypotheses:
- **Null Hypothesis \( H_0 \):** All population proportions of defective components are equal.
\( p_A = p_B = p_C \)
- **Alternative Hypothesis \( H_3 \):** Not all population proportions of defective components are equal.
#### (b) Conduct the Hypothesis Test:
Using a 0.05 level of significance:
1. **Test Statistic Calculation:**
\[
\chi^2 = 13.703
\]
2. **p-value:**
\[
p\text{-value} = 0.0011
\]
Conclusion:
- Since the p-value (0.0011) is less than 0.05, we reject the null hypothesis \( H_0 \).
- We conclude that the suppliers do not provide equal proportions of defective components.
#### (c) Multiple Comparison Test:
To determine if there is an overall best supplier or if one supplier can be eliminated due to poor quality, we perform multiple comparisons using a 0.05 level of significance.
| Comparison | \( |\hat{p}_i - \hat{p}_j| \) | \( CV_{ij} \) | Significant \( \text{Diff} > CV_{ij} \) |
|------------|-------------------------------|--------------|-----------------------------------------|
| A vs. B |
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