Question

A sample of n = 4 scores is selected from a population with LaTeX: \mu=40μ = 40 with LaTeX: \sigma=8,\:σ = 8 ,and the sample mean is M=43. What is the expected value for the sample mean

Answer 1

Answer:

The expected value for the sample mean is 40.

Step-by-step explanation:

The sample mean ($\bar x$) is a sample statistic used to estimate the population mean ($\mu$). It is computed using the sample values.

The sample mean is an unbiased estimator of the population mean.

As the sample size is increased according to the law of large numbers the sample mean gets closer and closer to the population mean. Hence it is known as the unbiased estimator of the population mean.

Now if many samples of same size are selected from a population and the sample mean is computed for each sample then the expected value of these sample means is same as the population mean.

That is, if i samples of same size are selected from a population then the expected value of $\bar x_{i}$ is:

$E(\bar x_{i})=\mu$

In this case it is provided that,

n = 4

μ = 40

$\bar x_{i}$ = 43

Then the expected value for the sample mean is:

$E(\bar x_{i})=\mu=40$

Thus, the expected value for the sample mean is 40.