Statistical Inference: Variance and Goodness of Fit Tests

Chi-square test

Statistical test for variance comparison.

Sample size (n)

Number of observations in a sample.

Sample variance (s²)

Variance calculated from sample data.

Population variance (σ²)

Variance of the entire population.

Degrees of freedom (v)

Calculated as n - 1 for variance tests.

Null hypothesis (H0)

Assumes no effect or difference exists.

Alternative hypothesis (H1)

Assumes a significant effect or difference exists.

Significance level (α)

Threshold for rejecting the null hypothesis.

Critical region

Values leading to rejection of H0.

Goodness of fit test

Compares observed and expected frequencies.

Observed frequency (oᵢ)

Actual count recorded in an experiment.

Expected frequency (eᵢ)

Theoretical count based on hypothesis.

Test for independence

Assesses if two categorical variables are related.

Expected value formula

eᵢ = (row total * column total) / grand total.

Test for homogeneity

Compares proportions across different groups.

Yates' correction

Adjusts Chi-square for continuity in small samples.

Fisher-Irwin exact test

Used for small sample sizes with low frequencies.

Chi-square statistic (χ²)

Sum of squared differences between observed and expected.

Critical value

Threshold value from Chi-square distribution table.

Insufficient evidence

Not enough data to reject the null hypothesis.

Sample standard deviation

Square root of sample variance.

Normal distribution

Symmetrical distribution where mean equals median.

Significance level (0.05)

Common threshold for statistical tests.

Observed outcomes

Results recorded from an experiment.

Expected outcomes

Results predicted based on theoretical models.