. Suppose we have the following sample data of reaction times: Right Hand Left H
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Question
. Suppose we have the following sample data of reaction times:
Right Hand
Left Hand
1.05
1
0.76
0.74
0.71
0.66
0.79
0.78
0.69
0.68
0.72
0.65
0.75
0.71
0.72
0.68
0.99
0.94
0.8
0.79
0.82
0.81
0.67
0.62
There are two ways to look at these data.
Scenario A: as 24 individuals randomly assigned to two groups
Scenario B: as 12 individuals who each had to use both their left and right hands
One of the scenarios above involves independent samples, and the other involves paired samples. The sample standard deviations are as follows: the standard deviation of the right hand measurements is 0.088, the standard deviation of the left hand measurements is 0.117, and the standard deviation of the paired differences is 0.021. So the standard error for one of the scenarios is 0.0061, and the standard error for the other scenario is 0.0423.
The data are treated as independent samples in Scenario _______. For that scenario, the correct standard error is ________.
Fill in the blanks in the following statements:
Treating the data as independent samples, the standard error is ________ [larger/smaller] than if the data were treated as paired samples.
Consequently, treating the data as independent samples would result in a ________ [larger/smaller] test statistic.
Therefore, a test for a difference of means will generally have a ________ [larger/smaller] p-value if the data are treated as independent samples.
Right Hand
Left Hand
1.05
1
0.76
0.74
0.71
0.66
0.79
0.78
0.69
0.68
0.72
0.65
0.75
0.71
0.72
0.68
0.99
0.94
0.8
0.79
0.82
0.81
0.67
0.62
Explanation / Answer
Treating data for independent samples, the standard error of the difference between sample means is:
SE(x1bar-x2bar)=sqrt[s1^2/n1+s2^2/n2], where, s denotes sample standard deviation, and n denotes sample size, 1 , 2 denote right hand and left hand respectively.
=sqrt[0.088^2/12+0.117^2/12]
=0.0423 [Independent samples]
Treating data as paired samples, the standard error of mean difference is as follows:
SE(dbar)=sd/sqrt n, where, sd is standard deviation of the difference.
=0.021/sqrt 12
=0.0061 [Paired data]
Larger (0.0423>0.0061)
Smaller [because, test statistic for independnet samples is t=(x1bar-x2bar)/SE(x1bar-x2bar), and test statistic for paired samples, t=dbar/SE(dbar), a higher denominator therefore, result in smaller test statistic]
Larger [smaller test statistic at higher degrees of freedom will result into higher p value compared to paired samples]
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