Post on 12-Jan-2016
Overview
• Two paired samples: Within-Subject Designs-Hypothesis test
-Confidence Interval
-Effect Size
• Two independent samples: Between-Subject Designs-Hypothesis test
-Confidence interval
-Effect Size
Comparing Two Populations
Until this point, all the inferential statistics we have considered involve using one sample as the basis for drawing conclusion about one population.
Although these single sample techniques are used occasionally in real research, most research studies aim to compare of two (or more) sets of data in order to make inferences about the differences between two (or more) populations.
What do we do when our research question concerns a mean difference between two sets of data?
Two kinds of studies
There are two general research strategies that can be used to obtain the two sets of data to be compared:
1. The two sets of data could come from two independent populations (e.g. women and men, or students from section A and from section B)
2. The two sets of data could come from related populations (e.g. “before treatment” and “after treatment”)
<- between-subjects design
<- within-subjects design
Part I
• Two paired samples: Within-Subject Designs-Hypothesis test
-Confidence Interval
-Effect Size
Paired T-Test for Within-Subjects Designs
Our hypotheses:
Ho: D = 0
HA: D 0
To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.
Paired Samples t
t = D - D sD = sD
s
n
Steps for Calculating a Test Statistic
Paired Samples T
1. Calculate difference scores
2. Calculate D
3. Calculate sd
4. Calculate T and d.f.
5. Use Table E.6
Confidence Intervals for Paired Samples
Paired Samples t
D t (sD)
General formula
X t (SE)
Effect Size for Dependent Samples
Paired Samples d
One Sample d
s
Xd H0ˆ
Ds
Dd ˆ
Exercise
In Everitt’s study (1994), 17 girls being treated for anorexia were weighed before and after treatment. Difference scores were calculated for each participant.
Change in Weight
n = 17 = 7.26sD = 7.16D
Test the null hypothesis that there was no change in weight.
Compute a 95% confidence interval for the mean difference.
Calculate the effect size
ExerciseChange in Weight
n = 17 = 7.26sD = 7.16D
74.117
16.7SE
17.474.1
026.7)16(
t
01.p
T-test
ExerciseChange in Weight
n = 17 = 7.26sD = 7.16D
12.2critt
)74.1(12.226.7 CI
)95.10,57.3(
Confidence Interval
ExerciseChange in Weight
n = 17 = 7.26sD = 7.16D
16.7
26.7d
01.1d
Effect Size
Part II
• Two independent samples: Between-Subject Designs-Hypothesis test
-Confidence Interval
-Effect Size
T-Test for Independent SamplesThe goal of a between-subjects research study is to evaluate the mean difference between two populations (or between two treatment conditions).
Ho: 1 = 2
HA: 1 2
We can’t compute difference scores, so …
T-Test for Independent Samples
We can re-write these hypotheses as follows:
Ho: 1 - 2 = 0
HA: 1 - 2 0
To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.
T-Test for Independent Samples
General t formula
t = sample statistic - hypothesized population parameter estimated standard error
?21 XXs
Independent samples t
21
)()( 2121
XXs
XXt
One Sample t
X
Htest s
Xt 0
T-Test for Independent Samples
Standard Error for a Difference in Means
The single-sample standard error ( sx ) measures how much error expected between X and .
The independent-samples standard error (sx1-x2) measures how much error is expected when you are using a sample mean difference (X1 – X2) to represent a population mean difference.
21 XXs
T-Test for Independent Samples
Standard Error for a Difference in Means
Each of the two sample means represents its own population mean, but in each case there is some error.
The amount of error associated with each sample mean can be measured by computing the standard errors.
To calculate the total amount of error involved in using two sample means to approximate two population means, we will find the error from each sample separately and then add the two errors together.
2
22
1
21
21 n
s
n
ss XX
T-Test for Independent Samples
Standard Error for a Difference in Means
But…
This formula only works when n1 = n2. When the two samples are different sizes, this formula is biased.
This comes from the fact that the formula above treats the two sample variances equally. But we know that the statistics obtained from large samples are better estimates, so we need to give larger sample more weight in our estimated standard error.
2
22
1
21
21 n
s
n
ss XX
T-Test for Independent Samples
Standard Error for a Difference in Means
We are going to change the formula slightly so that we use the pooled sample variance instead of the individual sample variances.
