Question

To determine how well a new method of teaching vocabulary is working in a certain elementary school, education researchers plan to give a vocabulary test to a sample of 40 sixth graders. It is known that scores on this test have a standard deviation of 8. The researchers found the sample mean to be 80. Construct a 99% confidence interval for the population mean. Round your answers to whole numbers.

Answer 1

To construct the 99% confidence interval for the population mean with a known standard deviation, use the sample mean, the z-score for the desired confidence level, and the known standard deviation. Calculating with these values, the 99% confidence interval for the population mean is approximately (77, 83).

Step-by-step explanation:

To construct a 99% confidence interval for the population mean when the standard deviation is known, we can use the formula for the confidence interval which is given by:

CI = µ ± (z * (σ/√n)),

where µ is the sample mean, z is the z-score corresponding to the given confidence level, σ is the population standard deviation, and n is the sample size.

In this case, the sample mean (µ) is 80, the population standard deviation (σ) is 8, and the sample size (n) is 40.

The z-score for a 99% confidence interval can be found using standard z-score tables or a statistical software. The z-score that corresponds to the middle 99% is approximately 2.576.

Now, we can calculate the margin of error (ME):

ME = z * (σ/√n) = 2.576 * (8/√40) ≈ 3.26.

Finally, the 99% confidence interval for the population mean is:

(µ - ME, µ + ME) = (80 - 3.26, 80 + 3.26) = (76.74, 83.26).

Rounding to whole numbers, the 99% confidence interval is (77, 83).