A survey has been conducted to estimate the usage of high-speed internet in the local area. 154 households were surveyed and asked whether they used high-speed internet at their home. 114 households surveyed responded that they do have a high-speed internet connection at home.

a)Calculate the margin of error for the 95% confidence interval for the proportion of households that have a high-speed internet connection. Give your answer as a decimal to 3 decimal places.

margin of error = 

b)Calculate the 95% confidence interval for the proportion of households that have a high-speed internet connection. Give your answers as decimals to 3 decimal places.

 ≤ p ≤ 

a)The margin of error for the 95% confidence interval for the proportion of households that have a high-speed internet connection is 0.069.

b) The 95% confidence interval for the proportion of households that have a high-speed internet connection is  0.671≤ p ≤ 0.810

Step-by-step explanation

Given that a survey has been conducted to estimate the usage of high-speed internet in the local area. 154 households were surveyed and asked whether they used high-speed internet at their home. 114 households surveyed responded that they do have a high-speed internet connection at home.

a)The margin of error for the 95% confidence interval for the proportion of households that have a high-speed internet connection is calculated as follows;

Margin of error ?=zc?⋅np^?(1−p^?)???

p?=114/154

n=154

At 95% confidence level, α=1-0.95=0.05

z_c=1.96

Therefore;

M.E?=1.96⋅154154114?(1−154114?)???

?=1.96⋅154154114?(7720?)???

=0.0693

The margin of error is therefore 0.069

b) The 95% confidence interval for the proportion of households that have a high-speed internet connection is determined as follows;

?C.I=p^?±M.E?

?p^?=154114?=0.7403?

?M.E=0.0693?

C.I?=0.7403±0.0693?

Upper confidence limit=0.7403+0.0693=0.8096

Lower confidence limit=0.7403-0.0693=0.6710

Therefore the 95% confidence interval for the proportion of households that have a high-speed internet connection is; 0.671≤ p ≤ 0.810