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

4

percentage points with

95​%

confidence if he uses an estimate of

48​%

obtained from a​ poll?

The sample size is ___? (Round up to the nearest​ integer.)​

(b) What sample size should be obtained if he wants to be within

4

percentage points with

95​%

confidence if he does not use any prior​ estimates?

The sample size is ____? ​(Round up to the nearest​ integer.)

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

A. The results are close because the margin of error 4​% is less than​ 5%.

B.The results are close because 0.48(1-0.48) =0.2496 is very close to 0.25.

C.The results are close because the confidence 95​% is close to​ 100%.

Answer 1

Solution :

Given that,

margin of error = E = 4% = 0.04

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

(a)

= 0.48

1 - = 1 - 0.48 = 0.52

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

= (1.96 / 0.04)² * 0.48 * 0.52

= 599.28

sample size = 599

(b)

= 0.5

1 - = 1 - 0.5 = 0.5

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

= (1.96 / 0.04)² * 0.5 * 0.5

= 600.25

sample size = 600

(c)

B)The results are close because 0.48(1-0.48) =0.2496 is very close to 0.25.

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