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

Answer a)

In Chi-Square goodness of fit test, the null hypothesis assumes that there is no significant difference between the observed and the expected value.

In this case, student wants to see if the correct answers to multiple choice problems are evenly distributed. So null hypothesis as follows:

H_o: p_A = p_B = p_C = p_D = 1/4

Answer b) Chi Square Statistics is 10.80

Following the table showing calculation of Chi Square Statistics:

Answer c) P value = 0.0129

In this case degrees of freedom df = n - 1 = 4 -1 = 3

Chi Square Statistics = 10.80

P value = 0.0129 (Obtained using P Value from Chi-square Calculator Screenshot attached)

Answer d) Fail to reject H₀

In this case p value (0.0129) is greater than 0.01 significance level so we fail to reject null hypothesis H₀.

We can answer 4 parts only