TWO-SAMPLE t-TESTS. Independent versus Related Samples Your two samples are independent if you randomly assign individuals into the two treatment

TWO-S

AMPLE t-

TESTS

TWO-SAMPLE t-TESTS

Independent versus Related Samples Your two samples are independent if you

randomly assign individuals into the two treatment groups.

Your samples are related if either▪Each person in sample A is matched to a partner in sample B (matched samples) OR▪Each person in the study is measured under both conditions (repeated measures)

INDEPENDENT SAMPLES t-TEST

The steps to conducting an independent samples t-test are:

State your research question hypothesesDetermine your rejection ruleCalculate the t-statisticUse your rejection rule to decide whether you

Reject the null hypothesisFail to reject the null hypothesis


Two-Tailed Test Hypotheses H0: = 0 H1: ≠ 0


Two-tailed Rejection Rule If you are using the t-table,

reject H0▪ if t(obt) > t(crit, , n1 + n2 - 2) OR ▪ if t(obt) < -t(crit, , n1 + n2 - 2)

If you are using the SPSS printout, reject H0 if p < .05


One-tailed test where you believe the scores in sample 1 will be greater than the scores in sample 2. The hypotheses are

H0: ≤ 0 H1: > 0

INDEPENDENT SAMPLES t-TEST One-Tailed Rejection Rule where

you believe the scores in sample 1 will be greater than the scores in sample 2.

If using the t-table, reject H0 if t(obt) > t(crit, , n1 + n2 - 2)

If using the SPSS printout, reject Ho if p< .05.

INDEPENDENT SAMPLES t-TESTOne-tailed test where you believe the scores in sample 1 will be less than the scores in sample 2. The hypotheses are

H0: ≥ 0H1: < 0

INDEPENDENT SAMPLES t-TESTOne-tailed Rejection Rule where

you believe the scores in sample 1 will be less than the scores in sample 2.

If using the t-table, reject H0 if t(obt) < -t(crit, , n1 + n2 - 2)

If using the SPSS printout, reject Ho if p < .05.


Calculating the independent samples t-statistic

1. Calculate the mean for each of the two samples

2. Calculate the sum of squares for each of the two samples. What’s a sum of squares? It’s the same as the formula for the variance, except don’t do the final step of dividing by n – 1. SS1 for the first sample and SS2 for the second

n

XXSS

22 )(

21, XX


Calculating the independent samples t-statistic:

Step 1. NOBODY PANIC!

2121

21

21

11

2 nnnn

SSSS

XXtobt


Step 2. Find the difference between the group means. Note: It doesn’t matter which group you designate as sample 1 and which as sample 2, AS LONG AS you take into consideration which group you mean when you set up your hypotheses and rejection rules. [Continued on next slide.]


Step 3a. Divide the number 1 by the number of observations in the first sample (n1).

Step 3b. Divide the number 1 by the number of observations in the second sample (n2).

Step 3c. Add the answers to Step 3a and Step 3b.

Refer to Step 1!


Step 4. Add the sum of squares for the first sample (SS1) to the sum of squares for the second sample (SS2). See slide #8 for information about calculating the sum of squares.

Step 5. Find the degrees of freedom by adding the number of observations in the first sample (n1) to the number of observations in the second sample (n2) and then subtracting the number 2. [Continued on next slide.]


Step 6. Divide your answer from Step 4 by the answer from Step 5.

Step 7. Multiply your answer from Step 6 to the answer from Step 3c.

Refer to Step 1.


Step 8. Take the square root of your answer in Step 7.


Compare your obtained t-statistic to the critical t-value from your rejection rule and decide the appropriate action.


Find the appropriate t (crit) from the t-table in the back of the book, using the correct bar at the top depending on a one-tailed or a two-tailed test, , and df = n1 + n2 - 2.

OR use the significance level shown on the SPSS printout. If it is less than .05, reject Ho.

Calculate t (obt) using the two independent samples t-test.

Make your decision based on your rejection rule.

