A sample of n = 36 individuals is drawn at random from a population that has a μ = 105 and a σ = 18. The probability is .0500 that the mean of the sample will be below what value?
A) 75.3
B) 134.7
C) 109.95
D) 100.05

Question

Quizplus · Accepted Answer

Using the formula for the z-score, $ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} $, and knowing that for a probability of .0500 in a normal distribution, the z-score is approximately -1.645 (looking at the negative side of the standard normal distribution), we can solve for $\bar{x}$ (the sample mean). Plugging in the values: $-1.645 = \frac{\bar{x} - 105}{18/\sqrt{36}}$, solving for $\bar{x}$ gives us approximately 100.05. This calculation uses the standard deviation of the sampling distribution of the mean, which is $\sigma/\sqrt{n}$, known as the standard error of the mean.

A Sample of N = 36 Individuals Is Drawn at Random