Suppose you have a random sample of $50$ measurements with a sample mean of $20$ and a known population standard deviation of $4$. You want to construct a $95\%$ confidence interval for the population mean. Which of the following is a correct analysis of this data set?

• A. The 95% confidence interval for the population mean is $20\pm 1.11$ because the margin of error is calculated as $1.96\times \frac{4}{\sqrt{50}}$
• B. The 95% confidence interval for the population mean is $20\pm 1.96$
• C. The 95% confidence interval for the population mean is $20\pm 1.11$
• D. The 95% confidence interval for the population mean is $20\pm 0.78$

Answer: (A)

Verified step by step guidance

1. Identify the given information: sample size \(n = 50\), sample mean \(\bar{x} = 20\), population standard deviation \(\sigma = 4\), and confidence level \$95\%$.
2. Since the population standard deviation is known and the sample size is large (\(n > 30\)), use the Z-distribution to construct the confidence interval for the population mean.
3. Find the critical Z-value for a 95% confidence level, which corresponds to the value \(Z_{\alpha/2}\) where \(\alpha = 0.05\). This value is commonly known as 1.96.
4. Calculate the standard error of the mean (SEM) using the formula: \(SEM = \frac{\sigma}{\sqrt{n}}\)
5. Compute the margin of error (ME) by multiplying the critical Z-value by the standard error: \(ME = Z_{\alpha/2} \times SEM = 1.96 \times \frac{4}{\sqrt{50}}\) Finally, express the 95% confidence interval as: \(\bar{x} \pm ME = 20 \pm ME\)