Construct a 95% confidence interval for the true average monthly salary earned by all employees of People Plus Pty, if it was found that the average monthly salary earned by a sample of 19 employees of the company was R18 500, with a standard deviation of R1 750. Interpret your answer

Construct a 95% confidence interval for the true average monthly salary earned by all employees of People Plus Pty, if it was found that the average monthly salary earned by a sample of 19 employees of the company was R18 500, with a standard deviation of R1 750. Interpret your answer

Question
Confidence intervals
asked 2021-02-24
Construct a 95% confidence interval for the true average monthly salary earned by all employees of People Plus Pty, if it was found that the average monthly salary earned by a sample of 19 employees of the company was R18 500, with a standard deviation of R1 750. Interpret your answer

Answers (1)

2021-02-25
Step 1 The sample size is 19, mean is R18500, standard deviation is R1750. The degrees of freedom is, \(\displaystyle{d}{f}={n}-{1}\)
\(\displaystyle={19}-{1}\)
\(\displaystyle={18}\) The degrees of freedom is 18. The confidence level is 95%, than the level of significance is 0.05. Computation of critical value: The critical value of t-distribution for 0.05 level of significance at 18 degrees of freedom can be obtained using the excel formula “=T.INV.2T(0.05,18)”. The critical value is 2.1009. Step 2 The 95% confidence interval for the true average monthly salary earned by all employees of People Plus Pty is, \(\displaystyle{C}{I}=\overline{{{x}}}\pm{t}_{{{\frac{{\alpha}}{{{2}}}}}}{\left({\frac{{{s}}}{{\sqrt{{{n}}}}}}\right)}\)
\(\displaystyle={18500}\pm{2.1009}{\left({\frac{{{1750}}}{{\sqrt{{{19}}}}}}\right)}\)
\(\displaystyle={18500}\pm{843.464}\)
\(\displaystyle={\left({18500}-{843.464},{18500}+{843.464}\right)}\)
\(\displaystyle={\left({17656.536},{19343.464}\right)}\)
\(\displaystyle\approx{\left({17657},{19343}\right)}\) Thus, the 95% confidence interval for the true average monthly salary earned by all employees of People Plus Pty is (17657,19343). Interpretation: When repeated samples of size 19 are taken, about 95% of those confidence intervals would contain the true average monthly salary earned by all employees of People Plus Pty that lies between 17657 and 19343. That is, there is 95% confidence that the true average monthly salary earned by all employees of People Plus Pty that lies between 17657 and 19343.
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