A study is going to be conducted in which a population mean will be estimated using a 92% confidence interval.The estimate needs to be within 12 of the actual population mean.The population variance is estimated to be around 2500.The necessary sample size should be at least _______.
A)15
B)47
C)54
D)638
E)700

Question

Quizplus · Accepted Answer

To determine the necessary sample size for estimating a population mean with a specific margin of error (E) at a certain confidence level, the formula is $n = \left(\frac{Z \cdot \sigma}{E}\right)^2$, where $Z$ is the Z-score corresponding to the confidence level, $\sigma$ is the population standard deviation, and $E$ is the desired margin of error. Given a 92% confidence interval, the Z-score (from a Z-table or calculator) is approximately 1.75. The population variance is 2500, so the standard deviation ($\sigma$) is the square root of 2500, which is 50. The desired margin of error (E) is 12. Plugging these values into the formula gives $n = \left(\frac{1.75 \cdot 50}{12}\right)^2 \approx 53.4$. Since the sample size must be a whole number, it is rounded up to the nearest whole number, which is 54.

A Study Is Going to Be Conducted in Which a Population