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 54​% 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 0.54 left parenthesis 1 minus 0.54 right parenthesis equals0.2484 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

z score for 94% confidence interval using the z distribution table is z = 1.88

(A) we have proportion p(hat)= 0.54 and margin of error = 2% = 2/100 = 0.02

Using the sample size formula

setting the given values, we get

this gives us

or

Rounding to next whole number, we get sample size n = 2195

(B) we have margin of error = 2% = 2/100 = 0.02

Sample size formula when no prior estimate for proportion is given

setting the given values, we get

this gives us

or n = 2209

we get sample size n = 2209

(C) We can see from the calculation that 0.54*(1-0.54) gives us 0.2484, which is very close to 0.25(used for no prior estimate)

Thats the only valid reason for sample sizes to be close.

Option B is incorrect because we have used same confidence interval in both part A and B.

option C is incorrect because we have used same margin of error 2% in both part A and B

Only correct answer is option A