# A firm employs 189 junior accountants. In a random sample of 50 of these, the mean number of hours overtime billed in a particular week was 9.7, and the sample standard deviation was 6.2 hours. Find a 95\% confidence interval for the mean number of hours overtime billed per junior accountant in this firm that week.

A firm employs 189 junior accountants. In a random sample of 50 of these, the mean number of hours overtime billed in a particular week was 9.7, and the sample standard deviation was 6.2 hours.
a. Find a $95\mathrm{%}$ confidence interval for the mean number of hours overtime billed per junior accountant in this firm that week.
b. Find a $99\mathrm{%}$ confidence interval for the total number of hours overtime billed by junior accountants in the firm during the week of interest.
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Given data,
The number of employee is 189.
The mean of the overtime is 9.7.
The standard deviation of overtime is 6.2
The sample size is 50.
Step 1
a) A $95\mathrm{%}$ confidence interval for the mean number of hours overtime,
Error margin $={z}_{\frac{0.05}{2}}\frac{\sigma }{\sqrt{n}}$
$={z}_{0.0025}\frac{6.2}{\sqrt{50}}$
$=1.96×0.8768$
$=1.72$
The lower bound of the interval is,
$L=\mu -$ Error
$=9.7-1.72$
$=7.98$
The upper bound of the interval is,
$U=\mu +$ Error
$=9.7+1.72$
$=11.42$
Step 2
b) A $99\mathrm{%}$ confidence interval for the mean number of hours overtime,
Error margin $={z}_{\frac{0.01}{2}}\frac{\sigma }{\sqrt{n}}$
$={z}_{0.005}\frac{6.2}{\sqrt{50}}$
$=2.576×0.8768$
$=2.2586$
The lower bound of the interval is,
$L=\mu -$ Error
$=9.7-2.2586$
$=7.4414$
The upper bound of the interval is,
$U=\mu +$ Error
$=9.7+2.2586$
$=11.9586$