Using a Z interval, construct a 90% confidence interval for 𝜇  if a random sample of size n = 20 with a sample mean = 110 is drawn from a normal population with 𝜎  = 25.

Sure, here are the steps to construct a 90% confidence interval for 𝜇 using a Z interval:

Step 1: Identify a sample statistic. Since we are trying to estimate a population mean, we choose the sample mean (110) as the sample statistic.

Step 2: Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 90% confidence level.

Step 3: Find the standard deviation or standard error. Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean. Instead, we compute the standard error (SE). SE of the sample mean is standard deviation of the population divided by the square root of sample size:

SE = 𝜎 / sqrt(n) = 25 / sqrt(20) = 5.59

Step 4: Find the margin of error. Elsewhere on this site, we show how to compute the margin of error when the sampling distribution is approximately normal. The key steps are shown below:

1. Find standard deviation or standard error.
2. Find the standard deviation or standard error.
3. Multiply it by the appropriate Z*-value (for the desired confidence level).

The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.

The Z value for a 90% confidence level is 1.645 (you can find this in a standard Z table).

So, the margin of error = Z * SE = 1.645 * 5.59 = 9.19

Step 5: Construct the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. The uncertainty is denoted by the confidence level.

So, the confidence interval would be:

Sample mean ± margin of error = 110 ± 9.19 = (100.81, 119.19)

So, we can be 90% confident that the population mean falls between 100.81 and 119.19.