Chapter 10 The t Test for Two Independent Samples

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
1
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.
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1
þ
þ
þ
3
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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2
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 (S p2)
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3
Logic and procedure:
t statistic for independent-measure design
• H0: µ1 - µ 2 = 0
• H1: µ1 - µ 2 <> 0
• overall t formula:
sample mean − population mean M − µ
t=
=
estimated standard error
sM
• the independent-measures t uses the difference
sample mean difference − population mean difference
estimated standard error
(M 1 − M 2 ) − ( µ1 − µ2 )
=
s( M 1 − M 2 )
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t=
Reminder
• The means of two groups are compared relative
to a sample distribution of:
– means (“is M 1 different from the population µ?”)
SS
s2 =
– sample variances (proved mean all s2 = σ2)
n −1
– mean differences (“is the difference between two
sample means, ? M 1-M 2, different?”)
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2
4
8
Sample n (M = ?)
.
.
Sample 1 (M= ?)
1
2
3
4
5
8
4
9
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 (M 1-M 2)
• Recall:
s2
sM =
n
2
s( M1 − M 2 )
2
s
s
= 1 + 2
n1 n2
• now we want to know the total error, but…
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5
t-statistic (contd.)
• But, what if the size of the two samples is not the
same (n1<>n2)?
• Recall:
SS
SS
2
s =
n −1
=
df
• we now have two SS and df values, thus:
sp =
2
sp
2
SS1 + SS2
or
df1 + df2
( n1 − 1) s12 + ( n2 − 1) s2 2
=
n1 + n2 − 2
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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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6
Computational procedure
t-test for means of two independent samples
• Step 1. State hypotheses and select alpha
H0: µ1 - µ 2 = 0
H1: µ1 - µ 2 <> 0
α =.05
• Step 2. Determine df
df = 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
(∑ X 1 ) 2
2
2
∑ X 1 ∑ X 1 and SS1 = ∑ X 1 −
n1
• Step 3b. Estimate the variance of the source population
sp =
2
SS1 + SS2
( n1 − 1) + (n2 − 1)
• Step 3c. Estimate the sd of the sampling distribution of
sample-mean differences
2
2
s( M1 − M 2 ) =
• Step 3d. Calculate t as
t=
sp
n1
+
sp
n2
( M 1 − M 2 ) − ( µ1 − µ2 )
s( M 1− M2 )
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Example (contd.)
• Step 3a.
• Step 3b.
sp =
2
SS1 + SS2
200 + 160
=
= 20
( n1 − 1) + (n2 − 1)
9 +9
• Step 3c. s( M1 − M 2 ) =
• Step 3d. t =
sp
2
n1
+
sp
2
n2
=
20 20
+
= 4 =2
10 10
( M 1 − M 2 ) − ( µ1 − µ 2 ) (25 − 19) − 0 6
=
= = 3.00
s( M 1−M 2 )
2
2
• Step 4. Make a decision. Table lookup df = 18
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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 size
Two methods (Cohen’s d or r2)
• Cohen’s d = mean difference/sd
=
M1 − M 2
s p2
=
25 − 19
6
=
= 1.34
4.47
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• 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
r2 =
t2
32
9
=
=
= 0. 333
t 2 + df 3 2 + 18 27
18
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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
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