A sample of n = 9 has a mean of M = 72 with SS = 162. Use the sample to make a point estimate of ì and to construct the 90% confidence interval for ì. (Round all answers to two decimal places
ì = ? ± ? ( ? )
The 90% confidence interval for ì:
( ? , ? )
To estimate the population mean (ì) and construct the confidence interval, we'll need to follow a few steps.
Step 1: Calculate the point estimate for ì.
The point estimate for ì (the population mean) is simply the mean of the sample. In this case, the given sample mean is M = 72. So ì (point estimate) = 72.
Step 2: Calculate the standard deviation (σ) using the sum of squares (SS) value.
Given SS = 162, we can calculate the variance (σ^2) by dividing SS by the sample size (n). So σ^2 = SS / n = 162 / 9 = 18.
Step 3: Calculate the standard deviation (σ) by taking the square root of the variance obtained in Step 2.
σ = √(18) ≈ 4.24 (rounded to two decimal places).
Step 4: Calculate the margin of error.
For a 90% confidence interval, the z-score (obtained from the standard normal distribution) is 1.645. The margin of error is given by z * (σ / √n), where z is the z-score and √n is the square root of the sample size.
Here, n = 9, so the margin of error = 1.645 * (4.24 / √9) = 1.645 * (4.24 / 3) = 1.645 * 1.41 ≈ 2.33 (rounded to two decimal places).
Step 5: Calculate the confidence interval.
The confidence interval is given by ì ± margin of error.
Using the point estimate obtained in Step 1 and the margin of error in Step 4, the confidence interval is:
(ì - margin of error, ì + margin of error)
So, the 90% confidence interval for ì is:
(72 - 2.33, 72 + 2.33) ≈ (69.67, 74.33) (rounded to two decimal places).
To summarize:
ì = 72 ± 2.33
The 90% confidence interval for ì is (69.67, 74.33).