A sample of size 50 will be drawn from a population with mean 76 and standard deviation 14 . Find the 69 th percentile of $\bar { x }$.
A) 77.0
B) 81.4
C) 73.1
D) 81.3

Question

Quizplus · Accepted Answer

The 69th percentile of $\bar{x}$ can be found using the formula for the mean of the sampling distribution of the sample mean, which is the population mean ($\mu$), and the standard error of the mean (SEM), which is the population standard deviation ($\sigma$) divided by the square root of the sample size ($n$). The population mean is given as 76, and the standard deviation is 14. The sample size is 50. The SEM is $14 / \sqrt{50}$. To find the 69th percentile, we use the Z-score associated with the 69th percentile, which is approximately 0.5 (from Z-tables, knowing that the 50th percentile corresponds to a Z-score of 0, and the 69th percentile would be slightly above that). The formula to find the specific value of $\bar{x}$ at the 69th percentile is $\mu + (Z \times SEM)$. Plugging in the values, we get $76 + (0.5 \times 14 / \sqrt{50})$, which simplifies to approximately 77. Hence, the 69th percentile of $\bar{x}$ is 77.0.

A Sample of Size 50 Will Be Drawn from a Population