e-Statistics
e-Statistics
For Students
For Contributors
Data in Statistical Studies
Textbook Data Sets
Worksheet Data Sets
Data and Story Library
Exploratory Graphics
Stem and Leaf Plot
Histogram
Boxplot
Scatter Plot
QQ plot
Bar Graph
Descriptive Statistics
Summary Statistics
Frequency Table
Grouped Data
Transformed Data
Normal Distribution
Normal Density Function
Z-score
Critical Value
Central Limit Theorem
Discrete Distributions
Binomial Distribution
Frequency Function
Normal Approximation
Inference on Mean
t-Distribution
Confidence Interval
Hypothesis Test
Power of Test
One-sided CI
Comparison of Two Groups
Test for Independent Groups
Nonparametric Test
Test for Paired Data
Simple Linear Regression
Regression Line
R Square
Residual Plot
Inference on Parameters
Predicting Responses
One-Way ANOVA Model
F-Distribution
Analysis of Variance
Strip Chart
Multiple Comparisons
Multiple Linear Regression
Model Estimate and Residuals
Model Selection and F-test
Two-Way Additive Model
Test of Effect
Analysis of Variance
Inference on Variances
Testing a Standard Deviation
Test for Two Groups
Test for Homogeneity
Inference on Proportions
Testing a Proportion
Comparison of Two Proportions
Chi-square Test
Chi-square Distribution
Goodness of Fit
Test for Homogeneity
Logistic Regression
Inference on parameters
Adjusted log OR

Test for Two Groups

In the comparison of population means in two independent groups, say “Group 1” and “Group 2,” test procedures (pooled t-test and Wilcoxon test) often assume that the two population variances are approximately equal. This assumption itself can be treated as hypothesis test, and F-test is introduced for the plausibility of equal variances.

Data can be arranged in two columns each of which contains data for the respective groups. The first column specifies data of Group 1, and the second column data of Group 2.

When data is arranged in a form of grouped data, one column (as identified at "SD") contains the whole data, and is "grouped by" the column of categorical values identifying Group 1 and 2.

The test is actually concerned with how Group 1 and Group 2 differ in terms of their respective population variance $ \sigma_1^2$ and $ \sigma_2^2$.

$ H_A:\hspace{0.05in}\sigma_1^2$ $ \sigma_2^2$

Test procedure is based on the sample standard deviations $ S_1$ and $ S_2$ from Group 1 and 2 with the respective sample sizes n and m, and test results are known to be sensitive to the appropriateness of normality assumption. The test statistic

$ F = \dfrac{S_1^2}{S_2^2}$ =

is distributed as F-distribution with $(n-1,m-1)$ = ( , ), and likely observed around unity under the null hypothesis “ $H_0: \sigma_1^2 = \sigma_2^2$.” The opposite of such an observation is expressed by the p-value = being less than $ \alpha$, suggesting an evidence against the null hypothesis $ H_0$ in favor of $ H_A$. In order to support the plausibility of equal variances, we may choose to accept $ H_0$ when we fail to reject it.

© TTU Mathematics