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Population

Entire group a researcher wants to study.

Sample

Subset of the population used to draw conclusions.

Parameter

Numerical value describing a population.

Statistic

Numerical value describing a sample.

Sampling variability

Natural variation in statistics across samples.

Variable

Any characteristic that can take different values.

Categorical variable

Places individuals into categories.

Quantitative variable

Numerical measurements.

Nominal variable

Categorical without order.

Ordinal variable

Categorical with order.

Interval variable

Numerical scale with equal intervals, no true zero.

Ratio variable

Numerical scale with true zero.

Distribution

Pattern of variation for a variable.

Mean

Arithmetic average.

Median

Middle value when ordered.

Standard deviation

Average distance from the mean.

IQR

Middle 50% of data (Q3 − Q1).

Z-score

Number of SDs a value is from the mean.

Outlier

Value far from the rest of the data.

Skewness

Asymmetry of distribution.

Unimodal

One peak.

Bimodal

Two peaks.

Resistant statistic

Unaffected by outliers (median, IQR).

Non-resistant statistic

Affected by outliers (mean, SD).

Boxplot

Graph showing median, quartiles, IQR, and outliers.

Probability

Long-run relative frequency of an event.

Law of Large Numbers

Long-run estimates converge to true probability.

Complement rule

P(not A) = 1 − P(A).

Addition rule

P(A or B) = P(A)+P(B)−P(A and B).

Independence

P(A and B) = P(A)P(B).

Mutually exclusive

Events that cannot occur together.

Random variable

Numeric outcome of a random process.

Expected value

Long-run average of a random variable.

Standard normal distribution

Normal(0,1).

Z-table

Provides P(Z < z).

P(Z > z)

1 − table value.

P(|Z| > z)

2 × tail probability.

Central Limit Theorem

Large samples → sampling distribution of x̄ becomes normal.

Normal approximation for p̂

Valid if np ≥ 10 and n(1−p) ≥ 10.

Standard error

Standard deviation of a statistic.

SE of p̂

sqrt[p̂(1−p̂)/n].

SE of p̂ under H₀

sqrt[p₀(1−p₀)/n].

SE of mean (unknown σ)

s/√n.

SE of mean (known σ)

σ/√n.

Larger n

Smaller SE, more precision.

Sampling distribution

Distribution of a statistic over repeated samples.

Unbiased estimator

Statistic whose mean equals the parameter.

Point estimate

Single best guess for parameter.

Interval estimate

Confidence interval.

Confidence interval

Range of plausible parameter values.

Confidence level

Long-run percentage of intervals that contain the parameter.

CI interpretation

"We are 95% confident the interval contains the true parameter."

Margin of error

Critical value × SE.

Higher confidence

Wider interval.

Larger sample size

Narrower interval.

z* for 95% CI

1.96.

t* critical value

Larger than z* for same confidence.

CI for mean

x̄ ± t*·SE.

CI for proportion

p̂ ± z*·SE.

CI excludes null value

Reject H₀.

CI includes null value

Fail to reject H₀.

Sample size formula (prop)

n = (z/ME)² p(1−p*).

Sample size formula (mean)

n = (z*σ/ME)².

SE vs SD

SE relates to the statistic; SD to raw data spread.

As n increases

Sampling distribution becomes narrower.

Hypothesis test

Statistical procedure for evaluating a claim.

Null hypothesis (H₀)

No effect, no difference, equality.

Alternative hypothesis (Hₐ)

Effect or difference.

H₀ always contains

"=".

One-tailed test

Directional claim.

Two-tailed test

Non-directional claim.

Test statistic

Standardized measure of difference.

P-value

Probability of getting results as extreme as observed, assuming H₀ true.

Small p-value

Strong evidence against H₀.

Large p-value

Weak evidence against H₀.

Significance level α

Cutoff for rejecting H₀.

Reject H₀

p < α.

Fail to reject H₀

p ≥ α.

Type I error

Rejecting a true H₀.

Type II error

Failing to reject a false H₀.

Power

1 − β (probability of detecting a true effect).

Increasing n

Increases power.

Increasing effect size

Increases power.

Lower α

Lowers Type I, raises Type II error.

Statistical significance

p < α.

Practical significance

Large real-world effect.

Random sample

Required for inference to population.

Random assignment

Required for causal conclusions.

Observational study

No assignment, no causation.

Experiment

Random assignment, causal conclusions.

Confounding

Outside variable affects both explanatory & response variables.

Matched pairs

Reduces variability, increases power.

z-test for mean (σ known)

(x̄−μ₀)/(σ/√n).

t-test for mean (σ unknown)

(x̄−μ₀)/(s/√n).

One-proportion z-test

(p̂−p₀)/√[p₀(1−p₀)/n].

Two-proportion z-test

Use pooled p̂.

Pooled p̂

total successes / total sample size.

Two-proportion z-statistic

(p̂1−p̂2)/√[p̂(1−p̂)(1/n1+1/n2)].

Independent two-sample t

(x̄1−x̄2)/√(s1²/n1 + s2²/n2).

Pooled t-test

Use when variances are equal.