Question

If n=23,xˉ(x-bar )=46, and s=13, construct a confidence interval at a 99% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place. <μ

Answer 1

The 99% confidence interval for the population mean, assuming a normally distributed population, with n=23, x-bar=46, and s=13, is approximately (39.0, 53.0). This is calculated using the z-score for 99% confidence and the standard error.

Step-by-step explanation:

To construct a confidence interval at a 99% confidence level for the population mean μ, when given a sample size n=23, sample mean μ=46, and standard deviation s=13, we need to use the appropriate z-score for a 99% confidence level. For a 99% confidence interval, the z-score is approximately 2.576.

The formula for the confidence interval is:

1. Calculate the standard error: SE = s / √n = 13 / √23
2. Multiply the standard error by the z-score: Margin of Error (MOE) = z × SE
3. Add and subtract the MOE from the sample mean to get the confidence interval: Lower Limit = μ - MOE, Upper Limit = μ + MOE

Here's the calculation step-by-step:

1. SE = 13 / √23 ≈ 2.7099
2. MOE = 2.576 × 2.7099 ≈ 6.9765
3. Lower Limit = 46 - 6.9765 ≈ 39.0, Upper Limit = 46 + 6.9765 ≈ 53.0

So the 99% confidence interval for the population mean is approximately (39.0, 53.0).