An administrator at a large university with 12,000 enrolled students wants to evaluate stu… — SAT

An administrator at a large university with 12,000 enrolled students wants to evaluate student satisfaction with campus services. The administrator designs two separate surveys:

* Survey A: A survey is sent to a random sample of 1,000 students selected from the registrar's list of all 12,000 enrolled students. In this sample, 60% of the students report being satisfied with campus services.
* Survey B: A survey is sent to a random sample of 250 students selected from the 3,000 students who live in on-campus dormitories. In this sample, 70% of the students report being satisfied with campus services.

Which of the following statements must be true?

I. The results of Survey B can be generalized to estimate the satisfaction of all 12,000 enrolled students at the university.
II. If the administrator wants to reduce the margin of error of Survey A to approximately half of its current value, the sample size of Survey A should be increased to 4,000 students.
III. Based on the results of Survey B, it is estimated that 2,100 students who live in on-campus dormitories are satisfied with campus services.

A
II only
II and III onlyAnswer
C
I and II only
D
I, II, and III

Answer

II and III only

The correct option is 'II and III only'. Statement II is true because the margin of error is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error by half (a factor of

\frac{1}{2}

), the sample size must be multiplied by

2^2 = 4

. Thus, increasing the sample size of Survey A from 1,000 to 4,000 students will halve its margin of error. Statement III is true because Survey B selected a random sample of 250 students from the population of 3,000 students living in on-campus dormitories. Since the sample is representative of this population, the sample proportion of 70% can be generalized to the 3,000 dormitory residents, yielding an estimate of

0.70 \times 3,000 = 2,100

satisfied students. Statement I is false because the sample for Survey B was drawn exclusively from students living in on-campus dormitories, who may have different satisfaction levels than off-campus students; therefore, the results cannot be generalized to all 12,000 enrolled students.

Step-by-Step Solution

Evaluate Statement I by checking the target population of Survey B's sampling frame.

Statement I is false.

Survey B's sample was drawn specifically from the 3,000 students living in on-campus dormitories, not from the entire student body of 12,000. Therefore, the findings of Survey B can only be generalized to on-campus dormitory residents, not to the entire university.

Evaluate Statement II by analyzing the relationship between sample size and margin of error.

Statement II is true.

The margin of error for a sample proportion is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error to

\frac{1}{2}

of its current value, the sample size must be increased by a factor of

2^2 = 4

. Since the original sample size of Survey A is 1,000, increasing it to

1,000 \times 4 = 4,000

will halve the margin of error.

Evaluate Statement III by calculating the estimated population value from the sample proportion.

Statement III is true.

Since Survey B used a random sample of the 3,000 dormitory residents, the sample proportion of 70% (

0.70

) can be used to estimate the number of satisfied dormitory residents in the population:

0.70 \times 3,000 = 2,100

students.

Combine the evaluations to select the correct option.

Only Statement II and Statement III must be true.

Since Statement I is false and both Statement II and Statement III are true, the correct choice is the option containing 'II and III only'.

Key Concept

Generalization limits of survey results based on sampling frames, and the mathematical relationship between sample size and margin of error.

Estimated Time:2m 30s

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Step-by-Step Solution

Key Concept

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