Question

Arthur is conducting a study on the preferred study options of students from East County College. He randomly selected 32 students from the college, and found that 25% of those surveyed preferred studying abroad. Assuming a 95% confidence level, which of the following statements holds true?
A. As the sample size is too small, the margin of error cannot be trusted.
B. As the sample size is too small, the margin of error is ±0.15.
C. As the sample size is appropriately large, the margin of error is ±0.178.
D. As the sample size is appropriately large, the margin of error is ±0.15.

Answer 1

The sample size of the confidence interval test of a proportion is considered appropriately large if np and n(1 - p) > 5.

Given that Arthur is conducting a study on the preferred study options of students from East County College and that 25% of those surveyed preferred studying abroad. Thus, p = 25% = 0.25 and 1 - 0.25 = 0.75

np = 32(0.25) = 8

n(1 - p) = 32(0.75) = 24

Thus, the sample size is appropriately large.

The margin of error is given by:

$\pm z_{ \alpha /2} \sqrt{ \frac{p(1-p)}{n} }$

The $z_{ \alpha /2}$ for 95% confidence interval is 1.96

Thus, the margin of error is given by:

$\pm1.96 \sqrt{ \frac{0.25(1-0.25)}{32} } =\pm1.96 \sqrt{\frac{0.25(0.75)}{32}} \\ \\ =\pm1.96 \sqrt{ \frac{0.1875}{32} } =\pm1.96 \sqrt{0.005859375} \\ \\ =\pm1.96(0.0765)=\pm0.15$

Therefore, the statement that holds true is "as the sample size is appropriately large, the margin of error is ±0.15".