Question

Grade distribution: A statistics teacher claims that, on the average, 10% of her students get a grade of A, 24% get a B, 38% get a C, 18% get a D, and 10% get an F. The grades of a random sample of 100 students were recorded. The following table presents the results. A. Compute the expected frequencies B. List all of the grades which were given more often than expected separate by commas. The grades which were given more often than expected are: List all of the grades which were given less then expected, separate by commas. C. What is the value of x^2? D. How many degrees of freedom are there. E. State the null and alternate hypotheses. Below PA represents the probability of earning an A. H0= A=,B=,C=,D=,F=. tHE HYPOTHESIS TEST IS A = Part 7: Find the critical value of using 0.05 level of significance. Part 8: Determine whether to reject H0 Part 9: State a conclusion.

Grade	Observed
A	3
B	23
C	44
D	24
F	6

Answer 1

As per the claims of the teacher, the probabilities are

• probability of earning an A : P(A) = 0.10
• probability of earning a B : P(B) = 0.24
• probability of earning a C : P(C) = 0.38
• probability of earning a D : P(D) = 0.18
• probability of earning an F : P(F) = 0.10

a) The total number of students in the sample is 100. The expected number of students getting a grade is

Hence the expected frequencies are

Grade	Observed Frequency (fo)	Probability	Expected frequency (fe)
A	3	0.1	0.1*100=10
B	23	0.24	0.24*100=24
C	44	0.38	0.38*100=38
D	24	0.18	0.18*100=18
F	6	0.1	0.1*100=10

ans: Expected frequencies are

Grade	Observed Frequency (fo)	Expected frequency (fe)
A	3	10
B	23	24
C	44	38
D	24	18
F	6	10

b) We will find the difference between observed and expected frequencies as below

Grade	Observed Frequency (fo)	Expected frequency (fe)	(observed-expected) - (fo-fe)
A	3	10	-7
B	23	24	-1
C	44	38	6
D	24	18	6
F	6	10	-4

The positive valued differences indicate that the observed grades are given more often than expected

The grades which were given more often than expected are: C,D

The negative valued differences indicate that the observed grades are given less often than expected

The grades which were given less than expected are: A,B,F

c) The chi-square value is calculated as

ans: The value of is 9.4890

The number of groups (grades) is 5. Hence the degrees of freedom is 5-1=4

ans: The degrees of freedom is 4

E) The hypotheses are

The hypothesis test is a chi-square goodness of fit test

part 7) Using the chi-square table for degrees of freedom df=4 and the area under the right tail = 0.05 we get the critical value=9.488

ans: the critical value of using 0.05 level of significance is 9.488

Part 8) We will reject the null hypothesis if the test statistics is greater than the critical value.

Here, the test statistics is 9.4890 and it is greater than the critical value 9.488. Hence we reject the null hypothesis.

ans: Reject

Part 9) At 0.05 level of significance, there is not sufficient evidence to support the claim of statistics teacher that, on the average, 10% of her students get a grade of A, 24% get a B, 38% get a C, 18% get a D, and 10% get an F.