Key Concepts

• The sample mean is a single value (point estimate) used to estimate the value of the population mean.
• An error bound is added and subtracted to the sample mean to form a confidence interval.
• The size of the error bound is based on the confidence level, standard deviation of the population and the sample size.
• The z-score for a confidence interval for a mean is based on a normal distribution when the population standard deviation is known.
• The confidence level refers to the long-run success rate of all possible confidence intervals from all possible random samples of a given sample size.
• For a given confidence level, intervals are narrower for larger sample sizes.
• For a given sample size, intervals are wider for larger confidence levels.
• The sample size can be calculated for a specified level of confidence and error bound.

Glossary

Confidence Interval (CI): an interval estimate for an unknown population parameter. This depends on:

• the desired confidence level
• the information that is known about the distribution (for example, known standard deviation),
• and the sample and its size.

Confidence Level (CL): the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.

Error Bound for a Population Mean (EBM): the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation.

inferential statistics: also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that four percent of the production is defective.

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

parameter:  a numerical characteristic of a population

point estimate: a single number computed from a sample and used to estimate a population parameter

standard deviation: a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation