Question 1177272
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.