A random sample of n = 16 scores is obtained from a normal population with µ = 40 and σ = 8.What is the probability that the sample mean will be within 2 points of the population mean?
A)0.3830
B)0.6826
C)0.8664
D)0.9544

Question

A random sample of n = 16 scores is obtained from a normal population with µ = 40 and σ = 8.What is the probability that the sample mean will be within 2 points of the population mean?​
A)​0.3830
B)​0.6826
C)​0.8664
D)​0.9544

Quizplus · Accepted Answer

We can start by finding the standard error of the mean (SEM), which is:

SEM = σ/√n = 8/√16 = 2

Next, we can use the z-score formula to find the probability of the sample mean being within 2 points of the population mean:

z = (sample mean - population mean) / SEM

z = (x̄ - µ) / SEM = (40 - 40) / 2 = 0

To find the probability, we need to look up the area under the normal distribution curve between z = -2 and z = 2. This can be done using a standard normal distribution table or calculator, and the result is:

P(-2 < z < 2) = 0.6826

Therefore, the probability that the sample mean will be within 2 points of the population mean is 0.6826, which corresponds to choice B.

A Random Sample of N = 16 Scores Is Obtained