STAT 13 Chapter 2: Generalization, Means, t-Tests, Errors & Significance Flashcards

Mean

The arithmetic average of a dataset; calculated by adding all values and dividing by the number of observations.

Median

The middle value of an ordered dataset; if there are two middle values, average them.

Resistant Statistic

A statistic that is not greatly affected by extreme values or outliers.

Nonresistant Statistic

A statistic strongly influenced by outliers or extreme values.

When to Use Mean

Use when a distribution is roughly symmetric with no major outliers.

When to Use Median

Use when a distribution is skewed or contains outliers.

Standard Deviation (SD)

A measure of spread representing the typical distance of data values from the mean.

Interquartile Range (IQR)

The range of the middle 50% of the data; calculated as Q3 − Q1.

Q1

The first quartile; the 25th percentile of ordered data.

Q3

The third quartile; the 75th percentile of ordered data.

Shape of Distribution

The overall pattern of data values, such as symmetric, skewed, unimodal, or bimodal.

Center of Distribution

A typical or middle value of a dataset, often measured by mean or median.

Spread of Distribution

How variable or dispersed the data values are.

Outlier

A value unusually far from the rest of the data.

Right-Skewed Distribution

A distribution with a long tail to the right; often mean > median.

Left-Skewed Distribution

A distribution with a long tail to the left; often mean < median.

Symmetric Distribution

A distribution where left and right sides are approximately mirror images.

Unimodal Distribution

A distribution with one main peak.

Bimodal Distribution

A distribution with two distinct peaks.

Population

The entire group of individuals or observations of interest.

Sample

A subset of the population used to collect data.

Simple Random Sample (SRS)

A sample where every possible group of size n has an equal chance of being selected.

Parameter

A numerical summary of a population, such as μ or p.

Statistic

A numerical summary of a sample, such as x̄ or p̂.

Population Mean (μ)

The true average value for an entire population.

Sample Mean (x̄)

The average value of a sample; used to estimate μ.

Population Standard Deviation (σ)

The true standard deviation of a population.

Sample Standard Deviation (s)

The standard deviation calculated from sample data; used to estimate σ.

Inference

Using sample data to draw conclusions about a population.

Generalization

Applying conclusions from a random sample to the broader population.

Hypothesis Test

A statistical procedure used to evaluate claims about a population parameter.

Null Hypothesis (H₀)

The default claim being tested, usually representing no effect or a stated value.

Alternative Hypothesis (Hₐ)

The competing claim supported when evidence against H₀ is strong.

Two-Sided Test

A hypothesis test where Hₐ uses ≠, checking for differences in either direction.

One-Sided Test

A hypothesis test where Hₐ uses < or >, checking one direction only.

Significance Level (α)

The chosen probability of making a Type I error; common values are 0.05 or 0.01.

p-value

The probability of obtaining results at least as extreme as observed if H₀ is true.

Reject H₀

Decision made when p-value ≤ α; indicates strong evidence against the null.

Fail to Reject H₀

Decision made when p-value > α; insufficient evidence against the null.

Why Not Accept H₀

Because failing to reject does not prove the null is true.

Type I Error

Rejecting H₀ when H₀ is actually true; false positive.

Type II Error

Failing to reject H₀ when H₀ is actually false; false negative.

Probability of Type I Error

Equal to the significance level α.

Probability of Type II Error

Denoted β; depends on effect size, variability, and sample size.

Power

The probability of correctly rejecting a false null hypothesis; equal to 1 − β.

Increase Power

Use larger sample size, lower variability, larger effect size, or larger α.

Sampling Distribution

The distribution of a statistic across many repeated random samples.

Sampling Distribution of x̄

The distribution of sample means from repeated samples of the same size.

Center of Sampling Distribution of x̄

Equal to the population mean μ.

Spread of Sampling Distribution of x̄

Equal to σ/√n.

Standard Error (SE)

The estimated standard deviation of a statistic’s sampling distribution.

SE of Sample Mean

s/√n when σ is unknown.

Central Limit Theorem (CLT)

For sufficiently large n, the sampling distribution of x̄ is approximately normal.

Why CLT Matters

It allows normal-based inference even when the population is not normal.

Sample Distribution vs Sampling Distribution

Sample distribution is raw data values; sampling distribution is values of a statistic across repeated samples.

z-Statistic for Mean

(x̄ − μ₀)/(σ/√n), used when population SD σ is known.

t-Statistic for Mean

(x̄ − μ₀)/(s/√n), used when population SD σ is unknown.

Why Use t Instead of z

Because σ is usually unknown and must be estimated with s.

t-Distribution

A bell-shaped distribution centered at 0 with thicker tails than normal.

Why t Has Thicker Tails

Extra uncertainty comes from estimating σ using s.

Degrees of Freedom (df)

For one-sample t-tests, df = n − 1.

As df Increases

The t-distribution approaches the standard normal distribution.

One-Sample t-Test

A hypothesis test for a population mean using one quantitative sample.

Conditions for One-Sample t-Test

Random sample, independent observations, sample size at least about 20, and no strong skewness.

Observed Statistic

The value of the sample statistic calculated from the collected data.

Null Distribution

The distribution of the test statistic assuming H₀ is true.

Extreme Statistic

A test statistic far from what is expected under H₀.

Evidence Against H₀

Smaller p-values indicate stronger evidence against the null.

Common α Values

0.10 is relaxed, 0.05 is common, 0.01 is strict.

Large Sample Effect

Larger n reduces standard error and improves precision.

Effect of Outlier on Mean

Can substantially pull the mean upward or downward.

Effect of Outlier on Median

Usually little or no effect unless near center.

Effect of Outlier on SD

Increases standard deviation substantially.

Effect of Outlier on IQR

Usually less affected than SD.

Right-Skew Mean vs Median

In right-skewed data, mean is usually greater than median.

Left-Skew Mean vs Median

In left-skewed data, mean is usually less than median.

Interpret p-value in Context

Assuming H₀ is true, it is the chance of seeing data this extreme or more extreme.

Practical Significance

Whether an effect is large enough to matter in real life, separate from statistical significance.

Statistical Significance

Result where p-value is less than or equal to α.

More Evidence in Testing

Usually means a larger sample size or more precise data.

False Positive

Another term for Type I error.

False Negative

Another term for Type II error.

Benefit of the Doubt in Testing

H₀ is assumed true until evidence strongly suggests otherwise.

Null Value

The parameter value stated in H₀, such as μ = 10.

Parameter of Interest

The population quantity being studied, such as μ.

Quantitative Variable

A variable measured numerically where arithmetic makes sense.

Categorical Variable

A variable describing group membership or labels.

Typical Value

A central value representing the dataset, often mean or median.

Variability

How much values differ from one another.

Precision

How close repeated estimates are to each other; improved by lower SE.

Bias

Systematic tendency of an estimate to miss the true parameter.

Unbiased Estimator

An estimator whose sampling distribution is centered at the true parameter.

Sample Size (n)

The number of observations in the sample.

Larger n Effect on SE

SE decreases because dividing by √n makes estimates more stable.

Decision Rule for Hypothesis Testing

Reject H₀ if p-value ≤ α; otherwise fail to reject H₀.

Contextual Conclusion

State the statistical decision in terms of the original real-world question.

Example H₀: μ = 10

Claims the population mean equals 10.

Example Hₐ: μ ≠ 10

Claims the population mean differs from 10.

Strong Evidence Against H₀

Usually indicated by a small p-value or large magnitude test statistic.

Weak Evidence Against H₀

Usually indicated by a large p-value or small magnitude test statistic.