Statistics word problem

Question

The ages of a group of 123 randomly selected adult females have a standard deviation of 17.1 years. Assume that the ages of female statistics students have less variation than ages of females in the general​ population, so let sigma= 17.1 years for the sample size calculation. How many female statistics student ages must be obtained in order to estimate the mean age of all female statistics​ students? Assume that we want 90​% confidence that the sample mean is within​ one-half year of the population mean.

Answer 1

To determine the sample size needed to estimate the mean age of all female statistics students with a 90% confidence level and a margin of error of one-half year, we can use the following formula for the sample size:

n = (Z*σ/E)^2

where:

1. n is the sample size
2. Z is the standard normal distribution value for the desired confidence level (90% confidence level corresponds to Z = 1.645)
3. σ is the standard deviation of the population (assumed to be 17.1 years)
4. E is the desired margin of error (0.5 years)

Plugging in the values, we get:

n = (1.645*17.1/0.5)^2 = 470.9

Since we cannot have a fractional number of participants, we need to round up the sample size to the nearest whole number. Therefore, we need at least 471 female statistics students to obtain a sample mean that is within one-half year of the population mean with 90% confidence.

Keep in mind that this formula assumes a few things: first, that we know the population standard deviation (or can make a reasonable assumption about it, as we did here). Second, that we are sampling from a normally distributed population (or that our sample size is large enough that the Central Limit Theorem applies). Finally, that we are using a simple random sample to select our participants. If any of these assumptions are violated, our estimate may not be as accurate as we would like.