Comparing Two Means: One-sample & Paired-sample

t-tests

Lesson 12

Inferential Statistics

Hypothesis testing Drawing conclusions about differences

between groups Are differences likely due to chance?

Comparing means t-test: 2 means Analysis of variance: 2 or more means ~

Comparing 2 means: t-tests

One-sample t-test Is sample likely from particular

population? Paired-Sample t-test

2 dependent (related) samples Independent-samples t-test

2 unrelated samples ~

The One-sample t-test

Evaluating hypothesis about population taking a single sample Does it likely come from population?

Test statistics z test if known t test if unknown ~

t statistic

Xs

Xt

1ndf

Example: One-sample t-test Survey: college students study 21 hr/wk

Do Coe students study 21 hrs/week? Select sample (n = 16)

unknown Nondirectional hypothesis:

H0 : = 21; H1 : 21

reject H0 if increase or decrease PASW/SPSS: Test value = 21

Assumed from H0 ~

PASW One Sample T Test

Menu Analyze Compare Means One-Sample T Test

Dialog box Test Variable(s) (DV) Test Value (value of testing against) Options (to change confidence intervals) ~

PASW Output

*1-tailed probability: divide Sig. 2-tailed by 2

Paired-Samples t-tests 2 samples are statistically related

Less affected by individual differences reduces variance due to error

Repeated-measures 2 measurements on same individual

Matched-subjects Match pairs on some variable(s) Split pairs into 2 groups ~

Difference Scores Find difference between each score

D = X2 - X1

Requires n1 scores equal n2 scores Calculate mean D

And standard deviation of D ~

N

DD

1

2

n

DDsD

Repeated-measures

2 measurements of same individual Pretest-posttest design

measure each individual twice pretest treatment posttest compare scores ~

Matched-subjects

Match individuals on important characteristic individuals that are related IQ, GPA, married, etc

Assign to different treatment groups each group receives different

levels of independent variable ~

Assumptions: Related Samples

Population of difference scores is normal

Observations within each treatment independent scores for each subject in a

group is independent of other subjects scores ~

Related-samples Hypotheses

Nondirectional H0: D = 0

H1: D 0 Directional

H0: D > 0

H1: D < 0 Remember: it depends on the direction of

the prediction ~

Sample Statistics

Mean difference

Mean for single sample

N

XX

DN

D

Standard Deviation: Related-samples Single sample

1

2

N

DDsD

1

2

N

XXs

1NdfD 1Ndf

Estimated Standard Error

Calculate same as single sample use standard deviation of

difference scores

Ds

N

sD

Test Statistic

Related-samples t test

Since D= 0

D

Dobs s

Dt

D

obs s

Dt

Example Does exercising longer have greater

health benefits? Participants

7 pairs of people matched on age, sex, & weight

Manipulation (IV) 1 of each pair exercised 2 hrs/week 1 of each pair exercised 5 hrs/week

Outcome (DV): Health rating ~

PASW Paired-Sample T Test Data entry

1 column each DV Menu

Analyze Compare Means Paired-Sample T Test

Dialog box Paired Variable(s) (DV) Options (to change confidence intervals) ~

PASW Output