Question 1187208: The following table gives the number of good and bad parts produced by each of three shifts in a
factory:
Shift Good Bad Total
Day. 900. 130 1030
Evening. 700. 170. 870
Night. 400. 200. 600
Total. 2000 500. 2500
Find x2 (x square) and test the null hypothesis of independence between the shift and the quality of parts
produced at 0.05 level of significance.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the chi-squared statistic (x²) and perform the hypothesis test:
**1. Calculate Expected Frequencies:**
The expected frequency for each cell in the table is calculated as:
(Row Total * Column Total) / Grand Total
* **Day - Good:** (1030 * 2000) / 2500 = 824
* **Day - Bad:** (1030 * 500) / 2500 = 206
* **Evening - Good:** (870 * 2000) / 2500 = 696
* **Evening - Bad:** (870 * 500) / 2500 = 174
* **Night - Good:** (600 * 2000) / 2500 = 480
* **Night - Bad:** (600 * 500) / 2500 = 120
**2. Calculate Chi-Squared Statistic (x²):**
x² is calculated as the sum of the squared difference between the observed and expected frequencies, divided by the expected frequency, for each cell:
x² = Σ [(Observed - Expected)² / Expected]
x² = (900-824)²/824 + (130-206)²/206 + (700-696)²/696 + (170-174)²/174 + (400-480)²/480 + (200-120)²/120
x² = (76²/824) + ((-76)²/206) + (4²/696) + ((-4)²/174) + ((-80)²/480) + (80²/120)
x² = 7.005 + 27.990 + 0.023 + 0.092 + 13.333 + 53.333
x² ≈ 101.776
**3. Degrees of Freedom:**
Degrees of freedom (df) are calculated as:
df = (Number of Rows - 1) * (Number of Columns - 1)
df = (3 - 1) * (2 - 1) = 2
**4. Critical Value:**
For a significance level of 0.05 and 2 degrees of freedom, you would consult a chi-squared distribution table. The critical value is approximately 5.991.
**5. Hypothesis Test:**
* **Null Hypothesis (H0):** The shift and the quality of parts produced are independent.
* **Alternative Hypothesis (H1):** The shift and the quality of parts produced are not independent (they are dependent).
Since our calculated x² value (101.776) is much greater than the critical value (5.991), we reject the null hypothesis.
**Conclusion:**
There is sufficient evidence at the 0.05 level of significance to conclude that the shift and the quality of parts produced are not independent. In other words, the shift seems to have an effect on the quality of parts being produced.
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