How to compute Confidence Interval for mean of a normal distribution with known/unknown variance?Question. Ask ChatGPT to give examples.

To compute the confidence interval for the mean of a normal distribution, we need to follow different steps depending on whether the variance is known or unknown. Let's go through each case:

1. Confidence Interval for the Mean with Known Variance:

   • Determine the significance level (usually denoted as α) for the confidence interval. For example, let's say α = 0.05, which corresponds to a 95% confidence level.
   • Calculate the sample mean (x̄) and the known population variance (σ^2).
   • Determine the critical value (z*) from the standard normal distribution table for the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.
   • Compute the margin of error (E) using the formula: E = z* * (σ / √n), where n is the sample size.
   • Finally, construct the confidence interval by subtracting and adding the margin of error to the sample mean: (x̄ - E, x̄ + E).

2. Confidence Interval for the Mean with Unknown Variance:

   • Determine the significance level (α) for the confidence interval, as before.
   • Calculate the sample mean (x̄) and the sample standard deviation (s).
   • Determine the critical value (t*) from the t-distribution table for the desired confidence level and degrees of freedom (df = n - 1), where n is the sample size.
   • Compute the margin of error (E) using the formula: E = t* * (s / √n).
   • Construct the confidence interval by subtracting and adding the margin of error to the sample mean: (x̄ - E, x̄ + E).

Examples:

1. Confidence Interval for the Mean with Known Variance: Let's say we have a sample of 100 students, and their test scores have a known population variance of 25. The sample mean is 80. Using a 95% confidence level, we can calculate the confidence interval as follows:

   • Critical value (z*) for a 95% confidence level is approximately 1.96.
   • Margin of error (E) = 1.96 * (√25 / √100) = 1.96 * (5 / 10) = 0.98.
   • Confidence interval = (80 - 0.98, 80 + 0.98) = (79.02, 80.98).

2. Confidence Interval for the Mean with Unknown Variance: Suppose we have a sample of 50 employees, and their monthly salaries have a sample mean of $3,000 and a sample standard deviation of$500. Using a 90% confidence level, we can calculate the confidence interval as follows:

   • Critical value (t*) for a 90% confidence level and 49 degrees of freedom is approximately 1.68.
   • Margin of error (E) = 1.68 * (500 / √50) = 1.68 * (500 / 7.07) = 119.19.
   • Confidence interval = (3000 - 119.19, 3000 + 119.19) = (2880.81, 3119.19).

Remember, the confidence interval provides a range of values within which we can be confident that the true population mean lies, based on the sample data and the chosen confidence level.