Chapter 7

Introduction to the t Test

Part 2: Dependent Samples

Oct. 3, 2013

t Test for Dependent Means

• Unknown population mean and variance

• Two scores for each person– Repeated measures design– aka “Paired Samples t-test” in SPSS

• Same procedure as t test for single sample, except– Use difference scores– Assume that the comparison mean is 0

t Test for Dependent Means• Difference scores

– For each person, subtract one score from the other – Carry out hypothesis testing with the difference scores

• Find S2 for difference scores, Find SM for difference scores

• Comparison population of difference scores will always have a mean of 0– That is, the relevant µ for the comparison with M will be

0.– This will always be stated in your null hypothesis for a

dependent samples t-test

Example• #5 in Ch. 7 – program to decrease litter:

City July 2001 July 2002

Fresno 9 2

Merced 10 4

Bakersfld 8 9

Stockton 9 1

Note: use alpha = .01

(cont.)• Research hyp: there will be a decrease in litter

from time1 to time 2 (2 < 1…or 1 - 2 > 0)

• Null hyp: there will be no difference/effect (2 = 1, or 1 - 2 = 0)

• Will need Difference scores for each city, need S2 and SM based on difference scores

• S2 = (X-M)2 / N-1

• SM = sqrt (S2 / N)

(cont.)

City July 01

July 02

Diff (01 – 02)

(X-M)2

Fresno 9 2 7 (7-5)2 = 4

Merced 10 4 6 (6-5)2 = 1

Bksfld 8 9 -1 (-1-5)2 = 36

Stockton 9 1 8 (8-5)2 = 9

M = 5 (X-M)2 = 50

(cont.)

• Find S2 and SM

• Find observed t from sample:

• Critical t? Draw distribution…

• Compare obtained t and critical…

• Conclusion?

MS

Mt

MS

Mt

Effect Size for t Test for Dependent Means

• If calculating before data collection, 2 will always be 0, 1 is the expected mean difference in our sample (pre/post-test), is expected SD of difference scores

• If calculating after data collection, 2 is still 0, 1 is the actual mean difference (pre/post-test), is actual SD of difference scores (use S)

• Use same effect size standards as earlier, small d = |.2|, medium d = |.5|, large d >= |.8|

21d

21d

Approximate Power for t Test for Dependent Means (.05 significance level)

Note: Table 7-9shows power inbody of table, you need to know N (rows), and effect size (columns)

Approximate Sample Size Needed for 80% Power

(.05 significance level – Table 7-10)

This table shows N needed for 80% power (rule of thumb)given different expected effect sizes.

SPSS: Dependent Means t-test• Using SATS data, assume ‘sats4’ is pre-semester

rating of difficulty of statistics, ‘sats5’ is post-semester rating of difficulty

• Is there a difference in pre/post semester?– Research hyp: Post should be lower than pre (diff >0)– Null hyp: No difference in pre/post (diff = 0)

• Analyze Compare Means Paired Samples t-test– Pop-up window, under ‘paired variables’, select‘Sats4’ for

var1, ‘Sats5’ for var2, OK

(cont.)• In output, 1st section is “Paired Samples Stats”,

look for means for ‘sats4’ and ‘sats5’ – this is what we’re comparing

• In 3rd section, “Paired Samples Test”, note mean difference score, t observed, df, and ‘sig (2-tail)’.– Mean difference score is compared to 0– Sig (2 tail) should be compared to alpha level

(e.g., .05). – If ‘sig’ value < alpha reject Null

• This example?