Problem

Sappose $\bar{x}=161.84$ and $\sigma=58.15$ and the sample size is given as 100 Determine the $95 \%$ CI

Answer

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Answer

\(\boxed{The \ 95\% \ confidence \ interval \ is \ (150.4426, \ 173.2374)}\)

Steps

Step 1 ：Given sample mean \(\bar{x} = 161.84\), standard deviation \(\sigma = 58.15\), and sample size \(n = 100\)

Step 2 ：Calculate the 95% confidence interval using the formula: \(CI = \bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\) with \(Z = 1.96\)

Step 3 ：Plug in the values and calculate the confidence interval: \(CI = 161.84 \pm 1.96 \frac{58.15}{\sqrt{100}}\)

Step 4 ：Calculate the margin of error: \(11.3974\)

Step 5 ：Find the lower and upper bounds: \(150.4426, 173.2374\)

Step 6 ：\(\boxed{The \ 95\% \ confidence \ interval \ is \ (150.4426, \ 173.2374)}\)

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