Question

A television sports commentator wants to estimate the proportion of citizens who follow professional football." Complete parts (a) through (c) Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). (a) What sample size should be obtained if he wants to be within 4 percentage points with 98% confidence if he uses an estimate of 62% 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 96% 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? O A. The results are close because the confidence 96% is close to 100% OB. The results are close because the margin of error 4% is less than 5% OC. The results are close because 0.52(1 -0.52)=0.2496 is very close to 0.25

Answer 1

Solution:

a)

Given,

E = 4% =  0.04

c = 96% = 0.96

$url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ or p = 52% = 0.52

1 - $url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ 1- p = 1 - 0.52 = 0.48

Now,

$url=https://latex.codecogs.com/gif.latex?%5Calpha&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Wx8odgvunvBz1glgcOJBqA--~D$ = 1 - c = 1 - 0.96 = 0.04

$url=https://latex.codecogs.com/gif.latex?%5Calpha&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Wx8odgvunvBz1glgcOJBqA--~D$/2 = 0.02

$url=https://latex.codecogs.com/gif.latex?%5Ctherefore&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=oMF_pyz5pCkYN9ix1tPExw--~D$$url=https://latex.codecogs.com/gif.latex?z_%7B%5Calpha/2%7D&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Lfx.nPTeIaSCsShKPc48lw--~D$ = 2.054 (using z table)

The sample size for estimating the proportion is given by

n = $url=https://latex.codecogs.com/gif.latex?%5Cfrac%7Bz_%7B%5Calpha/2%7D%5E%7B2%7Dp%281-p%29%7D%7BE%5E%7B2%7D%7D&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=aXB0GNJ5RnKBhzT1eeTfzw--~D$

= (2.054)² * 0.52 * 0.48/ (0.04²)

= 658.150896

= 659 ..(round to the next whole number)

n = 659

b)

Given,

E = 4% =  0.04

c = 96% = 0.96

here no any prior estimate is given si we it as p = 0.5

$url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ or p = 0.5

1 - $url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=rPhekId3_kRHW_GnIPHqIA--~D$ = 1- p = 1 - 0.5 = 0.5

Now,

$url=https://latex.codecogs.com/gif.latex?%5Calpha&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Wx8odgvunvBz1glgcOJBqA--~D$ = 1 - c = 1 - 0.96 = 0.04

$url=https://latex.codecogs.com/gif.latex?%5Calpha&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Wx8odgvunvBz1glgcOJBqA--~D$/2 = 0.02

$url=https://latex.codecogs.com/gif.latex?%5Ctherefore&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=oMF_pyz5pCkYN9ix1tPExw--~D$$url=https://latex.codecogs.com/gif.latex?z_%7B%5Calpha/2%7D&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=Lfx.nPTeIaSCsShKPc48lw--~D$ = 2.054 (using z table)

The sample size for estimating the proportion is given by

n = $url=https://latex.codecogs.com/gif.latex?%5Cfrac%7Bz_%7B%5Calpha/2%7D%5E%7B2%7Dp%281-p%29%7D%7BE%5E%7B2%7D%7D&t=1596154157&ymreqid=50870fb5-0ab3-84d7-1c0d-d80001010800&sig=aXB0GNJ5RnKBhzT1eeTfzw--~D$

= (2.054)² * 0.5 * 0.5/ (0.04²)

= 659.205625

= 660 ..(round to the next whole number)

n = 660

c)

The results are so close because,

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