The T-Test for Two Independent Samples

Introduction to StatisticsChapter 10

Oct 20-22, 2009Classes #18-19

A limitation of the t-test from chapter 9

Referred to as the one-sample t-test because can only test hypotheses concerning one sampleNeed to have a meaningful comparison

value for hypothesis testing

Another type of hypothesis

Have two groups of people, and want to compare them to see if they’re different or similarNull hypothesis = nothing’s going on, the

two groups are similar (i.e., the means of the two populations are the same)

Keys to keep in mind

Not interested in what the means of the two groups are; only interested in whether the means are different from each other

The two groups are separate, independent groups of peopleBetween subjects design

Research Designs

Independent Measures Between-subjectsMaking a comparison between two groups

Repeated MeasuresWithin-subjectsThe two sets of data are obtained from the

same sample

The t Test for Two Independent Samples

Compare means of two groups Experimental—treatment versus control Existing groups—males versus females

Notation—subscripts indicate group M1, s1, n1 M2, s2, n2

Null and alternative hypotheses translates into translates into

210 : H 0 : 210 H

211 : H 0 : 211 H

Same setup and logic

Compare what’s going on in data to what would be going if null hypothesis was true, taking into account variability from sample to sample

Larger the test statistic, less likely would get that by chance if the null hypothesis were true

Plugging in values

What’s going on in data = difference between means of each sample

What would be going on if null hypothesis were true = 0 (no difference between means)

Variability from sample to sample = standard error of the meanBut now we have two of them, since have

two different samples

Computing two standard errors of the mean n1 = n2

Normally: sM = s2/n

Now with two samples:

S(M1-M2) = s12/n + s22/n

Computing two standard errors of the mean n1 ≠ n2

First, need to deal with two sources of variance – variance in sample 1 and variance in sample 2Pool them together

Sp2 = SS total/df total

Referred to as pooled variance

Pooled Variance

Have two sources of df:Sample 1Sample 2

total df = dfsample 1 + dfsample 2

df = df1 + df2 = (n1-1) + (n2-1) = n1 + n2 - 2

Pooled Variance

S2 pooled = SS1 + SS2

df1 + df2

Computing two standard errors of the mean n1 ≠ n2

Second, compute standard error of the mean Normally: sM = √s2/n With two samples to

deal with:

2

2

1

2

21 ns

ns

s pooledpooledMM

t-test

t =

sample mean diff – population mean diff estimated standard error

2121

212121

MMMM sMM

sMMt

Hypothesis testing

Two-tailed H0: µ1 = µ2, µ1 - µ2 = 0 H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0

One-tailed H0: µ1 ≥ µ2, µ1 - µ2 ≥ 0 H1: µ1 < µ2, µ1 - µ2 < 0

Everything else is the same

As long as the calculated test statistic is more extreme than the critical t value, reject the null

Hypothesis testing

Determine αCritical value of t

df = n1 + n2 - 2

Hypothesis Testing with t Hypothesis Testing with t statisticstatistic

Step 1Step 1: : State the hypotheses. State the hypotheses. Step 2Step 2: : Set Set and locate the critical region. and locate the critical region.

– You will need to calculate the df to do this, and You will need to calculate the df to do this, and use the t distribution table. use the t distribution table.

Step 3Step 3: : Graph rejection regionsGraph rejection regions Step 4Step 4: : Collect sample data and compute t.Collect sample data and compute t.

– This will involve 3 calculations, given SS, n, This will involve 3 calculations, given SS, n, , , and and MM::

a) the sample variance (sa) the sample variance (s22)) b) the estimated standard error (sb) the estimated standard error (sMM)) c) the t statisticc) the t statistic

Hypothesis Testing with t Hypothesis Testing with t statisticstatistic

Step 5Step 5: : Make a decision Make a decision – Need to compare Need to compare ttcalculatedcalculated in Step 3 with in Step 3 with ttcriticalcritical found found

in the t tablein the t table– If its two-tailed:If its two-tailed:

If tIf tcalccalc > t > tCRITCRIT (ignoring signs) (ignoring signs) Reject HReject HOO If tIf tcalccalc < t < tCRITCRIT (ignoring signs) (ignoring signs) Fail to reject HFail to reject HOO

– If its one-tailed: You need to take the sign into If its one-tailed: You need to take the sign into consideration remembering to check back to the graphconsideration remembering to check back to the graph

Step 6Step 6: : Interpret decisionInterpret decision Step 7Step 7: : Find effect sizeFind effect size

Example 11X 2

1X 2X 22X

7 49 14 196 10 100 15 225 9 81 6 36 1 1 9 81 8 64 10 100

11 121 11 12 7 49 11 121 9 81 7 49 7 49

1X 76 21X 644 2X 76 2

2X 880

M1 = 7.6 1n 10 M2 = 10.857 2n 7

4.666.577644

10776,5644

1076644

2

1

212

11 nX

XSS

8571.541429.825880

7776,5880

776880

2

2

222

22 nX

XSS

Example 1

H0: µ1 = µ2, µ1 - µ2 = 0 H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0 df = n1 + n2 - 2 =10 + 7 – 2 = 15 =.05 t(15) = 2.131

