5. Sampling: Sampling Distributions
Sampling Distribution of the Sample Mean
When using a sample mean to estimate a population mean, the sampling distribution of the sample mean can be used to determine how much estimation error is reasonable to expect.
#\phantom{0}#
Sampling Distribution of the Sample Mean
The sampling distribution of the sample mean is the probability distribution of the sample means of every possible sample of a particular size #n# that can be drawn from a population.
The mean of the distribution of sample means is called the expected value of the sample mean and is denoted #\mu_{\bar{X}}#.
The standard deviation of the distribution of sample means is called the standard error of the mean and is denoted #\sigma_{\bar{X}}#. The standard error is a measure of how much discrepancy to expect between a sample mean #\bar{X}# and the population mean #\mu#.
Normally Distributed Population
If a random sample of size #n# is drawn from a population which is normally distributed with parameters #\mu# and #\sigma#, then the sampling distribution of #\bar{X}# will have the following characteristics:
- Shape. The sampling distribution of the sample mean will be normally distributed.
- Central tendency. The mean of the sampling distribution, #\mu_{\bar{X}}#, will be equal to the population mean #\mu#.
\[\mu_{\bar{X}} = \mu\] - Variability. The standard deviation of the sampling distribution, #\sigma_{\bar{X}}#, is calculated as follows:
\[\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}}\]
\[\bar{X} \sim N(\mu, \cfrac{\sigma}{\sqrt{n}})\]
#\mu_{\bar{X}} = 90#
#\sigma_{\bar{X}} = 2.3905#
Since the population from which the sample is drawn is normally distributed, we know that the sampling distribution of the sample mean is the normally distributed as well.
The expected value of the sample mean, #\mu_{\bar{X}}#, is equal to the population mean #\mu#:
\[\mu_{\bar{X}} = \mu = 90\]
The standard error of the sample mean, #\sigma_{\bar{X}}#, is calculated as follows:
\[\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{20}{\sqrt{70}}=2.3905\]
So #\bar{X} \sim N(90, 2.3905)#.
#\phantom{0}#
Suppose, however, that a random sample is drawn from a population that is not normally distributed. In these cases, the Central Limit Theorem can be invoked to determine the characteristics of the sampling distribution of the sample mean.
#\phantom{0}#
Central Limit Theorem
The Central Limit Theorem states that, as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution with a mean of #\mu_{\bar{X}}=\mu# and a standard deviation of #\sigma_{\bar{X}}=\sigma/\sqrt{n}#, regardless of the shape of the population from which the sample is drawn.
A sample size of #30# or more is large enough to consider the sampling distribution of the sample mean to be approximately normal.
Determine the expected value of the sample mean, #\mu_{\bar{X}}#, and the standard error of the sample mean, #\sigma_{\bar{X}}#.
#\mu_{\bar{X}} = 50#
#\sigma_{\bar{X}} = 0.5270#
A sample size of #n=90# is considered large enough for the Central Limit Theorem to apply.
This means that, although the sample in question comes from a population having a left-skewed distribution, the sampling distribution of the sample mean is approximately normal.
Now that we know that the sampling distribution of the sample mean is approximately normal, we can determine its characteristics:
- #\mu_{\bar{X}} = \mu = 50#
- #\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}} = \cfrac{5}{\sqrt{90}} = 0.5270#
So #\bar{X} \sim N(50, 0.5270)#.
