Respuesta :
Step 1: Find the standard error (SE)
The standard error is given by
[tex]SE=\frac{s}{\sqrt[]{n}}[/tex][tex]\begin{gathered} \text{ Where } \\ SE=\text{ the standard error} \\ s=\text{ the sample standard deviation} \\ n=\text{ the sample size} \end{gathered}[/tex]In this case,
[tex]n=74,s=0.76[/tex]Therefore,
[tex]SE=\frac{0.76}{\sqrt[]{74}}\approx0.0883[/tex]Step 2: Find the alpha level (α)
[tex]\alpha=1-\frac{(\text{Confidence level})}{100}[/tex][tex]\alpha=1-0.99=0.01[/tex]Step 3: Find the critical probability (P*)
[tex]P^{\prime}=1-\frac{\alpha}{2}[/tex]Therefore,
[tex]P^{\cdot}=1-\frac{0.01}{2}=0.995[/tex]Step 4: Find the critical value (CV)
The critical value the z-score having a cumulative probability equal to the critical probability (P*).
Using the cumulative z-score table we will find the z-score with value of 0.995
Hence,
[tex]CV=2.576[/tex]Step 5: Find the margin of error (ME)
[tex]ME=SE\times CV[/tex]Therefore,
[tex]ME=0.0883\times2.576=0.2275[/tex]Step 6: Find the confidence interval (CI)
[tex]\begin{gathered} CI\text{ is given by} \\ CI=(\bar{x}-ME,\bar{x}+ME) \\ \text{ In this case} \\ \bar{x}=17.1 \end{gathered}[/tex]Therefore,
[tex]CI=(17.1-0.2275,17.1+0.2275)=(16.8725,17.3275)[/tex]Hence there is a 99% probability that the true mean will lie in the confidence interval
(16.8725, 17.3275)