Example 1: t-test

t(15) = –2.325, p < .05 (precise p = 0.0345) Reject H0

8.0838 15

121.25714 6985714.544.66

21

212

dfdfSSSSsp

1.4011 1.9632 1.1548 0.8084 7

8.083810

8.0838

2

2

1

2

21 n

sns

s ppMM

325.24011.1257143.3

4011.18571.106.7

21

2121

MMs

MMt

Example 2Example 2 Mr. Fields owns construction companies in both Mr. Fields owns construction companies in both

Newport, Rhode Island and Miami, FloridaNewport, Rhode Island and Miami, Florida He feels that because of the warmer weather He feels that because of the warmer weather

more of his employees in Miami take days off more of his employees in Miami take days off from work (presumably to party on South Beach).from work (presumably to party on South Beach).

In an interesting attempt to find out if this were In an interesting attempt to find out if this were true, he simply surveyed his employees as to true, he simply surveyed his employees as to how many days they frequented the beach in the how many days they frequented the beach in the last year. He believes that the data will reveal last year. He believes that the data will reveal that the Miami workers spend more time at the that the Miami workers spend more time at the beach than do the Newport workers.beach than do the Newport workers.– On the next slide are the results of his surveyOn the next slide are the results of his survey– Sample 1 is Newport; Sample 2 is Miami Sample 1 is Newport; Sample 2 is Miami

Example 2

X1 X12 X2 X2

2 79 6241 71 5041 85 7225 82 6724 76 5776 96 9216 68 4624 58 3364 68 4624 98 9604 61 3721 72 5184 56 3136 108 11664 40 1600 73 5329 78 6084 79 6241 87 7569 75 5625 93 8649 67 4489 56 3136 73 5329 91 8281 61 3721 77 5929 83 6889 78 6084 71 5041

1X 1093 21X 82679 2X 1167 2

2X 93461 M1 = 72.867 1n 15 M2 = 77.8 2n 15

Example 2Example 2 Step 1: State hypothesesStep 1: State hypotheses Sep2: Find tSep2: Find tcriticalcritical

Step 3: Graph Critical Step 3: Graph Critical RegionRegion

Example 2Example 2

Step 4: FindStep 4: Find t tcalculatedcalculated

Step 5: Make decisionStep 5: Make decision

Step 6: Interpret decisionStep 6: Interpret decision

Step 7: Determine effect sizeStep 7: Determine effect size

Effect sizeCohen’s d =

Example 1 Cohen’s d

Example 2 Cohen’s d

221

deviation standarddifferencemean

psMM

-1.1462.843204

257142.30838.8

257142.3deviation standard

difference mean2

21

psMM

-0.34614.27302

4.93333-203.7194.93333-

deviation standarddifference mean

221

psMM

Effect size

r2: amount of information you have about someone’s value on the dependent variable by knowing whether that person is from group 1 or group 2

t2/t2+df

Effect size

Example 1:r2 = t2/t2+dfr2 = ???

Example 2:r2 = t2/t2+dfr2 = ???

Assumptions

Random and independent samplesNormalityHomogeneity of variance

SPSS—test for equality of variances, unequal variances t test

t-test is robust

SPSS Analyze

Compare Means Independent-Samples T Test

Dependent variable(s)—Test Variable(s) Independent variable—Grouping Variable

Define Groups Cut point value

Output Levene’s Test for Equality of Variances t Tests

Equal variances assumed Equal variances not assumed

Output Example 1

T-Test Group Statistics

10 7.6000 2.71621 .858947 10.8571 3.02372 1.14286

ClassClass 1Class 2

Test ScoreN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.145 .708 -2.325 15 .035 -3.25714 1.40115 -6.24362 -.27067

-2.278 12.116 .042 -3.25714 1.42965 -6.36879 -.14550

Equal variancesassumedEqual variancesnot assumed

Test ScoreF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

Homogeneity of Variance

Calculating the pooled variance across the two samples assumes that it’s ok to combine themAssumes that there is homogeneity of

variance

Testing the Homogeneity of Variance

This, itself, is a hypothesisNull hypothesis = no difference between

variance of sample 1 and variance of sample 2

When use SPSS to compute an independent samples t-test, this hypothesis is tested

Two t values to look at

SPSS then computes two t values, one in which the homogeneity of variance assumption is met, and the other in which it is not metFor the t in which the assumption is not met,

the df will not be a whole number Instead, df is lowered somewhat larger

critical t more stringent test of differences between two groups

Telling the world

Same APA style as for one-sample t-tests: t (df) = calculated t value, p informationDon’t forget to give the direction of

significant differences! Give the mean and standard deviation of each group

Credits http://myweb.liu.edu/~nfrye/psy801/ch10.ppt http://faculty.plattsburgh.edu/alan.marks/Stat%20206/The%20t%20Test

%20for%20Two%20Independent%20Samples.ppt