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	12	23	31	34

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	6.760
2	B	23		25.0	0.160
3	C	31	0.25		1.440
4	D	34	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 H₀fail 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

(b) There are 4 categories. So, degrees of freedom = 4-1 = 3

(c) Critical value = 7.815 (From chi-square table).

(d) Since it is observed that chi-square = 11.6 > critical value = 7.815, it is then concluded that the null hypothesis is rejected.

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