Answer :
Answer:
Step-by-step explanation:
Hello!
The objective is to test if the final grades for the current semester follow the grading guidelines for the final grades given for the course.
The theoretical frequencies are:
PA= 0.10; PB= 0.30; PC= 0.40; PD= 0.10; PF= 0.10
The observed frequencies are:
A: 100 students
B: 200 students
C: 100 students
D: 50 students
F: 50 students
n= 500 students
To test this you have to apply a Chi Square goodness to fit test. If the variable of interest is X: Final grade that a student received for the course, categorized: A, B, C, D, F.
The hypotheses are:
H₀: PA= 0.10; PB= 0.30; PC= 0.40; PD= 0.10; PF= 0.10
H₁: The observed frequencies do not follow the theoretical distribution
α: 0.05
This test statistic has k-1 degrees of freedom (k= number of categories of the variable) and the rejection region is one-tailed to the right:
[tex]X^2_{k-1; 1-\alpha /2}= X^2_{4; 0.975}= 11.143[/tex]
The rejection region is then X₄² ≥ 11.143
The expected frequencies for each category is calculated using the formula:
Ei= n*Pi
[tex]E_A= n*PA= 500*0.10= 50[/tex]
[tex]E_B= n*PB= 500*0.3= 150\\[/tex]
[tex]E_C= n*PC= 500*0.40= 200[/tex]
[tex]E_D= n*PD= 500*0.10= 50[/tex]
[tex]E_F= n*PF= 500*0.10= 50[/tex]
[tex]X^2= sum (\frac{(O_i-E_i)^2}{E_i})~~X^2_{k-1}[/tex]
[tex]X^2_{H_0}= \frac{(O_A-E_A)^2}{E_A} +\frac{(O_B-E_B)^2}{E_B} +\frac{(O_C-E_C)^2}{E_C} +\frac{(O_D-E_D)^2}{E_D} +\frac{(O_F-E_F)^2}{E_F}[/tex]
[tex]X^2_{H_0}= \frac{(100-50)^2}{50} +\frac{(200-150)^2}{150} +\frac{(100-200)^2}{200} +\frac{(50-50)^2}{50} +\frac{(50-50)^2}{50}= \frac{350}{3}= 116.67[/tex]
The value of the statistic is greater than the critical value, the decision is to reject the null hypothesis.
At a 5% significance level, there is significant evidence to reject the null hypothesis, so you can conclude that the distribution of the final grades of this semesters course doesn't follow the grading guidelines.
I hope this helps!