Question

An IQ test is designed so that the mean is 100, and the standard deviation is 14 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 99% confidence that the sample mean is within 4 IQ points of the true mean. Assume that σ = 14 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real-world calculation.

The required sample size is (round up to the nearest integer).

A. No. This number of IQ test scores is a fairly large number.

B. Yes. This number of IQ test scores is a fairly small number.

C. Yes. This number of IQ test scores is a fairly large number.

D. No. This number of IQ test scores is a fairly small number.

Answer 1

To estimate the mean IQ score of statistics students with 99% confidence that the sample mean is within 4 IQ points of the true mean, a sample size of 22 is required.

Step-by-step explanation:

To find the required sample size to estimate the mean IQ score of statistics students with 99% confidence that the sample mean is within 4 IQ points of the true mean, we can use the formula:

n = (Z * σ / E)^2

where n is the sample size, Z is the Z-score corresponding to the desired confidence level (in this case, 99% corresponds to a Z-score of approximately 2.576), σ is the population standard deviation (14), and E is the desired margin of error (4).

Substituting the values into the formula, we get:

n = (2.576 * 14 / 4)^2 = 21.16

Rounding up to the nearest integer, the required sample size is 22.

This is a reasonable sample size for a real-world calculation, as it is not overly large or small and should provide a representative sample for estimating the population mean.