Question

Multiple Choice Strategy: 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	16
B	20
C	37
D	17

(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) What is the critical value of z (denoted z_α/2) for a 99% confidence interval? Use the value from the table or, if using software, round to 2 decimal places.
z_α/2 =

(d) What is the margin of error (E) for a 99% confidence interval? Round your answer to 3 decimal places.
E =

(e) Construct the 99% confidence interval for the proportion of all multiple-choice questions with a correct answer of C? Round your answers to 3 decimal places.


(f) Can Carl be 99% confident that the correct answer of C shows up more frequently than the theoretical value found in part (a) would suggest?

No, because 0.25 is within the confidence interval limits.Yes, because 0.25 is below the lower limit of the confidence interval.     No, because 0.25 is below the lower limit of the confidence interval.Yes, because 0.25 is within the confidence interval limits.

Answer 1

a) Given that all the answer choices are equally likely, the hypothesized proportion for the test is computed here as:
= 1 / Total answer choices

= 1/ 4 = 0.25

Therefore 0.25 is the required point estimate for the proportion of all multiple-choice questions with a correct answer of C.

b) The point estimate for the sample proportion here is computed as:
= Total frequency who answered C / Total frequency

= 37/90

= 0.4111

Therefore 0.4111 is the required proportion here.

c) For 99% confidence level, we have from the standard normal tables, we have here:
P( -2.576 < Z < 2.576) = 0.99

Therefore the critical value here is given as:

d) the margin of error here is computed as:

Therefore 0.1176 is the required margin of error here.

e) The confidence interval for the proportion here is computed as:

This is the required 99% confidence interval here.

f) As the whole confidence interval here lies above 0.25, therefore the test is significant here and we can reject the null hypothesis here. therefore we have sufficient evidence here that C shows up more frequently than the theoretical value