Introduction
If you don't know what the population mean is, you can use one of numerous statistics as an approximation. The sample mean and sample median are two that come to mind right away. The sample mean has advantageous qualities as an estimator in many cases that are not shared by other competing estimators, hence it is more often utilized.
The Sampling Distribution of the Sample Mean
The STANDARD ERROR of the MEAN
In most practical situations we can determine how close a sample mean might be to the mean of the population from which the sample came, by referring to Central Limit Theorems which express essential facts about sampling distributions.
Definition:
If all possible random samples of size n are drawn without replacement from a finite population of size N with a population mean and standard deviation, the sampling distribution of the sample mean will be approximately normally distributed with a mean and standard deviation given by the formula below:
The standard deviation of a sampling distribution is called the standard error of the mean. When N is large relative to the sample size n, the
is approximately equal to 1, and the standard deviation of the sample mean is
Example 1:
Assume you have taken 100 samples of size 50 each from a population. The population variance is 36. What is the standard deviation of each and every sample mean?
Solutions: The population Standard deviation is
And the the standard deviation of the sampling distribution of the means are
Example 2:
Let x̄ be the mean of a random sample of size 60 drawn from a population with mean 110 and standard deviation 30.
1. Find the mean and standard distribution of x̄.
2. Find the probability that x̄ assumes a value between 109 and 112.
The solution is to leave it as an exercise.
I'd love to hear your thoughts about the Sampling Distribution of the Sample Mean. Feel free to leave your comment below.
0 Comments