Question

1. a sample is selected from a population with a mean of μ = 100 and a standard deviation of σ = 20.

a. if the sample has n = 16 scores, what is the expected value of m and the standard error of m?
b. if the sample has n = 100 scores, what is the expected value of m and the standard error of m?

Explanation:

Step1: Recall the formula for the expected value of the sample mean

The expected value of the sample mean \(M\), \(E(M)\) is equal to the population mean \(\mu\).
\(E(M)=\mu\)

Step2: Recall the formula for the standard error of the mean

The standard error of the mean \(\sigma_M=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

Part a:

• Expected value of \(M\):

Since \(E(M)=\mu\), and \(\mu = 100\), so \(E(M)=100\)

• Standard error of \(M\):

Given \(\sigma = 20\), \(n = 16\)
\(\sigma_M=\frac{\sigma}{\sqrt{n}}=\frac{20}{\sqrt{16}}=\frac{20}{4} = 5\)

Part b:

• Expected value of \(M\):

Since \(E(M)=\mu\), and \(\mu=100\), so \(E(M) = 100\)

• Standard error of \(M\):

Given \(\sigma = 20\), \(n = 100\)
\(\sigma_M=\frac{\sigma}{\sqrt{n}}=\frac{20}{\sqrt{100}}=\frac{20}{10}=2\)

Answer:

a. The expected value of \(M\) is \(100\) and the standard error of \(M\) is \(5\).
b. The expected value of \(M\) is \(100\) and the standard error of \(M\) is \(2\).