Question

A television sports commentator wants to estimate the proportion of citizens who​ "follow professional​ football." Complete parts​ (a) through​ (c). ​(a) What sample size should be obtained if he wants to be within 2 percentage points with 94​% confidence if he uses an estimate of 52​% obtained from a​ poll? The sample size is nothing. ​(Round up to the nearest​ integer.) ​(b) What sample size should be obtained if he wants to be within 2 percentage points with 94​% confidence if he does not use any prior​ estimates? The sample size is nothing. ​(Round up to the nearest​ integer.) ​(c) Why are the results from parts​ (a) and​ (b) so​ close? A. The results are close because 0.52 left parenthesis 1 minus 0.52 right parenthesis equals0.2496 is very close to 0.25. B. The results are close because the confidence 94​% is close to​ 100%. C. The results are close because the margin of error 2​% is less than​ 5%.

Answer 1

Solution:

Given: A television sports commentator wants to estimate the proportion of citizens who​ "follow professional​ football."

Part a) What sample size should be obtained if he wants to be within 2 percentage points with 94​% confidence if he uses an estimate of 52​% obtained from a​ poll?

E = Margin of Error = 2% = 0.02

c = confidence level = 94%

p = 52% = 0.52

Sample Size n is given by:

Find Area = ( 1 + c ) / 2 = ( 1 + 0.94) /2 = 1.94 / 2 = 0.9700

Look in z table for Area = 0.9700 or its closest area and find z value

Area 0.9699 closest to 0.9700 and it corresponds to 1.8 and 0.08

thus Z_c = 1.88

Thus

( Note: Sample size is always rounded up. so that margin of error should not exceed specified level.)

Part b) What sample size should be obtained if he wants to be within 2 percentage points with 94​% confidence if he does not use any prior​ estimates?

If no prior​ estimates is given , then we use p = 0.5

Thus

Part c) Why are the results from parts​ (a) and​ (b) so​ close?

A. The results are close because 0.52 x ( 1 - 0.52 ) = 0.2496 is very close to 0.25.