Construct a 90% confidence interval for 𝜇 if a random sample of size n = 30 results in a sample mean = 45 and s = 2.2. The value of 𝜎 is unknown.

To construct a 90% confidence interval for the population mean (μ), we will use the formula for a confidence interval when the population standard deviation (σ) is unknown:

CI = x̄ ± t * (s/√n)

where:

• x̄ is the sample mean
• t is the t-score from the t-distribution table
• s is the sample standard deviation
• n is the sample size

Given:

• x̄ = 45
• s = 2.2
• n = 30

First, we need to find the t-score. For a 90% confidence interval and degrees of freedom (df) = n - 1 = 30 - 1 = 29, the t-score is approximately 1.699 (you can find this value in a t-distribution table or using a statistical calculator).

Now, we can substitute these values into the formula:

CI = 45 ± 1.699 * (2.2/√30)

Calculate the value of the standard error (SE = s/√n = 2.2/√30 ≈ 0.401).

Then, calculate the margin of error (ME = t * SE = 1.699 * 0.401 ≈ 0.681).

Finally, calculate the confidence interval:

CI = 45 ± 0.681

So, the 90% confidence interval for μ is (44.319, 45.681).