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 | 29 | 36 |
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: pA = 0.12, pB = 0.23, pC = 0.29, pD = 0.36.
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.
H0: pA = pB = pC = pD = 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.
a)H0: pA = pB = pC = pD = 1/4.
b)
| relative | observed | Expected | residual | Chi square | |
| category | frequency(p) | Oi | Ei=total*p | R2i=(Oi-Ei)/√Ei | R2i=(Oi-Ei)2/Ei |
| 1 | 1/4 | 12.000 | 25.00 | -2.60 | 6.760 |
| 2 | 1/4 | 23.000 | 25.00 | -0.40 | 0.160 |
| 3 | 1/4 | 29.000 | 25.00 | 0.80 | 0.640 |
| 4 | 1/4 | 36.000 | 25.00 | 2.20 | 4.840 |
| total | 1.000 | 100 | 100 | 12.4000 | |
| test statistic X2 = | 12.400 | ||||
c) p value =0.0061
d) reject H0
e)
The data supports the claim that correct answers for all multiple-choice questions are not evenly distributed.
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