This pooled variance is going to be a weighted estimate of the variance derived from the two samples.
sp2
SS1 SS2
df1 df2
2
2
1
2
21 n
s
n
ss pp
XX
Steps for Calculating a Test Statistic
One-Sample T
1. Calculate sample mean
2. Calculate standard error
3. Calculate T and d.f.
4. Use Table D
Independent Samples T
1. Calculate X1-X2
2. Calculate pooled variance
3. Calculate standard error
4. Calculate T and d.f.
5. Use Table E.6
sp2
SS1 SS2
df1 df2
sp2
n1
sp
2
n2
t (X 1 X 2) (1 2 )
sx 1 x 2 d.f. = (n1 - 1) + (n2 - 1)
Steps for Calculating a Test Statistic
Illustration
A developmental psychologist would like to examine the difference in verbal skills for 8-year-old boys versus 8-year-old girls. A sample of 10 boys and 10 girls is obtained, and each child is given a standardized verbal abilities test. The data for this experiment are as follows:
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
STEP 1: get mean difference
621 XX
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
Illustration
STEP 2: Compute Pooled Variance
sp2
SS1 SS2
df1 df2
150 210
(10 1) (10 1)
360
1820
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
Illustration
STEP 3: Compute Standard Error
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
SE s p
2
n1
s p
2
n2
20
10
20
10 4 2
Illustration
STEP 4: Compute T statistic and df
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
t (X 1 X 2) (1 2 )
sx 1 x 2
(37 31) 0
23
d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18
Illustration
STEP 5: Use table E.6
Girls Boys
n1 = 10 = 37SS1 = 150
X 1
n2 = 10 = 31SS2 = 210
X 2
T = 3 with 18 degrees of freedom
For alpha = .01, critical value of t is 2.878
Our T is more extreme, so we reject the null
There is a significant difference between boys and girls
Illustration
T-Test for Independent Samples
Sample Data
Hypothesized Population Parameter
Sample Variance
Estimated Standard
Error
t-statistic
Single sample
t-statistic
Independent samples t-statistic
X 1 X 2
1 2
sp2
n1
sp
2
n2
sp2
SS1 SS2
df1 df2
X
s2
n
s2 SS
df
t X
sx
t (X 1 X 2) (1 2 )
sx 1 x 2
Confidence Intervals for Independent Samples
One Sample t
X t (sx)
General formula
X t (SE)
Independent Sample t
(X1-X2) t (sx1-x2)
Effect Size for Independent Samples
One Sample d
Independent Samples d
s
Xd H0ˆ
ps
XXd 21ˆ
Exercise
Subjects are asked to memorize 40 noun pairs. Ten subjects are given a heuristic to help them memorize the list, the remaining ten subjects serve as the control and are given no help. The ten experimental subjects have a X-bar = 21 and a SS = 100. The ten control subjects have a X-bar = 19 and a SS = 120.
Test the hypothesis that the experimental group differs from the control group.
Give a 95% confidence interval for the difference between groups
Give the effect size
ExerciseExperimental Control
n1 = 10 = 21SS1 = 100
X 1
n2 = 10 = 19SS2 = 120
X 2
221 XX
2.1218
220
)110()110(
120100
21
212
dfdf
SSSSsp
56.144.210
2.12
10
2.12
2
2
1
2
n
s
n
sSE pp
T-test
Exercise
28.156.1
02)()(
21
2121
xxs
XXt
d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18
20.p
T-test
Experimental Control
n1 = 10 = 21SS1 = 100
X 1
n2 = 10 = 19SS2 = 120
X 2
Exercise
101.2critt
)56.1(101.22 CI
Confidence Interval
)28.5,28.1(
Experimental Control
n1 = 10 = 21SS1 = 100
X 1
n2 = 10 = 19SS2 = 120
X 2
Exercise
57.d
Effect Size
ps
XXd 21
2.12
2d
Experimental Control
n1 = 10 = 21SS1 = 100
X 1
n2 = 10 = 19SS2 = 120
X 2
Summary
Hypothesis Tests
Confidence Intervals
Effect Sizes
1 Sample
2 Paired Samples
2 Independent Samples
Review
Sample Data
Hypothesized Population Parameter
Sample Variance
Estimated Standard
Error
t-statistic
One sample
t-statistic
Paired samples t-
statistic
Independent samples t-statistic
X
X 1 X 2
1 2
s2
n
sp2
n1
sp
2
n2
s2 SS
df
sp2
SS1 SS2
df1 df2
t X
sx
t (X 1 X 2) (1 2 )
sx 1 x 2
D D
s2
ndf
SSs D2
D
D
s
Dt
Confidence Intervals
Paired Samples t
D t (sD)
One Sample t
X t (SE)
Independent Sample t
(X1-X2) t (sx1-x2)
Effect Sizes
Paired Samples d
One Sample d
Independent Samples d
Ds
Dd ˆ
s
Xd H0ˆ
ps
XXd 21ˆ