The distribution of final grades given by a certain biometry department in the past was 5% A’s, 20% B’s, 30% C’s, 25% D’s and 20% F’s. A new lecturer gave the following number of students for the different grades:
Grade A B C D E
Number 12 20 26 14 8
You have to test whether there is sufficient evidence to suggest that the new lecturer’s grading policy is different from that of the department. The test statistic for this hypothesis test is
The test statistic for this hypothesis test is the chi-square statistic (χ^2).
To calculate the chi-square statistic, we need to compare the observed frequencies (the number of students given each grade by the new lecturer) with the expected frequencies (the expected number of students given each grade according to the department's distribution of grades).
The expected frequencies can be calculated by multiplying the total number of students (80 in this case) by the corresponding proportions for each grade from the department's distribution (0.05 for A, 0.2 for B, 0.3 for C, 0.25 for D, and 0.2 for F).
Using these values, we can set up the following table:
Grade Observed Frequency Expected Frequency
A 12 (0.05) * 80 = 4
B 20 (0.2) * 80 = 16
C 26 (0.3) * 80 = 24
D 14 (0.25) * 80 = 20
F 8 (0.2) * 80 = 16
To calculate the chi-square statistic, we can use the formula:
χ^2 = Σ((Observed Frequency - Expected Frequency)^2 / Expected Frequency)
So, plugging in the observed and expected frequencies, we have:
χ^2 = (12-4)^2/4 + (20-16)^2/16 + (26-24)^2/24 + (14-20)^2/20 + (8-16)^2/16
Calculating this sum gives us the chi-square statistic, which we can then compare to the chi-square distribution with (number of categories - 1) degrees of freedom (in this case, 5-1 = 4) to determine whether there is sufficient evidence to suggest that the new lecturer's grading policy is different from that of the department.
To test whether the new lecturer's grading policy is different from that of the department, we can use a chi-square test. The test statistic for this hypothesis test is the chi-square statistic.
The chi-square statistic (χ²) is calculated using the formula:
χ² = Σ[(O - E)² / E]
Where:
- O represents the observed frequencies in each category (number of students for each grade given by the new lecturer)
- E represents the expected frequencies in each category (percentage of students in each grade according to the department's distribution)
First, we need to calculate the expected frequencies for each grade category based on the department's distribution. Since we are given the percentages of each grade, we can calculate the expected frequencies as follows:
Expected_A = Total_Students * Percentage_A
Expected_B = Total_Students * Percentage_B
Expected_C = Total_Students * Percentage_C
Expected_D = Total_Students * Percentage_D
Expected_E = Total_Students * Percentage_E
Given:
- Total_Students = Σ(number of students in all grades) = Σ(Number), where Σ represents summation.
Next, we can substitute the observed frequencies (O) and expected frequencies (E) into the chi-square formula and calculate the chi-square statistic.
Once we have the chi-square statistic, we can compare it to the chi-square distribution with (n - 1) degrees of freedom, where n is the number of categories (grades). In this case, since we have 5 grades, the degrees of freedom will be (5 - 1) = 4.
Finally, we can compare the calculated chi-square statistic with the chi-square critical value at the desired significance level (e.g., α = 0.05). If the calculated chi-square statistic is greater than the critical value, we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the new lecturer's grading policy is different from that of the department.
In order to test whether the new lecturer's grading policy is different from that of the department, you can use the chi-square test for goodness-of-fit. This test will compare the observed frequencies (number of students in each grade) to the expected frequencies based on the distribution of grades from the department.
To calculate the test statistic for the chi-square test, you can follow these steps:
1. State the null hypothesis (H0) and the alternative hypothesis (H1):
- Null hypothesis (H0): The new lecturer's grading policy is the same as that of the department.
- Alternative hypothesis (H1): The new lecturer's grading policy is different from that of the department.
2. Calculate the expected frequencies for each grade based on the distribution of grades given by the department. Multiply the total number of students (80 in this case) by the corresponding percentage for each grade:
- Expected number of A's = 0.05 * 80 = 4
- Expected number of B's = 0.20 * 80 = 16
- Expected number of C's = 0.30 * 80 = 24
- Expected number of D's = 0.25 * 80 = 20
- Expected number of F's = 0.20 * 80 = 16
3. Calculate the chi-square statistic:
- The chi-square statistic is calculated using the formula: chi-square = (Σ((O - E)^2 / E), where O is the observed frequency and E is the expected frequency for each grade.
- Calculate this value for each grade and sum them up:
- For grade A: ((12 - 4)^2 / 4) = 2.25
- For grade B: ((20 - 16)^2 / 16) = 0.25
- For grade C: ((26 - 24)^2 / 24) = 0.167
- For grade D: ((14 - 20)^2 / 20) = 1.2
- For grade F: ((8 - 16)^2 / 16) = 4
- Summing all these values: chi-square = 2.25 + 0.25 + 0.167 + 1.2 + 4 = 7.867
4. Determine the degrees of freedom (df) for the chi-square test:
- The degrees of freedom for this test is (number of categories - 1).
- In this case, the number of categories is 5 (A, B, C, D, F), so df = 5 - 1 = 4.
5. Find the critical value for the chi-square statistic:
- The critical value depends on the significance level (alpha) chosen for the test. Let's assume a significance level of 0.05 (5%).
- Look up the critical value for chi-square with 4 degrees of freedom and a significance level of 0.05 in a chi-square distribution table, or use statistical software.
- The critical value for this case is approximately 9.488.
6. Compare the chi-square statistic to the critical value:
- If the chi-square statistic is greater than the critical value, we reject the null hypothesis.
- If the chi-square statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
In this case, the chi-square statistic is 7.867, which is less than the critical value of 9.488. Therefore, we fail to reject the null hypothesis and conclude that there is not sufficient evidence to suggest that the new lecturer's grading policy is different from that of the department.