AP Stats must-know term

Alternative Hypothesis

states that a treatment has had an effect or caused a change in the population

Bias

describes a study which systematically favors certain outcomes

Binomial Distribution

the distribution of the probabilities of X successes out of n trials, calculated using p as the probability of any single success - B(n, p)

Blind

describes an experiment in which the subjects do not know which treatment they are getting

Blocking

a statistical design which creates groups that are similar in some way, and then randomizes the treatments within each block

Central Limit Theorem

states that when an SRS is drawn from a population with mean µ and standard deviation σ, the sampling distribution for the sample mean will be approximately normally distributed, and have a mean µ and a standard deviation σ/√n

Chi-Square Distributions

a family of skewed-right distributions which take on only positive values and are defined by their degrees of freedom - the specific shape of the Chi-Square Distribution changes as the sample size changes

Chi-Square Goodness-of-Fit Test

used to determine if a population has a certain hypothesized distribution

Chi-Square Test for Homogeneity

used to determine if every category in the population has the same population

Chi-Square Test for Independence/Chi-Square Test for Association

used to determine if there is a relationship between two categorical variables

Coefficient of Determination

tells what percent of the change in the response variable can be attributed to the change in the explanatory variable - symbolized as r 2

Complement of an Event

the set of all outcomes not defined as successful outcomes for any event

Conditional Probability

the probability of an event occurring if it is known that another specific event has already occurred

Confidence Interval

an interval estimate of a parameter calculated using a sample from that population

Confidence Level

the probability that the desired parameter will fall into a confidence interval if many intervals were calculated from samples of the same size

Confounding Variable

a variable which could affect the result of a statistical test but has not been controlled for

Continuous Random Variable

a random variable which takes on all values in an interval of numbers

Control Group

any group of subjects who receive either a placebo or no treatment at all during an experiment

Correlation

measures the direction and strength of the linear relationship between two quantitative variables - symbolized as r

Critical Value

a value (z-score, t-score, or χ2 value) used in a hypothesis test to help determine if the null hypothesis should be rejected

Cumulative Distribution Function

A function which calculates the sum of the probabilities for each possible value for any random variable X

Degrees of Freedom

a value used to help determine significance for a t-test or a Chi-Square test - measured as n-1 in most cases, or (r-1)(c-1) when dealing with two-way tables

Dependent Trials

trials whose probability is affected by the outcome of previous trials

Density Curve

a curve used to represent a distribution - a density curve is always on or above the horizontal axis and has a total area of exactly 1 underneath it

Discrete Random Variable

a random variable with countable outcomes

Mutually Exclusive Events (disjoint)

events which cannot occur at the same time

Distribution

a list of what values a variable takes on and how often it takes on each one of those values

Double Blind

describes an experiment in which neither the subjects nor the researcher know which treatment each subject is getting

Empirical Rule

(the 68-95-99.7 rule)

is used as an approximation for what percent of the data falls within 1, 2, or 3 standard deviations of the mean in any normal distribution

Experimental Units

the individuals on which an experiment is conducted - if the test is being conducted on humans, the units are called Subjects

Explanatory Variable/Independent Variable

attempts to explain the observed outcomes in a statistical study

Exploratory Data Analysis

uses graphs and numerical summaries to describe the variables in a data set and the relationships among them

Factor

any explanatory variable in an experiment

Five Number Summary

a method to describe a data set using the minimum, first quartile, median, third quartile, and maximum points in the data set

Geometric Distribution

a distribution of probabilities of when the first successful outcome occurs in a probability experiment

Hypothesis Test/Significance Test

a type of inference used to determine the feasibility of an assumed population parameter

Independent Trials

trials whose probabilities are not affected by the outcome of previous trials

Individuals

people or objects described by a set of data

Inference

the statistical process of drawing conclusions about a population by examining data from a sample

Influential Point

a point which, if removed from the data set, would markedly change the regression equation for that data set

Interquartile Range (IQR)

the difference between the third and first quartiles of a data set

Law of Large Numbers

states that as increased numbers of observations are drawn from any population, the mean of the observations eventually approaches the mean of the population as closely as we would like to estimate it, and remains that close or closer

Least Squares Regression Line

a regression line which makes the sum of the squares of the vertical distances from the data points to the line as small as possible

Level

a numerical value of a factor of an experiment

Matched Pairs

a statistical design which compares two treatments - this is usually done with one sample receiving each treatment over a different time period

Mean (expected value)

the "average" of a data set

Median

the point at which 50% of the data is above and 50% of the data is below

Mutually Exclusive Events

see Disjoint Events

Nonresponse

a type of bias that occurs when an individual chosen for a sample cannot be contacted or chooses not to participate

