t-Tests

Overview of t-Tests

How a t-Test Works Single-Sample t Independent Samples t Paired t Effect Size

How a t-Test Works The t-test is used to compare

means. The difference between means is

divided by a standard error The t statistic is conceptually

similar to a z-score.

How a t-Test Works

diff. oferror standard

diff. expected - diff. observed t

How a t-Test Works

variationicunsystemat

variationsystematic t

The t-Test as Regression

bo is the mean of one group b1 is the difference between means

If b1 is significant, then there is a significant difference between means

i1o e (IV)b b DV

Single Sample t-test

Compare a sample mean to a hypothesized population mean (test value based on previous research or norms)

Assumptions for Single-Sample t

1. Independent observations. 2. Population distribution is

symmetrical. 3. Interval or ratio level data.

Sampling Distribution of the Mean

The t distribution is symmetrical but flatter than a normal distribution.

The exact shape depends on degrees of freedom

normal distribution

t distribution

Degrees of Freedom

Amount of information in the sample Changes depending on the design

and statistic For a one-group design, df = N-1 The last score is not “free to vary”

Independent Samples t-test

Also called: Unpaired t-test Use with between-subjects,

unmatched designs

Sampling Distribution of the Difference Between Means

We are collecting two sample means and finding out how big the difference is between them.

The mean of this sampling distribution is the Ho difference between population means, which is zero.

m1- m2

x1-x2

sampling distribution of the difference between means

Independent Samples t -test Assumptions

Interval/ratio data Normal distribution or N at least 30 Independent observations Homogeneity of variance - equal

variances in the population

Levene’s Test

Test for homogeneity of variance If the test is significant, the variances

of the two populations should not be assumed to be equal

Independent Samples t-testInterpretation

Sign of t depends on the order of entry of the two groups

df = N1 + N2 - 2 Use Bonferroni correction for multiple

tests Divide alpha level by the number of tests

Paired t-Test

Also called: Dependent Samples or Related Samples t-test

Compares two conditions with paired scores: Within subjects design Matched groups design

Paired Samples t-Test Assumptions

Interval/ratio data Normal distribution or N at least 30 Independent observations

Paired Samples t-test - Interpretation

The sign of the t depends on the order in which the variables are entered

df = N-1 Use Bonferroni correction for multiple

tests

Effect Size

Statistical significance is about the Null Hypothesis, not about the size of the difference

A small difference may be significant with sufficient power

A significant but small difference may not be important in practice

Effect Size with r2

Compute the correlation between the independent and dependent variables

This will be a point-biserial correlation

Square the r to get the proportion of variance explained

Computing r2 from t

r 2 t2

t2 + df

Example APA Format Sentence

A paired samples t-test indicated a significant difference between the number of incorrect items (M = 2.64, SD = 2.54) and the number of lures recalled (M = 3.30, SD = 1.83), t(97) = 2.54, p = .013, r2 = .06.

Take-Home Points

Every t-test compares a systematic difference to a measure of error

Effect size should be reported along with whether a difference is significant