Question

In order to determine what proportion of desks in each classroom should be left handed, the administration wants to know what proportion of students at Hope College are left-handed. In a preliminary sample of 12 students, 2 of them are left handed.

(A) If they want the quick-and-dirty 95% confidence interval for the true proportion of all students who are left-handed to have a margin of error of less than 3%, about how large of a sample will they need?

(B) If they use the sample size in (a) will the confidence interval necessarily have a margin of error less than 3%. Choose the best answer out of the following four options:

(i) Yes, it will necessarily have a margin of error less than 3%.

(ii) No, it will not necessarily have a margin of error less than 3% because the proportion of students in the sample who are left-handed will not necessarily be equal to the proportion we used to estimate the sample size.

(iii) No, it will not necessarily have a margin of error less than 3% because the proportion of students at Hope College who are left-handed is not necessarily equal to the proportion we used to estimate the sample size.

(iv) No, it will not necessarily have a margin of error less than 3% because the quick-and-dirty 95% confidence interval is only an approximate 95% confidence interval.

Answer 1

A) p = 2/12 = 0.167

At 95% confidence level, the critical value is z0.025 = 1.96

Margin of error = 0.03

Or, z0.025 * sqrt(p(1 - p)/n) = 0.03

Or 1.96 * sqrt(0.167 * (1 - 0.167)/n) = 0.03

Or, n = (1.96 * sqrt(0.167 * 0.833)/0.03)^2

Or, n = 594

B) ii) No, it will not necessarily have a margin of error less than 3% because the proportion of students in the sample who are left-handed will not necessarily be equal to the proportion we used to estimate the sample size.