T-Tests, ANOVA, and Correlation in Statistical Analysis
T-Tests
Understanding T-Distributions
True Statement: The larger the sample size, the more a t distribution resembles a normal curve.
Estimating Population Variance
When estimating the variance of a population from a sample, the sample variance cannot be used directly because it tends to be slightly too small—it underestimates the population variance.
T-Test vs. Z-Test
The difference between a t test for a single sample and a Z test for a single sample lies in how the variance of the known population is determined.
Calculating T-Score
Example: If 15 participants take a pretest and a posttest and have a mean difference score of 1.5, and if the standard deviation of the comparison distribution is 0.5, the t score is calculated as follows:
1.5 / 0.5 = 3
Effect Sizes in Difference Score Studies
Studies using difference scores tend to have larger effect sizes than studies using other research designs because the standard deviation of difference scores is usually low.
T-Test for Independent Means
The comparison distribution for a t test for independent means is a distribution of differences between means.
Reducing Variances
The best way to reduce the variances in the distributions of means when conducting a t test for independent means is to increase the size of the samples.
Degrees of Freedom
When using a t table, the degrees of freedom used for a t test for independent means is the sum of the degrees of freedom for the two samples.
Assumptions of T-Test for Independent Means
Key Assumption: The variance of each of the parent populations is the same.
Assumption that Doesn’t Apply: Unweighted averages.
Analysis of Variance (ANOVA)
When to Conduct ANOVA
Analysis of Variance should be conducted only when population variances can be assumed to be equal.
Rejecting the Null Hypothesis
In an analysis of variance, you reject the null hypothesis when the F ratio is much larger than 1.
Calculating the F Ratio
Example: In an analysis of variance with a between-groups population variance estimate of 30 and a within-groups estimate of 25, the F ratio is calculated as follows:
30 / 25 = 1.20
Cohen’s Conventions for R-Squared
According to Cohen’s conventions, an R2 of .14 would represent a large effect.
Estimating Population Variance from Sample Variance
When carrying out an analysis of variance with equal sample sizes, the estimated variance of the distribution of means is converted to an estimated variance of the population of individual scores by multiplying the estimated variance of the distribution of means by the number of scores in each sample.
Two-Way Factorial Design
Example: A consumer psychologist is interested in the effects of Annual Income and Motivations to Shop on shopping patterns of consumers. If Annual Income is divided into two levels (High and Moderate) and Motivation to Shop is divided into three levels (Escape, Necessity, and Socializing), and both are considered in one study, the number of cells will be 6.
In a two-way factorial design, there can be one interaction and two main effects.
Marginal Means
Marginal means are the means of one grouping variable alone.
Identifying Interaction Effects
An interaction effect can be identified by looking at the pattern of cell means.
F Ratio for Column Effects
In a 2 × 2 analysis of variance, the basis for estimating the numerator of the F ratio for the column effects is the variance between the two column marginal means.
Correlation
Scatter Diagrams
A scatter diagram shows the relation of two variables as dots in a two-dimensional graph.
Positive vs. Negative Correlation
The difference between a positive correlation and a negative correlation is that in a negative correlation, high scores on one variable tend to go with low scores on the other, and vice-versa; in a positive correlation, high scores on one variable tend to go with high scores on the other, and low scores tend to go with low scores.
Summed Cross-Products
When figuring a correlation coefficient, the absolute value of the summed cross-products gets larger when the scores of more people are included in the analysis.
Average of Cross-Products of Z Scores
The average of the cross-products of Z scores is a better indicator of the relationship between two variables than the sum of the cross-products of Z scores because the average appropriately measures the strength of the relationship, whereas the sum does not.
Significance Testing of Correlation Coefficient
When testing the significance of the correlation coefficient, the null hypothesis is usually that in the population, the true correlation is zero.
Linear Prediction
Criterion Variable
Example: If a counseling psychologist wants to predict college grades from high school grades, college grades are the criterion variable.
Regression Constant
When making predictions using a linear prediction rule, the baseline number that is added to each prediction is a.
Predicting with Linear Prediction Rule
Example: If a person’s score on a questionnaire has been found to predict observed social skills, and the linear prediction rule uses a regression constant of 16 and a regression coefficient of 3, the predicted level of social skills for a person with a score of 10 on the questionnaire is 46 (16 + 3 * 10 = 46).
The Best Linear Prediction Rule
Considering the number of possible linear prediction rules for predicting Y from X, for any particular set of scores, there is only one best rule.
Error in Linear Prediction
On a scatter diagram, the vertical distance between the dot for the actual score and the regression line represents the error.