Chapter 6: Q17 (page 264)

SAT and ACT Tests Because they enable efficient procedures for evaluating answers, multiple choice questions are commonly used on standardized tests, such as the SAT or ACT.
Such questions typically have five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of “a.”
a. After listing the 25 different possible samples, find the proportion of correct answers in each sample, then construct a table that describes the sampling distribution of the sample proportions of correct responses.
b. Find the mean of the sampling distribution of the sample proportion.
c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of correct responses? Does the mean of the sampling distribution of proportions always equal the population proportion?

Short Answer

Expert verified

a. The following table is constructed describing the sampling distribution of the sample proportions of correct answers:

Sample proportion	Probability
0	$\frac{16}{25}$
0.5	$\frac{8}{25}$
1	$\frac{1}{25}$

b. The mean of the sampling distribution of sample proportion is equal to 0.2.

c. The mean of the sampling distribution of sample proportion of correct answers is equal to the population proportion of correct answers. Yes, the mean of the sampling distribution of sample proportions is always equal to the population proportion.

Step by step solution

Given information

2 multiple choice questions are randomly answered. Each question has 5 possible choices.

Sampling distribution of sample proportions

Let the possible choices for the two questions be a, b, c, d and e.

All possible samples for the answer choices of the 2 questions are given as follows:

aa	ab	ac	ad	ae	ba	bb	bc	bd
be	ca	cb	cc	cd	ce	da	db	dc
dd	de	ea	eb	ec	ed	ee

The correct answer for each of the two questions is “a”.

The sample proportion of correct answers for each sample has the following formula:

$\hat{p} = \frac{Number of Correct Answers}{2}$

The following table shows all possible samples of size equal to 2 and the corresponding sample proportions:

Sample	Sample proportion of girls
aa	${\hat{p}}_{1} = \frac{2}{2} = 1$
ab	${\hat{p}}_{2} = \frac{1}{2} = 0.5$
ac	${\hat{p}}_{3} = \frac{1}{2} = 0.5$
ad	${\hat{p}}_{4} = \frac{1}{2} = 0.5$
ae	${\hat{p}}_{5} = \frac{1}{2} = 0.5$
ba	${\hat{p}}_{6} = \frac{1}{2} = 0.5$
bb	${\hat{p}}_{7} = \frac{0}{2} = 0$
bc	${\hat{p}}_{8} = \frac{0}{2} = 0$
bd	${\hat{p}}_{9} = \frac{0}{2} = 0$
be	${\hat{p}}_{10} = \frac{0}{2} = 0$
ca	${\hat{p}}_{11} = \frac{1}{2} = 0.5$
cb	${\hat{p}}_{12} = \frac{0}{2} = 0$
cc	${\hat{p}}_{13} = \frac{0}{2} = 0$
cd	${\hat{p}}_{14} = \frac{0}{2} = 0$
ce	${\hat{p}}_{15} = \frac{0}{2} = 0$
da	${\hat{p}}_{16} = \frac{1}{2} = 0.5$
db	${\hat{p}}_{17} = \frac{0}{2} = 0$
dc	${\hat{p}}_{18} = \frac{0}{2} = 0$
dd	${\hat{p}}_{19} = \frac{0}{2} = 0$
de	${\hat{p}}_{20} = \frac{0}{2} = 0$
ea	${\hat{p}}_{21} = \frac{1}{2} = 0.5$
eb	${\hat{p}}_{22} = \frac{0}{2} = 0$
ec	${\hat{p}}_{23} = \frac{0}{2} = 0$
ed	${\hat{p}}_{24} = \frac{0}{2} = 0$
ee	${\hat{p}}_{25} = \frac{0}{2} = 0$

Combining the values of proportions that are the same, the following probability values are obtained:

Sample proportion	Probability
0	$\frac{16}{25}$
0.5	$\frac{8}{25}$
1	$\frac{1}{25}$

Mean of the sample proportions

The mean of the sample proportions of correct answers is computed below:

$Mean of \hat{p} = \frac{{\hat{p}}_{1} + {\hat{p}}_{2} + . . . . . + {\hat{p}}_{25}}{25} = \frac{1 + 0.5 + . . . . . . . . + 0}{25} = 0.2$

Thus, the mean of the sampling distribution of the sample proportion is equal to 0.2.

Population proportion

The population can be described as all possible choices: {a, b, c, d, e}.

The population proportion of correct answers is computed below:

$p = \frac{1}{5} = 0.2$

Thus, the population proportion of correct answers is equal to 0.2.

Comparison

Here, the population proportion of correct answers (0.2) is equal to the mean of the sampling distribution of sample proportion of correct answers (0.2).

Yes, the mean of the sampling distribution of sample proportions is always equal to the population proportion.