EXAMPLE OF INDEPENDENT SAMPLES t-TESTIndependent Variable is brand of oven

(two brands)

Dependent Variable is hours it worked until failure.Brand A Brand B

237 208

254 178

246 187

178 146

179 145

183 141

EXAMPLE OF INDEPENDENT SAMPLES t-TEST

Stuff we’ll need

Brand A Brand B

X 1,277 1,005

n 6 6

1,277 ÷ 6 = 212.833

1,005 ÷ 6 = 167.500

278,415 172,139

SS =

X2X

n

XX

22 )( 833.626,6

6

)277,1(415,278

2

500.801,36

)005,1(139,172

2

EXAMPLE OF INDEPENDENT SAMPLES t-TESTNow, for it!

432.2635.18

333.45

)333.0)(833.042,1(

333.45

61

61

266500.801,3833.626,6

500.167833.212

112 2121

21

21

nnnnSSSS

XXtobt

RELATED SAMPLES t-TEST

Two-Tailed TestHypotheses

Rejection Rule--Reject H0 if t(obt) > t(crit, , N-1) OR if t(obt) < -t(crit, , N-1) where N is the number of differences

OR if the significance level on the SPSS printout is less than .05

0:

0:

1

0

D

D

H

H

RELATED SAMPLES t-TESTOne-tailed test where you believe the scores in sample 1 will be less than the scores in sample 2 (which means that the differences will tend to be less than 0).Hypotheses

Rejection Rule--Reject H0 if t(obt) < -t(crit, , N-1) OR if the significance level on the SPSS printout is less

than .05

0:

0:

1

0

D

D

H

H

RELATED SAMPLES t-TESTOne-tailed test where you believe the scores in sample 1 will be greater than the scores in sample 2 (which means that the differences will tend to be greater than 0).Hypotheses

Rejection Rule--Reject H0 if t(obt) > t(crit, , N-1) OR if the significance level on the SPSS printout is less

than .05.

0:

0:

1

0

D

D

H

H


Calculating the related samples t-statistic

Step 1. Find the difference between each pair of scores. Again, it doesn’t matter which sample you designate as 1 and which is 2, AS LONG AS you (a) consistently subtract sample 2 from sample 1, and (b) keep the order in mind as you set up your hypotheses and rejection rule.

From this point on, ignore the original scores and use only the difference scores (designed with a subscript D). The test is conducted pretty much the same as if it were a one-sample test. [Continued on next slide.]


Step 4. Divide your answer from Step 3 by N(N – 1).

Step 5. Take the square root of your answer from Step 4.


Compare this obtained t-value against the critical t-value from the rejection rule and decide the appropriate action.

)1(

NNSS

Dt

D

D Step 2. Find the mean of the difference scores.

Step 3. Find the sum of squares of the difference scores. Be sure to use N as the number of pairs of scores (or the number of difference scores) to do the division.


Find the appropriate t (crit) from the t-table in the back of the book, using the correct bar at the top depending on a one-tailed or a two-tailed test, , and df = N - 1.

OR use the significance level on the SPSS printout.

Calculate t (obt) using the related samples t-test.

Make your decision based on your rejection rule.

EXAMPLE OF RELATED-MEASURES t-TEST

Same data as before

Brand A Brand B Difference

Size 1 237 208 29

Size 2 254 178 76

Size 3 246 187 59

Size 4 178 146 32

Size 5 179 145 34

Size 6 183 141 42

EXAMPLE OF RELATED-MEASURES T-TESTStuff we’ll need

Differences

D 272

N 6

272 ÷ 6 = 45.333

14,042

SS =

D2D

N

DD

22 )( 333.711,1

6

)272(042,14

2

EXAMPLE OF RELATED-MEASURES t-TEST

And now

002.6553.7

333.45

)16(6333.711,1

0333.45

)1(

NNSS

Dt

D

D

Documents

TWO-SAMPLE t-TESTS. Independent versus Related Samples Your two samples are independent if you randomly assign individuals into the two treatment