Normal Distribution

a symmetric, bell-shaped distribution in which approximately 68% of the data lies within one standard deviation of the mean, 95% lies within two standard deviations of the mean, and 99.7% lies within three standard deviations of the mean - all normal distributions can be defined by their mean and standard deviation

Null Hypothesis

states that either a treatment has had no effect on a population, or that the population has not changed

Observation

any single point from a data set

Outlier

an individual observation that falls outside the pattern of the data set - often defined as any number that is 1.5(IQR) outside of Q1 or Q3

P-value

the probability that the observed outcome would take on a value as extreme or more extreme than observed if the null hypothesis were true

Parameter

a number that describes a population

Percentile

tells what percent of a data set falls below the given observation

Placebo

a false treatment which should have no effect on an experiment - placebos should appear to be the same as the actual treatment

Pooled Procedures

occurs when separate samples are combined into a single sample for analysis - this should only be done if it is known that the variances of the two populations are equal

Population

the entire group of individuals that we want information about

Power of a Hypothesis Test

the probability that the test will reject the null hypothesis when the null hypothesis is false - the power is equal to 1 minus (probability of a Type II error for the given alternative)

Probability

the proportion of times an outcome would occur over a large number of trials

Probability Distribution Function

a function which assigns a probability for each possible value for any discrete random variable X

Proportion

tells what percent of a data set falls into a given category

Qualitative Variable

a variable which takes on a non-numeric description

Quantitative Variable

a variable which takes on a numeric value

Quartiles

observations which fall at the 25th, 50th, and 75th percentiles of a data set

Percentiles

Fall at the 25th, 50th, and 75th percentiles of a data set.

Range

The difference between the maximum and minimum values of a data set.

Random

When individual outcomes are uncertain, but there is a pattern to the distribution of the outcomes over time.

Random Variable

A variable whose value is a numeric outcome of a random phenomenon.

Randomization

Using the laws of probability - this is done to select members for a sample and also to assign treatments to specific samples in experiments.

Regression Line

A straight line that describes how a response variable changes as the explanatory variable changes.

Residual

The difference between an observed value of a response variable and its predicted value from a regression equation.

Response Variable/Dependent Variable

Measures the outcome of a statistical study

Robustness

A measure of how much the P-value of a test is affected if the conditions of the hypothesis test are not met.

Sample

A part of the population used to gather information about the entire population.

Sample Space

A list of all possible outcomes for a random event.

Sampling Distribution

A distribution of values taken by a statistic in all possible samples of the same size from the same population.

Sampling Frame

A list from which a sample is chosen - ideally the sampling frame consists of the entire population.

Significance Level

The point at which it will be determined that a result is statistically significant.

Simple Random Sample (SRS)

A sample in which every member of the population has the same probability to be chosen, and every group of size n has the same probability to be chosen.

Simulation

A method for collecting data which uses the laws of probability to represent all possible outcomes of an experiment.

Skewed

Describes a distribution whose histogram extends much farther to one side of the mean than the other - the distribution is said to be skewed in the direction of this 'tail'.

Standard Deviation

Square root of the variance - used as a common measure of spread for a data set.

Standard Error

The standard deviation of a sampling distribution - measures the amount of expected error per standard deviation from the mean of the distribution.

Standard Normal Distribution

A normal distribution with a mean of zero and a standard deviation of one.

Statistic

A number that describes a sample.

Statistically Significant

An observed effect so far removed from the mean that it would be unlikely to occur by chance alone.

Stratified Random Sample

A sample chosen by splitting the population into several well-defined groups, then taking an SRS from each group.

Symmetric

Describes a distribution whose histogram has its left and right sides as mirror images of each other.

t-Distributions

A family of symmetric, bell-shaped distributions with a standard deviation larger than that of the standard normal distribution - the specific shape of the t-distribution changes as the sample size changes - this distribution is defined by its degrees of freedom.

Treatment

A specific experimental condition applied to an experimental unit or subject.

Treatment Group

A group of subjects who receive an actual treatment during an experiment.

Type I Error

When the null hypothesis is rejected but it is in fact true - the probability of a Type I Error is the significance value for that test.

Type II Error

When the null hypothesis is not rejected but it is in fact false - the probability of a Type II Error must be calculated for a specific alternative test value.

Unbiased Statistic

A statistic from a sampling distribution whose mean must be equal to the mean of the population.

Undercoverage

A type of bias that occurs when some groups of a population are left out of the selection process for the sample.

Variability

Describes the spread of a data set.

Variable

Any characteristic of an individual.