Question

A simple random sample of size n=250 individuals who are currently employed is asked if they work at home at least once per week. Of the 250 employed individuals​ surveyed, 41 responded that they did work at home at least once per week. Construct a​ 99% confidence interval for the population proportion of employed individuals who work at home at least once per week.

Answer 1

99% confidence interval for the population proportion of employed individuals is [0.104 , 0.224].

Explanation:

We are given that a simple random sample of size n=250 individuals who are currently employed is asked if they work at home at least once per week.

Of the 250 employed individuals​ surveyed, 41 responded that they did work at home at least once per week.

Firstly, the pivotal quantity for 99% confidence interval for the population proportion is given by;

P.Q. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of individuals who work at home at least once per week =
`(41)/(250)` = 0.164

n = sample of individuals surveyed = 250

Here for constructing 99% confidence interval we have used One-sample z proportion statistics.

So, 99% confidence interval for the population proportion, p is ;

P(-2.5758 < N(0,1) < 2.5758) = 0.99 {As the critical value of z at 0.5%

level of significance are -2.5758 & 2.5758}

P(-2.5758 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 2.5758) = 0.99

P(
`-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

P(
`\hat p-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

99% confidence interval for p = [
`\hat p-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ,
`\hat p+2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.164-2.5758 * {\sqrt{(0.164(1-0.164))/(250) } }` ,
`0.164+2.5758 * {\sqrt{(0.164(1-0.164))/(250) } }` ]

= [0.104 , 0.224]

Therefore, 99% confidence interval for the population proportion of employed individuals who work at home at least once per week is [0.104 , 0.224].