stats midterm

Simulation

A method used to imitate a real-world statistical situation using random processes

Random process

A process that uses chance to determine outcomes, often with random numbers

Repeated trials

Running a simulation many times to observe the distribution of outcomes

Response variable (simulation)

The outcome recorded for each trial

Sampling variability

The natural differences that occur from sample to sample

Population

The entire group we want information about

Sample

A subset of the population used to draw conclusions

Census

A study that includes every member of the population

Sample size

The number of individuals in a sample

Representative sample

A sample that accurately reflects the population

Simple random sample (SRS)

Every possible sample of the same size has an equal chance of being chosen

Sampling frame

The list of individuals from which a sample is drawn

Stratified random sample

Population divided into similar groups (strata), then SRS taken from each

Cluster sample

Population divided into clusters, randomly select clusters and survey everyone in them

Systematic sample

Selecting every nth individual after a random start

Pilot survey

A trial run of a survey used to improve the final version

Voluntary response bias

Bias caused when individuals choose to participate

Convenience sampling

Choosing individuals because they are easy to reach

Undercoverage

Some groups in the population are left out or undersampled

Nonresponse bias

Bias from differences between respondents and nonrespondents

Response bias

Bias caused by the wording or design of survey questions

Observational study

Researchers observe but do not impose treatments

Retrospective study

Uses data from past events

Prospective study

Collects data going forward in time

Experiment

A study that imposes treatments to determine cause and effect

Explanatory variable

The variable that explains or causes changes

Response variable

The variable that is measured as an outcome

Control group

A group that receives no treatment

Randomization

Randomly assigning treatments to reduce bias

Replication

Using enough subjects to reduce variability

Blocking

Grouping similar subjects to reduce variability

Matched pairs design

A design comparing two treatments on the same or similar subjects

Factor

A variable manipulated in an experiment

Levels

The different values of a factor

Treatment

A specific combination of factor levels

Confounding variable

A variable that affects the response but is not controlled

Lurking variable

A hidden variable that affects both explanatory and response variables

Blinding

Subjects do not know which treatment they receive

Double blinding

Neither subjects nor evaluators know the treatment

Placebo

A fake treatment used for comparison

Placebo effect

A response caused by belief in treatment rather than the treatment itself

Sample space

The set of all possible outcomes

Law of large numbers

As trials increase, empirical probability approaches true probability

Complement rule

P(A) + P(not A) = 1

Mutually exclusive events

Events that cannot happen at the same time

Independent events

The outcome of one event does not affect another

Conditional probability

The probability of an event given another has occurred

Tree diagram

A diagram that shows all possible outcomes and probabilities

Probability model

A list of outcomes and their probabilities

Random variable

A variable whose value depends on chance

Expected value

The long-run average outcome (weighted mean)

Variance

The average squared distance from the mean

Standard deviation

The square root of variance, measures spread

Bernoulli trials

Trials with two outcomes, constant probability, and independence

Geometric distribution

Probability the first success occurs on the nth trial

Binomial distribution

Probability of a certain number of successes in fixed trials

10% condition

The sample is less than 10% of the population

Normal model

A bell-shaped distribution defined by mean and standard deviation

Success/failure condition

np ≥ 10 and nq ≥ 10

z-score

The number of standard deviations a value is from the mean

Scatterplot

A graph that shows the association between two quantitative variables

Explanatory variable

The variable on the x-axis that explains or predicts

Response variable

The variable on the y-axis that responds or is predicted

Form

The overall shape of a scatterplot (linear or non-linear)

Direction

Whether the association is positive, negative, or unclear

Strength

How closely the points follow a pattern; less scatter means stronger association

Unusual features

Outliers, clusters, or gaps in a scatterplot

Correlation coefficient (r)

A unitless measure of the strength and direction of a linear relationship between −1 and 1

Linear condition

The requirement that a scatterplot be linear in order to use correlation

Correlation

A numerical description of linear association that must include r

Association

A relationship between variables that does not require mentioning r

Least squares regression line

The line that minimizes the sum of squared residuals and best fits the data

Regression equation

An equation that models the relationship between explanatory and response variables

Slope (b₁)

The expected change in y for each one-unit increase in x

Intercept (b₀)

The predicted value of y when x = 0

Residual

The difference between an observed value and a predicted value

Positive residual

Observed value is greater than predicted value

Negative residual

Observed value is less than predicted value

Slope formula

b₁ = r × SD(y) ÷ SD(x)

Intercept formula

b₀ = mean(y) − b₁(mean(x))

Residual plot

A graph of residuals versus explanatory variable

Good residual plot

Residuals randomly scattered around zero, indicating a linear model is appropriate

Coefficient of determination (R²)

The proportion of variability in y explained by x

Interpretation of R²

If R² = 0.80, then 80% of the variation in y is explained by x

Extrapolation

Using a regression line to predict values outside the range of observed data

Why extrapolation is risky

Predictions far from the mean of x are less reliable

High leverage point

A point with an x-value far from the mean of x

Outlier

A point with a y-value far from its predicted value

Influential point

A point that significantly affects the regression model

Non-influential point

A point that is neither high leverage nor an outlier

Grouped data point

A point representing an average of several values, reducing variability

Regression output (constant)

The y-intercept of the regression line

Regression output (coefficient)

The slope associated with the explanatory variable

s (standard deviation of residuals)

The average distance from observed points to their predicted values

Interpretation of s

The model’s predictions are typically within s units of the actual values