Question

1. a sample is selected from a population with a mean of (mu = 100) and a standard deviation of (sigma = 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 expected value of 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 standard error of sample mean

The standard error of the sample mean \(\sigma_M=\frac{\sigma}{\sqrt{n}}\)

Part a

• Expected value of \(M\):

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

• Standard error of \(M\):

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

Part b

• Expected value of \(M\):

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

• Standard error of \(M\):

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

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 \(2\)