Question

1) Out of 200 people sampled, 14 had kids. Based on this, construct a 95% confidence interval for the true population proportion of people with kids.

Give your answers as decimals, to three places

< p <

2)

If n=450n=450 and ˆpp^ (p-hat) =0.91, find the margin of error at a 90% confidence level.

Round to 4 places.

z-scores may be rounded to 3 places or exact using technology.

3) In a recent poll, 110 people were asked if they liked dogs, and 41% said they did. Find the margin of error of this poll, at the 90% confidence level.

Give your answer to three decimals

Answer 1

Solution :

1 ) Given that,

n = 200

x = 14

= x / n = 14 / 200 = 0.070

1 - = 1 - 0.070 = 0.930

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 = 1.960

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.960 * (((0.070 * 0.930) / 200) = 0.035

A 95 % confidence interval for population proportion p is ,

- E < P < + E

0.070 - 0.035 < p < 0.070 + 0.035

0.035 < p < 0.105

The 95% confidence interval for the population proportion p is : ( 0.035 , 0.105)

2 ) Given that,

n = 450

= 0.91

1 - = 1 - 0.91 = 0.09

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

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * (((0.91 * 0.09) / 450) = 0.0222

A 90 % confidence interval for population proportion p is ,

- E < P < + E

0.9100 - 0.0222 < p < 0.9100 + 0.0222

0.8878< p < 0.9322

The 90% confidence interval for the population proportion p is : ( 0.8878 , 0.9322)

3 ) Given that,

n = 110

= 41 % = 0.410

1 - = 1 - 0.410 = 0.590

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

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * (((0.410 * 0.590) / 110) = 0.077

A 90 % confidence interval for population proportion p is ,

- E < P < + E

0.410 - 0.077 < p < 0.410 + 0.077

0.333 < p < 0.487

The 90% confidence interval for the population proportion p is : ( 0.333 , 0.487)