Question

A television sport commentator wants to estimate the proportion of citizens who "follow professional football."

a. What sample size should be obtained if he wants to be within 4 percentage points with 94% confidence if he uses an estimate of 54% obtained from a poll?

b. What sample size should be obtained if he wants to be within 4 percentage points with 94% confidence if he does not use any prior estimates?

c. Why are the results from parts a and b so close?

1. the results are close because the margin of error 4% is less than 5%

2. The results are close because 0.54(1-0.54)=0.2484 is very close to 0.25

3. the results are close because the confidence 94% is close to 100%

Answer 1

a)

Sample size = Z²/2 * p( 1 - p) / E²

= 1.8808² * 0.54 * 0.46 / 0.04²

= 549.18

Sample size = 550 (Rounded up to nearest integer)

b)

When prior estimate or proportion is not specified then p = 0.50

Sample size = Z²/2 * p( 1 - p) / E²

= 1.8808² * 0.50 * 0.50 / 0.04²

= 552.72

Sample size = 553 (Rounded up to nearest integer)

c)

The results are close because 0.54(1-0.54)=0.2484 is very close to 0.25