Question

Suppose that 275 students are randomly selected from a local college campus to investigate the use of cell phones in classrooms. When asked if they are allowed to use cell phones in at least one of their classes, 40% of students responded yes. Using these results, with 95% confidence, the margin of error is 0.058 . How would the margin of error change if the sample size decreased from 275 to 125 students? Assume that the proportion of students who say yes does not change significantly. As the sample size decreases, the margin of error remains unchanged. Cannot be determined based on the information provided. As the sample size decreases, the margin of error increases. As the sample size decreases, the margin of error decreases.

Answer 1

Answer:

Correct option: As the sample size decreases, the margin of error increases.

Step-by-step explanation:

The (1 - α) % confidence interval for population proportion is:

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

The margin of error in this confidence interval is:

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

The sample size n is inversely related to the margin of error.

An inverse relationship implies that when one increases the other decreases and vice versa.

In case of MOE also, when n is increased the MOE decreases and when n is decreased the MOE increases.

Compute the new margin of error for n = 125 as follows:

$\\ MOE=z_{\alpha /2}\sqrt{\frac{\hat p(1-\hat p)}{n} }=1.96\times \sqrt{\frac{0.40(1-0.40)}{125} }=0.086$

*Use z-table for the critical value.

For n = 125 the MOE is 0.086.

And for n = 275 the MOE was 0.058.

Thus, as the sample size decreases, the margin of error increases.