Question

# In order to determine what proportion of desks in each classroom should be left handed, the...

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.

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.

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