Question

A certain region would like to estimate the proportion of voters who intend to participate in upcoming elections. A pilot sample of 25 voters found that 17 of them intended to vote in the election. Determine the additional number of voters that need to be sampled to construct a 99​% interval with a margin of error equal to 0.07 to estimate the proportion.

The region should sample ___________ additional voters. ​(Round up to the nearest​ integer.)

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Determine the sample size n needed to construct a 90​% confidence interval to estimate the population proportion for the following sample proportions when the margin of error equals 4​%.

a. p overbar=0.20

b. p overbar=0.30

c. p overbar=0.40

a. n=___________(Round up to the nearest​ integer.)

Answer 1

Solution :

Given that,

1)

n = 25

x = 17

= x / n = 0.68

1 - = 0.32  

margin of error = E = 0.07

At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

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

= (2.576 / 0.07)² * 0.68 * 0.32

= 294.68

sample size = 295

The region should sample 295 additional voters.

2)

a)

margin of error = E = 0.04

= 0.20

1 - = 0.80

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= (1.645 / 0.04)² *0.20 * 0.80

= 270.60

sample size = 271

b)

= 0.30

1 - = 0.70

Z_/2 = Z_0.05 = 1.645

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

= (1.645 / 0.04)² *0.30* 0.70

= 355.17

sample size = 356

c)

= 0.40

1 - = 0.60

Z_/2 = Z_0.05 = 1.645

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

= (1.645 / 0.04)² *0.40* 0.60

= 405.90

sample size = 406