Question

The 2008 Workplace Productivity Survey, commissioned by LexisNexis and prepared by World One Research, included the question, "How many hours do you work at your job on a typical workday." Let x = the number of hours a legal professional works on a typical workday. Suppose that x is normally distributed with a known standard deviation of 12.6.

A sample of 250 legal professionals was surveyed, and the sample's mean response was 9 hours. Use the sample information to estimate μ, the mean number of hours a legal professional works on a typical workday. Develop a 95% confidence interval estimate of the mean number of hours a legal professional works on a typical workday.

Answer 1

A 95% confidence interval estimate of the mean number of hours a legal professional works on a typical workday is [7.44 hours, 10.56 hours].

Explanation:

We are given that x is normally distributed with a known standard deviation of 12.6.

A sample of 250 legal professionals was surveyed, and the sample's mean response was 9 hours.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

P.Q. =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\bar X` = sample average mean response = 9 hours

`\sigma` = population standard deviation = 12.6

n = sample of legal professionals = 250

`\mu` = mean number of hours a legal professional works

Here for constructing a 95% confidence interval we have used One-sample z-test statistics as we know about population standard deviation.

So, 95% confidence interval for the population mean,
`\mu` is ;

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level

of significance are -1.96 & 1.96}

P(-1.96 <
`(\bar X-\mu)/((\sigma)/(√(n) ) )` < 1.96) = 0.95

P(
`-1.96 * {(\sigma)/(√(n) ) }` <
`{\bar X-\mu}` <
`-1.96 * {(\sigma)/(√(n) ) }` ) = 0.95

P(
`\bar X-1.96 * {(\sigma)/(√(n) ) }` <
`\mu` <
`\bar X+1.96 * {(\sigma)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X-1.96 * {(\sigma)/(√(n) ) }` ,
`\bar X+1.96 * {(\sigma)/(√(n) ) }` ]

= [
`9-1.96 * {(12.6)/(√(250) ) }` ,
`9+1.96 * {(12.6)/(√(250) ) }` ]

= [7.44 hours, 10.56 hours]

Therefore, a 95% confidence interval estimate of the mean number of hours a legal professional works on a typical workday is [7.44 hours, 10.56 hours].