Question

Answers to Multiple-Choice Problems: A student wants to see if the correct answers to multiple choice problems are evenly distributed. She heard a rumor that if you don't know the answer, you should always pick C. In a sample of 100 multiple-choice questions from prior tests and quizzes, the distribution of correct answers are given in the table below. In all of these questions, there were four options {A, B, C, D}.

Correct Answers (n = 100)

	A	B	C	D
Count	22	25	30	23

The Test: Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.05 significance level.

(a) The table below is used to calculate the test statistic. Complete the missing cells.
Round your answers to the same number of decimal places as other entries for that column.

	Correct	Observed	Assumed	Expected
i	Answer	Frequency (Oi)	Probability (pi)	Frequency (Ei)	(Oi − Ei)2 Ei
1	A	(?)	0.25	25.0	0.360
2	B	25	(?)	25.0	0.000
3	C	30	0.25	(?)	1.000
4	D	23	0.25	25.0	(?)
Σ		n = 100			χ2 =(?)

(b) What is the value for the degrees of freedom?

(c) What is the critical value of

χ²? Use the answer found in the χ²-table or round to 3 decimal places.
t_α =  

(d) What is the conclusion regarding the null hypothesis?

reject H₀

fail to reject H₀    
(e) Choose the appropriate concluding statement.

We have proven that correct answers for all multiple-choice questions are evenly distributed.

The data supports the claim that correct answers for all multiple-choice questions are not evenly distributed.    

There is not enough data to support the claim that correct answers for all multiple-choice questions are not evenly distributed.

Answer 1

Solution:

the distribution of correct answers are given in the table below.

	A	B	C	D
Count	22	25	30	23

The Test: Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.05 significance level.

Part a) The table below is used to calculate the test statistic. Complete the missing cells.

	Correct	Observed	Assumed	Expected
i	Answer	Frequency (Oi)	Probability (pi)	Frequency (Ei)	(Oi − Ei)2 Ei
1	A	(?)	0.25	25.0	0.360
2	B	25	(?)	25.0	0.000
3	C	30	0.25	(?)	1.000
4	D	23	0.25	25.0	(?)
Σ		n = 100			χ2 =(?)

From given table of the distribution of correct answers , Observed frequency for A is 22

Thus first blank cell is 22

Second blank is Probability and we assume all frequencies are equally distributed

thus P(B) =1/4 = 0.25

Thus second blank = 0.25

Third blank is for Expected count = 100 X P(C) = 100 X 0.25 = 25.0

Fourth blank is for contribution of D in chi-square test statistic and is given by:

Thus fourth blank = 0.16

Fifth blank is total of last column = 0.360+0.000+1.000 + 0.160 = 1.520

	Correct	Observed	Assumed	Expected
i	Answer	Frequency (Oi)	Probability (pi)	Frequency (Ei)	(Oi − Ei)2 Ei
1	A	22	0.25	25.0	0.360
2	B	25	0.25	25.0	0.000
3	C	30	0.25	25.0	1.000
4	D	23	0.25	25.0	0.160
Σ		n = 100			χ2 =1.520

Part b) What is the value for the degrees of freedom?

df = k - 1 =   4 - 1 = 3

Part c) What is the critical value of χ²?

df= 3 and level of significance = 0.05

Using Chi-square critical value table we get:

critical value of χ²= 7.815

Part d) What is the conclusion regarding the null hypothesis?

Since χ² =1.520 < critical value of χ²= 7.815 , we fail to reject H0.

Part e) Choose the appropriate concluding statement.

There is not enough data to support the claim that correct answers for all multiple-choice questions are not evenly distributed.