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

44

percentage points with

9494​%

confidence if he uses an estimate of

4848​%

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

44

percentage points with

9494​%

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 the margin of error

44​%

is less than​ 5%.

B.The results are close because

0.48 left parenthesis 1 minus 0.48 right parenthesis equals0.48(1−0.48)=0.24960.2496

is very close to 0.25.

C.The results are close because the confidence

9494​%

is close to​ 100%.

Answer 1

Solution :

Given that,

a) = 48% = 0.48

1 - = 1 - 0.48 = 0.52

margin of error = E = 4% = 0.04

At 94% confidence level

= 1 - 94%

=1 - 0.94 =0.06

/2 = 0.03

Z/2 = 1.881

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

= (1.881 / 0.04)² * 0.48 * 0.52

= 551.95

sample size = n = 552

b) =  1 - =   0.5

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

= (1.881 / 0.04)² * 0.5 * 0.5

= 552.83

sample size = n = 553

c) correct option is = B.

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