The t Test for Two Independent Samples - University of...
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1 1 Statistics for the Behavioral Sciences (5 th ed.) Gravetter & Wallnau Chapter 10 The t Test for Two Independent Samples University of Guelph Psychology 3320 — Dr. K. Hennig Winter 2003 Term 2 Once or Twice? Within Subjects Design • If the subjects are used more than once or matched, this design is called a Within Subjects Design or a Repeated Measures Design. Advantages of Repeated Measures Designs: • They take fewer participants. • They typically have more statistical power (like a matched t-test). • Disadvantages of Repeated Measures Designs: • You have to worry about practice effects and carryover effects. We will return to this.
Transcript of The t Test for Two Independent Samples - University of...
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Statistics for the Behavioral Sciences (5 th ed.)Gravetter & Wallnau
Chapter 10
The t Test for Two Independent Samples
University of GuelphPsychology 3320 — Dr. K. Hennig
Winter 2003 Term
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Once or Twice?Within Subjects Design
• If the subjects are used more than once or matched, this design is called aWithin Subjects Design or a Repeated Measures Design.
Advantages of Repeated Measures Designs:• They take fewer participants. • They typically have more statistical power (like a matched t-test). • Disadvantages of Repeated Measures Designs:• You have to worry about practice effects and carryover effects. We
will return to this.
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þ
þ
þ
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Between vs. within designs
• Between-subjects– comparison of separate groups (men cf. women;
ethnicity1 cf. ethnicity2; married cf. unmarried); or matched groups
• between-subjects/repeated measures – two data sets from the same sample (patients before
therapy cf. after; drug use Grade 9 cf. grade 11; time1 cf. time2)
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Figure 10-2 (p. 310)Do the achievement scores for children taught by method A differ from the scores for children taught by method B? In statistical terms, are the two population means the same or different? Because neither of the two population means is known, it will be necessary to take two samples, one from each population. The first sample will provide information about the mean for the first population, and the second sample will provide information about the second population.
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Table 10-1 (p. 316) Three major differences: mean difference of the two samples, different ns and thus pooled variance (Sp
2)
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Logic and procedure:t statistic for independent-measure design
• H0: µ1 - µ 2 = 0• H1: µ1 - µ 2 <> 0• overall t formula:
• the independent-measures t uses the differenceMs
Mt
µ−=
−=
errorstandardestimatedmeanpopulationmeansample
)(
)()(errorstandardestimated
differencemeanpopulationdifferencemeansample
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2121
MMsMM
t
−
−−−=
−=
µµ
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Reminder• The means of two groups are compared relative
to a sample distribution of:– means (“is M1 different from the population µ?”)– sample variances (proved mean all s2 = σ2)
– mean differences (“is the difference between two sample means, ?M1-M2, different?”)
Sample 1 (M= ?)
Sample n (M = ?)..
1 2 3 4 5
2
4
6
8
12
−=
nSS
s
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t-statistic (contd.)
• (Review) standard error: measures how accurately the sample statistic represents the population parameter– single sample = the amount of error expected for a
sample mean– independent measures formula = amount of error
expected when you use the mean difference (M1-M2)
• Recall:
• now we want to know the total error, but…2
22
1
21
21 ns
ns
s MM +=− )(
ns
sM
2
=
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t-statistic (contd.)
• But, what if the size of the two samples is not the same (n1<>n2)?
• Recall:
• we now have two SS and df values, thus:
dfSS
nSS
s =−
=1
2
211
21
222
2112
21
212
−+−+−
=
++
=
nnsnsn
s
ordfdfSSSS
s
p
p
)()(
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• Example 10.1 (n = 10 for both groups): A list of 40 pairs of nouns (e.g., dog/bicycle, grass/door) are given two (independent) groups for 5 min.:– group 1 is asked to memorize– group 2 uses mental imagery to aid memorization (e.g., dog
riding a bicycle) = a Tx effect
group 2
group 1
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Computational proceduret-test for means of two independent samples• Step 1. State hypotheses and select alpha
H0: µ1 - µ 2 = 0H1: µ1 - µ 2 <> 0α =.05
• Step 2. Determine dfdf = df1 + df2
=(n1 - 1) + (n2-1) = 9+9 = 18
• Step 3. Obtain data& calculate t
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(contd.)
• Step 3a. For the two samples, A and B, of sizes n1 and n2 respectively, calculate
• Step 3b. Estimate the variance of the source population
• Step 3c. Estimate the sd of the sampling distribution of sample-mean differences
• Step 3d. Calculate t as
1
212
112
11 nX
XSSandXX)(∑
−∑=∑∑
)()( 11 21
212
−+−+
=nn
SSSSsp
2
2
1
2
21 n
s
n
ss pp
MM +=− )(
)(
)()(
21
2121
MMsMM
t−
−−−=
µµ
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Example (contd.)
• Step 3a.• Step 3b.
• Step 3c.
• Step 3d.
• Step 4. Make a decision. Table lookup df = 18
2099160200
11 21
212 =++
=−+−
+=
)()( nnSSSS
sp
241020
1020
2
2
1
2
21==+=+=− n
s
n
ss pp
MM )(
00326
201925
21
2121 .)()()(
)(
==−−
=−−−
=−MMs
MMt
µµ
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Table look up for df = 18
Obtained value of t is three times greater than would be expected by chance (the standard error): t = 3.00
The Tx moved the mean from M = 22 to M = 25
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Figure 10-4 (p. 318)The t distribution with df = 18. The critical region for α= .05 is shown.Two things are important: significance and effect size
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Calculating effect sizeTwo methods (Cohen’s d or r2)
• Cohen’s d = mean difference/sd
• the distance between the two means is slightly more than 1 sd, thus d should be slightly larger than 1.00
• Can also calculate variance accounted for by Tx
3414746
201925
2
21 ..
==−=−=ps
MM
3330279
1833
2
2
2
22 .==
+=
+=
dftt
r
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Calculating r2 directly:
Tx contribution to variability=Tx variability/ total variability=180/540 = .333 = 33.3%
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Final comparison
