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	24	30	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) What is the null hypothesis for this test in terms of the probabilities of the outcomes?

H₀:  p_A = p_B = p_C = p_D = 1/4.H₀:  p_A = 0.12, p_B = 0.24, p_C = 0.30, p_D = 0.34.     H₀: At least one of the probabilities doesn't equal 1/4. H₀: The probability of a correct answer being C is greater than the others.

(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 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

a)H₀:  p_A = p_B = p_C = p_D = 1/4.

b)Applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
A	0.250	12	25.00	-2.60	6.760
B	0.250	24	25.00	-0.20	0.040
C	0.250	30	25.00	1.00	1.000
D	0.250	34	25.00	1.80	3.240
total	1.000	100	100		11.040

value of the test statistic X² =11.040

c)

p value =0.0115

d)fail to reject H₀     

e)

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