KEY TERMS

Normal Distribution
a continuous random variable (RV) with pdf [latex]f(x) =  \dfrac{1}{\sigma \sqrt{2\pi}}e^{\frac{(x-\mu)^2}{2\sigma^2}}[/latex], where [latex]\mu[/latex] is the mean of the distribution and [latex]\sigma[/latex] is the standard deviation; notation: [latex]X \sim N(\mu,\sigma)[/latex]. If [latex]\mu=0[/latex] and [latex]\sigma=1[/latex] , the RV is called the standard normal distribution.

Standard Normal Distribution
a continuous random variable (RV) [latex]X \sim N(0,1)[/latex] ; when [latex]X[/latex] follows the standard normal distribution, it is often noted as [latex]Z \sim N(0,1)[/latex]

z-score
the linear transformation of the form [latex]z=\dfrac{x-\mu}{\sigma}[/latex]; if this transformation is applied to any normal distribution [latex]X \sim N(\mu,\sigma)[/latex] the result is the standard normal distribution [latex]Z \sim N(0,1)[/latex]. If this transformation is applied to any specific value [latex]x[/latex] of the RV with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex] , the result is called the [latex]z-[/latex]score of [latex]x[/latex] . The [latex]z-[/latex]score allows us to compare data that are normally distributed but scaled differently.

Average
a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

Mean
a number that measures the central tendency of the data; 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 [latex]\bar x[/latex]) is [latex]\bar x = \dfrac{\text{Sum of all values in the sample }}{\text{Number of values in the sample }}[/latex], and the mean for a population (denoted by [latex]\mu[/latex]) is [latex]\mu = \dfrac{\text{Sum of all values in the population }}{\text{Number of values in the population }}[/latex].

Central Limit Theorem for the Mean
Given a random variable (RV) with known mean [latex]\mu[/latex] and known standard deviation, [latex]\sigma[/latex], we are sampling with size [latex]n[/latex], and we are interested in a RV: the sample mean, [latex]\bar X[/latex]. If the size [latex]n[/latex] of the sample is sufficiently large, then [latex]\bar X \sim N\left(\mu,\frac{\sigma}{\sqrt n}\right)[/latex]. If the size [latex]n[/latex] 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, [latex]\dfrac{\sigma}{\sqrt n}[/latex], is called the standard error of the mean.

Central Limit Theorem for the Proportion
If the population proportion is [latex]p[/latex] and we are sampling with size [latex]n[/latex], and we are interested in a RV: the sample proportion, [latex]\hat P[/latex]. If [latex]np \ge 5[/latex] and [latex]nq \ge 5[/latex] where [latex]q = 1 - p[/latex],  then the distribution of sample proportions [latex]\hat p[/latex]‘s will be approximately normal with [latex]\hat P \sim N\left(p,\sqrt {\dfrac{pq}{n}}\right)[/latex]. The standard deviation of the distribution of the sample proportions, [latex]\sqrt {\dfrac{pq}{n}}[/latex], is called the standard error of the proportion.

Sampling Distribution
Given simple random samples of size [latex]n[/latex] 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 [latex]\dfrac{\sigma}{\sqrt n}[/latex].

Standard Error of the Proportion
the standard deviation of the distribution of the sample means, or [latex]\sqrt {\dfrac{p(1-p)}{n}}[/latex] .