Skip to main content
-
Central Limit Theorem
-
Given a random variable with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean, \(\overline X\). If the size (\(n\)) of the sample is sufficiently large, then \(\overline{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\). If the size (\(n\)) of the sample is sufficiently large, then the distribution of the sample means will approximate a normal distributions regardless of the shape of the population. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, \(\frac{\sigma}{\sqrt{n}}\), is called the standard error of the mean.
-
Mean
-
A number that measures the central tendency; a common name for mean is "average." The term "mean" is a shortened form of "arithmetic mean." By definition, the mean for a sample (denoted by \(\overline x\)) is \(\overline x =\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\), and the mean for a population (denoted by \(\mu\)) is \(\mu=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\).
-
Normal Distribution
-
Notation: \(X \sim N(\mu, \sigma)\) for a continuous random variable, where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation. If \(\mu = 0\) and \(\sigma = 1\), the random variable, \(z\), is called the
standard normal distribution
.
-
Sampling Distribution
-
Given simple random samples of size \(n\) from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.
-
Standard Error of the Mean
-
The standard deviation of the distribution of the sample means, or \(\frac{\sigma}{\sqrt{n}}\)