Question 1177272: Grades assigned by an economics lecturer for his postgraduate coursework students at a oxford have historically followed a symmetrical distribution: 5% HDs, 25% Ds, 40% Cs, 25% Ps and 5% Fs. This year, a sample of grades revealed 11 HDs, 32 Ds, 62 Cs, 29 Ps and 16 Fs. Can you conclude, at the 1% level of significance, that this year’s marks are distributed differently from marks in the past?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! We will use a chi-square goodness-of-fit test to determine if this year's grades are distributed differently from the historical distribution.
**1. Define Hypotheses:**
* **Null Hypothesis (H0):** The distribution of grades this year is the same as the historical distribution.
* **Alternative Hypothesis (H1):** The distribution of grades this year is different from the historical distribution.
**2. Set Significance Level:**
* α = 0.01
**3. Observed and Expected Frequencies:**
* Total number of students: 11 + 32 + 62 + 29 + 16 = 150
* Historical percentages: 5% HD, 25% D, 40% C, 25% P, 5% F
* Expected frequencies:
* HD: 150 * 0.05 = 7.5
* D: 150 * 0.25 = 37.5
* C: 150 * 0.40 = 60
* P: 150 * 0.25 = 37.5
* F: 150 * 0.05 = 7.5
* Observed frequencies:
* HD: 11
* D: 32
* C: 62
* P: 29
* F: 16
**4. Calculate the Chi-Square Statistic:**
* χ² = Σ [(Observed - Expected)² / Expected]
* χ² = [(11 - 7.5)² / 7.5] + [(32 - 37.5)² / 37.5] + [(62 - 60)² / 60] + [(29 - 37.5)² / 37.5] + [(16 - 7.5)² / 7.5]
* χ² = (3.5² / 7.5) + (-5.5² / 37.5) + (2² / 60) + (-8.5² / 37.5) + (8.5² / 7.5)
* χ² = (12.25 / 7.5) + (30.25 / 37.5) + (4 / 60) + (72.25 / 37.5) + (72.25 / 7.5)
* χ² = 1.6333 + 0.8067 + 0.0667 + 1.9267 + 9.6333
* χ² ≈ 14.0667
**5. Determine Degrees of Freedom:**
* Degrees of freedom (df) = number of categories - 1
* df = 5 - 1 = 4
**6. Find the Critical Chi-Square Value:**
* Using a chi-square distribution table or calculator, with df = 4 and α = 0.01, the critical chi-square value is approximately 13.277.
**7. Compare the Calculated Chi-Square and Critical Value:**
* Calculated χ² (14.0667) > Critical χ² (13.277)
**8. Make a Decision:**
* Since the calculated chi-square value is greater than the critical chi-square value, we reject the null hypothesis.
**9. Conclusion:**
* At the 1% level of significance, we can conclude that this year's marks are distributed differently from marks in the past.
Final Answer: Yes, we can conclude at the 1% level of significance that this year’s marks are distributed differently from marks in the past.
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