One-Way ANOVA makeup exam

Descriptive Statistics

Summarize and describe the main features of a dataset (e.g., mean, median, SD, range). They describe what is observed in a sample.

Inferential Statistics

Use sample data to draw conclusions or make generalizations about a population. Their goal is to make inferences about the population using a sample.

Population

The entire group of individuals of interest.

Parameter

A numerical characteristic of a population (usually unknown). Example: If a sample of 206 students yields a mean of 9.7, the mean that would be obtained from all students.

Sample

A subset of the population

Statistic

A numerical summary of the sample, used to estimate the population parameter. Example: The observed mean of 9.7 college applications from 206 students.

Types of Inference

Statistical inference involves hypothesis testing, interval estimation (Confidence Intervals), and point estimation.

Assumed Hypothesis

In hypothesis testing, the null hypothesis (H0​) is assumed true at the start. The test evaluates whether sample data provide enough evidence to reject H0​ in favor of the alternative (HA​).

Test Statistic (General Form)

A standardized value computed from sample data used to decide whether to reject H0​. It measures how far the sample mean deviates from the null hypothesis mean in standard error units.

P-value

The probability of obtaining data as extreme or more extreme than the observed result, assuming H0​ is true. A low p-value suggests the observed data are unlikely under H0​, suggesting evidence against it.

Decision Rule (p-value method)

If p<α→ reject H0​. If p≥α→ fail to reject H0​.

Significance Level (α)

The maximum acceptable probability of making a Type I error (rejecting a true H0​). It sets the upper limit for the Type I error rate.

Statistical Significance

Occurs when the observed result is unlikely under the null hypothesis. It is obtained when the p-value is lower than the significance level

Rejection Region

A range of test statistics (determined by critical value(s)) such that if the observed test statistic falls within it, H0​ is rejected.

Elements Changing (p-value method)

The test statistic and p-value change with each sample (they depend on data); α is fixed.

Elements Changing (Critical-value method)

The test statistic changes with each sample (depends on data); α and its corresponding critical value(s) are fixed.

One-Tailed Test

Looks for an effect in one direction only (e.g., μ>μ0​). The p-value is the area in one tail beyond the observed test statistic.

Two-Tailed Test

Examines both directions (e.g., μ=μ0​). The p-value is the combined area in both tails.

Type I Error (α)

Rejecting H0​ when it is actually true (a false positive). This probability is set by the significance level α.

Type II Error (β)

Failing to reject H0​ when it is actually false (i.e., when HA​ is true) (a false negative).

Power

The probability of correctly rejecting a false H0​. Power =1−β. Higher power means a greater ability to detect a true effect.

Factors Increasing Power

Larger sample size (reduces standard error), higher α level (easier to reject H0), and larger effect size.

Purpose

A CI gives a plausible range of values for the population parameter, providing a "net" rather than a "spear" (point estimate).

General Form

Point estimate ± margin of error. Margin of error is zα/2​×SE or tα/2,df​×SE.

Interpretation

Correct: We are 95% confident that the average number of college applications of HU students is between 8.5 and 10.9

Confidence Level Meaning

If the same sampling and construction of a CI were performed repeatedly, with 95% probability the resulting interval would cover the true population parameter.

Effect of Sample Size

Increasing the sample size (n) makes the standard error (SE) smaller, which makes the margin of error smaller and the CI narrower.

Effect of Confidence Level

A higher confidence level (e.g., 99% instead of 95%) involves a greater critical value (z∗/t∗), making the margin of error larger, and the CI wider.

When to Use

Use ANOVA when comparing three or more group means with one continuous dependent variable (DV) and a categorical independent variable (IV).

ANOVA vs. t-test

When comparing exactly two groups, a one-way ANOVA yields the same p-value as an independent-samples t-test (t² = F for df between​= 1).

ANOVA Hypothesis

H0​: μ1 ​= μ2​ = μ3​ = ⋯= μk​ (All group means are equal). HA​: At least one mean differs among groups.

Idea Behind ANOVA

ANOVA partitions total variability into between-group (treatment) and within-group (error) variability. If between-group variance is much larger than within-group variance, the result is significant

F-statistic

The test statistic for ANOVA. It is the ratio of Mean Square Between (MSG) to Mean Square Within (MSE) (F = MSG/MSE)

MSG (Mean Square Between)

Measures variance due to differences between group means, reflecting the treatment effect. Under H0​, it is an estimate of the common population variance based on group mean differences.

MSE (Mean Square Within)

Measures variance within groups, reflecting random error or noise. Under H0​, it is an estimate of the common population variance based on within-group variability.

F-distribution Properties

F ≥ 0 (never negative). It is a right-skewed distribution. Its shape is determined by its numerator and denominator degrees of freedom (dfG​ and dfE​)

F-test p-value

p= P(F ≥ Fobt​). It is the probability of obtaining the observed or larger F-statistic if H0​ is true (i.e., all group means are equal).

Degrees of Freedom

df between​= k−1 (k = number of groups); df within​= N−k (N = total sample size); df total​= N−1.

ANOVA Conclusion

If p<α, we reject H0​ and conclude that at least one mean is different

Inflation of Type I Error

Conducting multiple t-tests to find specific differences after ANOVA increases the probability of making a Type I Error across the series of tests.

Bonferroni Correction

A procedure used for multiple pairwise comparisons that suggests using a more stringent significance level: α^* =α/K, where K is the number of comparisons.

ANOVA Assumptions

Independence of observations, normality of residuals, and homogeneity of variances.

1. Independence of observations

2. Normality

3. Equal Variances

Two-Way ANOVA

Analyzes data from a study with two factors (or Independent Variables).

Factors/IVs

The independent variables being manipulated or grouped by the researcher.

Interaction Effect

Occurs when the effect of one factor on the DV changes depending on the level of the other factor.

Interpreting Main Effects (with interaction)

If an interaction effect exists, the main effects of each factor should not be interpreted independently; they must be interpreted conditionally (i.e., conditional on a level of the other factor).

Visualizing Interaction

If cell means are plotted on a line graph, non-parallel lines indicate an interaction; parallel lines indicate no interaction.