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. Normal distribution or large N. 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