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 .25 years of the population mean u?

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To determine the sample size required to estimate the population mean age of women at the time of their first marriage, with a 99% confidence level and a margin of error of 0.25 years, we can use the formula for determining sample size for estimating a population mean.

The formula is as follows:

n = ((Z * σ) / E)²

Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (in this case, 99% confidence level)
σ = population standard deviation
E = desired margin of error

In this case, the population standard deviation is given as 2.3 years, and the margin of error is 0.25 years.

Step 1: Determine the Z-score
To determine the Z-score corresponding to a 99% confidence level, you can use a Z-table or a statistical software. The Z-score for a 99% confidence level is approximately 2.576.

Step 2: Plug the values into the formula
Now, we can plug the values into the formula:

n = ((2.576 * 2.3) / 0.25)²

Calculating this expression:

n = (5.9288 / 0.25)²
n = (23.7152)²
n ≈ 562.06

Step 3: Round up the sample size
Since the sample size must be a whole number, we round up the calculated result to the nearest whole number:

n ≈ 563

Therefore, the sociologist would need a sample size of at least 563 in order to be 99% confident that the sample mean ages of the women in the rural community in Denmark is within 0.25 years of the population mean age.