Unit 7: Inference for Quantitative Data: Means

Inference

Using sample data to make a justified claim about a population parameter, accounting for sampling variability (uncertainty).

Parameter

A numerical characteristic of a population (e.g., the population mean μ).

Statistic

A numerical characteristic computed from a sample (e.g., the sample mean x̄), used to estimate a parameter.

Population Mean (μ)

The true average of a quantitative variable for the entire population of interest.

Sample Mean (x̄)

The average of the sample data; a point estimate of the population mean μ.

Population Standard Deviation (σ)

The true standard deviation of the population; usually unknown in real applications.

Sample Standard Deviation (s)

The standard deviation computed from sample data; used to estimate the unknown population standard deviation σ.

Standard Error (SE)

An estimate of the standard deviation of a statistic; for a one-sample mean, SE = s/√n.

z Statistic

Standardized statistic for a mean when σ is known: z = (x̄ − μ)/(σ/√n), which follows the standard normal model under conditions.

t Statistic

Standardized statistic for a mean when σ is unknown: t = (x̄ − μ)/(s/√n).

Student’s t Distribution

A family of symmetric, bell-shaped distributions centered at 0 with heavier tails than the normal; used for inference about means when σ is unknown.

Degrees of Freedom (df)

A parameter that determines the exact shape/spread of a t distribution; smaller df gives heavier tails.

One-Sample Degrees of Freedom (n − 1)

For one-sample t procedures, df = n − 1.

Heavier Tails

A distribution feature giving more probability to extreme values; t distributions have heavier tails than normal because s varies from sample to sample.

Critical Value (t*)

The cutoff from a t distribution used to create a confidence interval, based on confidence level and df.

Confidence Interval

An interval of plausible values for a population parameter, computed from sample data.

Confidence Level (C%)

The long-run success rate of the interval method: about C% of intervals from repeated random samples would capture the true parameter.

Margin of Error (ME)

How far the confidence interval extends from the point estimate; for a one-sample mean, ME = t*·(s/√n).

One-Sample t Confidence Interval

Interval for a population mean μ (σ unknown): x̄ ± t*·(s/√n), with df = n − 1.

Significance Test

A procedure that uses sample data to evaluate evidence against a claim about a population parameter (e.g., a claim about μ).

Null Hypothesis (H0)

The default claim tested, stated with equality (e.g., H0: μ = μ0).

Alternative Hypothesis (Ha)

The competing claim you seek evidence for (e.g., Ha: μ ≠ μ0, μ > μ0, or μ < μ0).

Two-Sided Alternative (μ ≠ μ0)

An alternative hypothesis looking for a difference in either direction; leads to a two-sided p-value.

One-Sided Alternative (μ > μ0 or μ < μ0)

An alternative hypothesis specifying a direction; p-value is computed in the corresponding tail.

p-Value

Assuming H0 is true, the probability of getting a test statistic at least as extreme as the observed one (in the direction(s) of Ha).

Significance Level (α)

The cutoff probability for deciding statistical significance (commonly 0.05); compared to the p-value.

Reject H0

Decision when p ≤ α; conclude the data provide convincing evidence against H0 (supporting Ha).

Fail to Reject H0

Decision when p > α; conclude the data do not provide convincing evidence against H0 (not the same as “accept H0”).

Type I Error

Rejecting a true null hypothesis (a “false positive”).

Type II Error

Failing to reject a false null hypothesis (a “false negative”).

Power (1 − β)

The probability of rejecting H0 when H0 is false; power = 1 − (probability of Type II error).

Random Condition

Requirement that data come from a random sample or randomized experiment; supports generalization (random sampling) or cause-and-effect (random assignment).

Independence

Condition that observations do not influence each other; often supported by sampling design and the 10% condition when sampling without replacement.

10% Condition

When sampling without replacement, require n ≤ 0.10(population size) to justify independence.

Normal/Large Sample Condition

For t procedures, the sampling distribution is approximately normal if the population is roughly normal or n is large enough for CLT; watch for strong skewness/outliers.

Central Limit Theorem (CLT)

For sufficiently large n, the sampling distribution of x̄ is approximately normal even if the population is not (often summarized by n ≥ 30, but outliers/skewness can still matter).

Outlier

An extreme data value that can strongly affect x̄ and s, especially when n is small, potentially undermining t procedures.

Robustness

The idea that t procedures often work reasonably well even when normality is not perfect, especially for moderate/large n without extreme outliers.

Statistical Significance

A result considered unlikely under H0 (typically p ≤ α); indicates evidence against H0, not necessarily a large or important effect.

Practical Importance

Whether an effect size is meaningful in context; a result can be statistically significant but practically trivial (especially with large n).

Two-Sample t Procedures

Inference methods for comparing means of two independent groups using x̄1 − x̄2 and an unpooled (Welch) standard error when σ1 and σ2 are unknown.

Parameter (μ1 − μ2)

The true difference in population means between group 1 and group 2 (often defined as population 1 minus population 2).

Statistic (x̄1 − x̄2)

The observed difference between sample means, used to estimate μ1 − μ2.

Two-Sample Standard Error

Estimated SD of x̄1 − x̄2 when σ’s are unknown: SE = √(s1²/n1 + s2²/n2).

Welch–Satterthwaite Approximation

Technology-based method for estimating df in two-sample t procedures; df depends on s1, s2, n1, and n2.

Paired (Matched-Pairs) Data

Data where observations come in natural pairs (e.g., before/after on the same person); the two measurements within a pair are not independent.

Difference Variable (d)

For paired data, compute one value per pair: d = (first measurement) − (second measurement); inference is then done on the d’s.

Mean of Differences (μd)

The population mean of the paired differences; the parameter for paired t procedures.

Simulation-Based p-Value

An estimated p-value found by simulating the null model many times and counting the proportion of simulated statistics at least as extreme as the observed statistic.

Median Absolute Deviation (MAD)

A variability measure: the median of the absolute deviations from the median; can be used with simulations to assess unusual variability.