If a population is normally distributed with µ = 500 and σ = 50, and a sample of size 20 is selected, the probability that the sample mean falls between 450 and 550 will be ________ the probability that an individual value falls between 450 and 550.
A) the same as
B) less than
C) greater than
D) less than or the same as
E) not enough information given

Question

Quizplus · Accepted Answer

The Central Limit Theorem tells us that the distribution of sample means will be approximately normal with a mean of µ=500 and a standard error of σ/√n=50/√20 ≈ 11.18. To find the probability that the sample mean falls between 450 and 550, we need to standardize the sample mean as follows:

Z = (X̄ - µ) / (σ/√n) = (450 - 500) / 11.18 and Z = (X̄ - µ) / (σ/√n) = (550 - 500) / 11.18

We can use a standard normal distribution table or calculator to find that the probability of Z falling between these two values is about 0.7887.

To find the probability that an individual value falls between 450 and 550, we need to standardize the values as follows:

Z = (X - µ) / σ = (450 - 500) / 50 and Z = (X - µ) / σ = (550 - 500) / 50

Again, using a standard normal distribution table or calculator, we can find that the probability of Z falling between these two values is about 0.3829.

Since 0.7887 is greater than 0.3829, the answer is (C) greater than.

If a Population Is Normally Distributed with µ = 500