Question

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	24	33	31

The Test: Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.01 significance level.

(a) What is the null hypothesis for this test in terms of the probabilities of the outcomes?

H₀: The probability of a correct answer being C is greater than the others.

H₀: At least one of the probabilities doesn't equal 1/4.     

H₀:  p_A = 0.12, p_B = 0.24, p_C = 0.33, p_D = 0.31.

H₀:  p_A = p_B = p_C = p_D = 1/4.

(b) What is the value of the test statistic? Round to 3 decimal places unless your software automatically rounds to 2 decimal places.

χ² =  

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places unless your software automatically rounds to 3 decimal places.
P-value =  

(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:

Given: 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}.

	A	B	C	D
Count	12	24	33	31

We have to test if the correct answers to multiple choice problems are evenly distributed or not.

Claim: The correct answers for all multiple-choice questions are not evenly distributed.

Level of significance = 0.01

Part a) What is the null hypothesis for this test in terms of the probabilities of the outcomes?

Evenly distributed means all frequencies are equal.

That is each option has equal chance = 1/4

Thus null hypothesis is:

H₀:  p_A = p_B = p_C = p_D = 1/4.

Part b ) What is the value of the test statistic?

Where

k = Number of categories = Number of options = 4

Oi = Observed counts

Ei = Expected counts = N / k

N = total counts = 100

Thus expected counts = 100 / 4 = 25

Thus each Ei = 25

We need to make following table:

Options	Oi : Observed Counts	Ei : Expected Count	Oi^2/Ei
A	12	25	5.760
B	24	25	23.040
C	33	25	43.560
D	31	25	38.440
	N = 100

Thus we get:

Part (c) Use software to get the P-value of the test statistic.

Use Excel command to find P-value:

=CHISQ.DIST.RT( x , df )

where x = and df = k - 1 = 4 - 1 = 3

=CHISQ.DIST.RT( 10.800 , 3 )

=0.01286

=0.0129

=0.013

Thus P-value = 0.013

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

Decision Rule: Reject H0, if P-value < 0.01 significance level, otherwise we fail to reject H0.

Since P-value = 0.013 > 0.01 significance level, 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.