A sample of size 50 will be drawn from a population with mean 73 and standard deviation 8. Find the 19th percentile of $\bar { x }$
A) 73.8
B) 75.8
C) 72.0
D) 75.7

Question

Quizplus · Accepted Answer

The central limit theorem states that the distribution of sample means is approximately normal, with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of sample size when the sample size is large enough (n ≥ 30). In this case, n = 50, which is large enough.

To find the 19th percentile of the sample means, we need to find the corresponding z-score using the standard normal distribution table.

z-score = invNorm(0.19) = -0.88

Then, we use the formula for the sample mean:

sample mean = population mean + z-score * (population standard deviation / sqrt(sample size))

sample mean = 73 - 0.88 * (8 / sqrt(50))

sample mean ≈ 72.01

Therefore, the 19th percentile of the sample means is approximately 72.01. The closest answer choice is C.

A Sample of Size 50 Will Be Drawn from a Population