A sociologist is studying marriage customs in a rural community in Denmark to determine the age of the woman at the time of her first marriage. The sample standard deviation of these ages was 2.3 years. The sociologist wants to estimate the population mean age of a woman at the time of her first marriage. How large a sample is required to be 99% confident that the sample mean of ages is within 0.25 years of the population mean?

Question

A sociologist is studying marriage customs in a rural community in Denmark to determine the age of the woman at the time of her first marriage. The sample standard deviation of these ages was 2.3 years. The sociologist wants to estimate the population mean age of a woman at the time of her first marriage. How large a sample is required to be 99% confident that the sample mean of ages is within 0.25 years of the population mean?

Answer 1

Use a formula to find sample size.

Here is one:
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be found using a z-table to represent the 99% confidence interval, sd = 2.3, E = 0.25, ^2 means squared, and * means to multiply.

Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.

Hope this helps.

Answer 2

To determine the sample size required, we can use the formula for sample size estimation for estimating population means:

n = (z * σ / E)^2

where:
n = sample size
z = z-value corresponding to the desired confidence level (in this case, 99% confidence, so z = 2.33)
σ = population standard deviation (2.3 years)
E = maximum error tolerance (0.25 years)

By substituting the given values into the formula, we can calculate the required sample size:

n = (2.33 * 2.3 / 0.25)^2
n = (5.359 / 0.25)^2
n = 21.436^2
n ≈ 459.34

Therefore, a sample size of approximately 460 would be required to be 99% confident that the sample mean of ages is within 0.25 years of the population mean.

Answer 3

To determine the required sample size, we can use the formula for the sample size estimation for estimating population means:

n = [Z * (σ / E)]²

Where:
n = sample size
Z = Z-score representing the desired confidence level (in this case, for a 99% confidence level, Z ≈ 2.576)
σ = sample standard deviation
E = maximum error allowed (in this case, 0.25 years)

Plugging in the given values:

n = [2.576 * (2.3 / 0.25)]²
n = [2.576 * 9.2]²
n ≈ (23.6912)²
n ≈ 559.17

Therefore, a sample size of at least 560 is required to be 99% confident that the sample mean age is within 0.25 years of the population mean.