Question # 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.

Confidence intervals 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 $$\displaystyle{95}\%$$ confidence interval for the mean number of hours overtime billed per junior accountant in this firm that week.
b. Find a $$\displaystyle{99}\%$$ confidence interval for the total number of hours overtime billed by junior accountants in the firm during the week of interest. 2021-08-16

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 $$\displaystyle{95}\%$$ confidence interval for the mean number of hours overtime,
Error margin $$\displaystyle={z}_{{\frac{{0.05}}{{2}}}}{\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={z}_{{{0.0025}}}{\frac{{{6.2}}}{{\sqrt{{{50}}}}}}$$
$$\displaystyle={1.96}\times{0.8768}$$
$$\displaystyle={1.72}$$
The lower bound of the interval is,
$$\displaystyle{L}=\mu-$$ Error
$$\displaystyle={9.7}-{1.72}$$
$$\displaystyle={7.98}$$
The upper bound of the interval is,
$$\displaystyle{U}=\mu+$$ Error
$$\displaystyle={9.7}+{1.72}$$
$$\displaystyle={11.42}$$
Step 2
b) A $$\displaystyle{99}\%$$ confidence interval for the mean number of hours overtime,
Error margin $$\displaystyle={z}_{{\frac{{0.01}}{{2}}}}{\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={z}_{{{0.005}}}{\frac{{{6.2}}}{{\sqrt{{{50}}}}}}$$
$$\displaystyle={2.576}\times{0.8768}$$
$$\displaystyle={2.2586}$$
The lower bound of the interval is,
$$\displaystyle{L}=\mu-$$ Error
$$\displaystyle={9.7}-{2.2586}$$
$$\displaystyle={7.4414}$$
The upper bound of the interval is,
$$\displaystyle{U}=\mu+$$ Error
$$\displaystyle={9.7}+{2.2586}$$
$$\displaystyle={11.9586}$$