Question

Suppose that 250 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 0.061.How would the margin of error change if the sample size decreased from 250 to 125 students? Assume that the proportion of students who say yes does not change significantly. a. As the sample size decreases, the margin of error remains unchanged. b. As the sample size decreases, the margin of error decreases. c. Cannot be determined based on the information provided. d. As the sample size decreases, the margin of error increases.

Answer 1

d. As the sample size decreases, the margin of error increases.

Explanation:

Hello!

The population proportion of students that are allowed to use cell phones in classrooms was estimated using a random sample of n=250 college students obtaining a margin of error of d= 0.061

The formula of the CI interval for the population proportion is:

p' ±
`Z_(1-\alpha /2)` *
`\sqrt{(p'(1-p'))/(n) }`

Where

p'= sample proportion ⇒ point estimation of the population proportion

d=
`Z_(1-\alpha /2)` *
`\sqrt{(p'(1-p'))/(n) }` ⇒ margin of error of the interval

As you can see in the formula, the sample size (n) and the margin of error (d) have an indirect relationship, this means, that when the sample size increases, the margin of error decreases and vice versa.

So if you decrease the sample size from 250 to 125 students, you will expect the margin of error to increase:

Using p'= 0.40 and Z= 1.96

d''=
`Z_(1-\alpha /2)` *
`\sqrt{(p'(1-p'))/(n) }`

d''= 1.96 *
`\sqrt{(0.4*0.6)/(125) }`

d''= 0.086

As expected, the margin of error of the CI is greater when the sample size is reduced. This means that the estimation of the population proportion is less accurate when a smaller sample is taken.

I hope this helps!