Respuesta :
If X1, X2, ..., XE is a random sample, then the sample mean is an unbiased estimator of the population mean: TRUE
What is a random sample?
- In statistics, a simple random sample is a subset of individuals chosen at random from a larger set, with all individuals having the same probability.
- It is the process of selecting a sample at random.
Given that X1, X2, X3,.....XN are independently selected random variables from the population with the mean [tex]\mu[/tex] and variance [tex]\sigma^{2}[/tex].
Then the expected value of the sample mean is given by:
[tex]\begin{aligned}&E[\bar{X}]=E\left[\frac{X 1+X 2+X 3+X 4 \ldots . \ldots n}{n}\right]=E\left[\frac{1}{n}(X 1+X 2+X 3 \ldots . . X n)\right] \\&E[\bar{X}]=\frac{1}{n} E[X 1+X 2+X 3+\ldots .+X n] \\&E[\bar{X}]=\frac{1}{n}[E[X 1]+E[X 2]+E[X 3]+\ldots E[X n]]\end{aligned}[/tex]
But since the expected value of each of the above random samples is [tex]\mu[/tex] we can write:
[tex]E[\bar{X}]=\frac{1}{n}[\mu+\mu+\mu+\ldots \ldots \mu]=\frac{1}{n}[n * \mu]=\mu[/tex]
This implies that, if X1, X2, ..., XE is a random sample, then the sample mean is an unbiased estimator of the population mean is true.
Therefore, the statement "If X1, X2, ..., XE is a random sample, then the sample mean is an unbiased estimator of the population mean" is TRUE.
Know more about random samples here:
https://brainly.com/question/24466382
#SPJ4
The correct question is given below:
If X1, X2, ..., XE is a random sample, then the sample mean is an unbiased estimator of the population mean. TRUE or FALSE.
