A random sample of size 36 is taken from a population with mean µ = 17 and standard deviation σ = 6. The probability that the sample mean is less than 15 is ________.
A) 0.0228
B) 0.3707
C) 0.6293
D) 0.9772

Question

Quizplus · Accepted Answer

The sample mean ($\bar{x}$) distribution is normally distributed (by the Central Limit Theorem) with mean $\mu = 17$ and standard error $SE = \frac{\sigma}{\sqrt{n}} = \frac{6}{\sqrt{36}} = 1$. To find the probability that the sample mean is less than 15, we calculate the Z-score: $Z = \frac{15 - 17}{1} = -2$. Looking up -2 in the Z-table gives us a probability of 0.0228.

A Random Sample of Size 36 Is Taken from a Population