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
asked 2021-08-10
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.

Answers (1)

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
\(\displaystyle={9.7}+{2.2586}\)
\(\displaystyle={11.9586}\)

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