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 3 percentage points with 96​% confidence if he uses an estimate of 48​% obtained from a​ poll?

(b) What sample size should be obtained if he wants to be within 3 percentage points with a 96% confidence if he does not use any estimate?

(c) Why are the results so close?

Answer 1

Solution :

Given that,

margin of error = E = 3% = 0.03

Z_/2 = 2.05

(a)

= 0.48

1 - = 0.52

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.05 / 0.03)² * 0.48 * 0.52

= 1166

sample size = n = 1166

(b)

= 0.5

1 - = 0.5

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (2.05 / 0.03)² * 0.5 * 0.5

= 1168

sample size = n = 1168

(c)

0.48 is near by 0.5

Having an estimate of the population proportion reduces the minimum sample size is needed