Question

Some students have suggested that if you have to guess on a multiple-choice question, you should always choose C. Carl, the student, wants to investigate this theory. He is able to get a sample of past tests and quizzes from various teachers. In this sample there are 90 multiple-choice questions with four options (A, B, C, D). The distribution of correct answers from this sample is given in the frequency table below.

Correct Answer Frequency:

A 19

B 23

C 27

D 21

(a) If the correct answers for all multiple-choice problems are uniformly distributed across the four options (A, B, C, D), what is the theoretical proportion of those which should have the answer C? Express your answer as an exact decimal, not a percentage.

(b) Based on the sample that Carl collected, what is the point estimate for the proportion of all multiple-choice questions with a correct answer of C? Round your answer to 3 decimal places.

(c) Construct the 95% confidence interval for the proportion of all multiple-choice questions with a correct answer of C? Round your answers to 3 decimal places.

(d) Can Carl be 95% confident that the correct answer of C shows up more frequently than the theoretical value found in part (a) would suggest?

a. No, because 0.25 is within the confidence interval limits.

b. Yes, because 0.25 is below the lower limit of the confidence interval.

c. No, because 0.25 is below the lower limit of the confidence interval.

d. Yes, because 0.25 is within the confidence interval limits.

Answer 1

(a) If the correct answers for all multiple-choice problems are uniformly distributed across the four options (A, B, C, D), what is the theoretical proportion of those which should have the answer C? Express your answer as an exact decimal, not a percentage.

1/4 = 0.25

(b) Based on the sample that Carl collected, what is the point estimate for the proportion of all multiple-choice questions with a correct answer of C? Round your answer to 3 decimal places.

27/90 = 0.3

(c) Construct the 95% confidence interval for the proportion of all multiple-choice questions with a correct answer of C? Round your answers to 3 decimal places.

Observed
0.3	p (as decimal)
27/90	p (as fraction)
27.	X
90	n
0.0527	std. error
0.205	confidence interval 95.% lower
0.395	confidence interval 95.% upper
0.0947	margin of error

(0.205, 0.395)

(d) Can Carl be 95% confident that the correct answer of C shows up more frequently than the theoretical value found in part (a) would suggest?

d. Yes, because 0.25 is within the confidence interval limits.