Statistical Inference: TwoSample Case

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Statistical Inference Two-Sample Case

• Chapter 11

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Statistical Comparison of Two Independent Samples

• Often, in psychology, researchers wish to compare
  two samples to determine if they have been drawn
  from different populations (i.e. to see if the
  two samples are significantly different).
• E.g., Doctors wanted to see which of two drugs
  should be given to patients after an organ
  transplant in order to prevent transplant
  rejection. One sample was given was given
  cyclosporine whereas the other was given FK506.
  The samples were then compared.

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• E.g., Lets imagine we are investigating drinking
  habits of university students in North America.
  Specifically, we want to compare students who
  live in dorms to those who live off-campus.
• We will draw every possible pair of samples (N
  10) from the population. One sample will be 10
  students that live in a dorm, whereas the other
  will be 10 students who live off-campus. We will
  calculate the mean number of alcoholic drinks
  consumed per week by each sample and compare
  them.

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First Pair Dorm Off-campus
Difference ? Mean 10
6 4
Second Pair Dorm Off-campus
Difference ? Mean 8
5 3
Third Pair Dorm Off-campus
Difference ? Mean 5
8 -3
We continue this process until every possible
pair of samples has been drawn from the
population. We obtain a difference score (?) and
plot these scores on a distribution called the
sampling distribution of the difference between
means.
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Sampling Distribution of the Difference Between
Means
Population in which ?D ?OC
Positive Numbers XD gt XOC
Negative Numbers XD lt XOC
0
We end up with a normal distribution with a mean
of 0.
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Going a Step Further
We have a normal distribution with a mean
difference between two populations and a standard
deviation. What statistic can we use to carry
out two-sample statistics?
Z-Scores
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• Here is the Z-formula...

Note, this formula assumes that we know the
population means and population standard
deviation. However, this is very rarely the
case. Therefore, this formula is hardly
ever used.
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The Independent t-test

• Thus, if we do not know the population means or
  standard deviations, we test hypotheses with the
  t formula.

S X1 - X2 is the standard error of the
difference between the means.
Usually, according to our H0 we assume that the
samples are drawn from the same population of
means (i.e., ? 1 ? 2). So we state that ? 1 -
? 2 0.
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Independent t-test (Equal N)

• E.g., A psychologist believes short-term memory
  capacity is reduced by sleep loss. Twelve
  subjects are randomly assigned to a group that
  receives a normal amount of sleep or to a group
  that is sleep deprived for 24 hours. The
  subjects are then presented with a list of nine
  numbers to be remembered over a short period.
  The percentage of numbers correctly recalled by
  each subject are provided below. Is the
  researcher correct.? Use ? 0.01.
• Note, we will use the standard error formula for
  equal N.

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• H0 ?1 ?2
• H1 ?1 gt ?2 (one-tailed, directional
  hypothesis)

This is the formula for standard error when we
have equal N
25 698 - (392)2 6
29 591 - (421)2 6
50.8333
87.333
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2.146
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Remember, according to H0 ?1 - ?2 0?
t (70.167 - 65.333) - 0 2.25
2.146
df N1 N2 -2 6 6 - 2 10
t crit(10) 2.764
Since tobs 2.25 lt tcrit 0.01 2.764, we do
not reject the H0. We can not conclude that
sleep deprivation has an effect on short-term
memory (t10 2.25, p gt 0.01).
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Another Example

• A clinician believes that depression may affect
  sleep. She decides to test the idea. The sleep
  of nine depressed patients and eight nondepressed
  subjects is monitored for three nights. The
  average number of hours slept by each subject is
  provided below. Is the clinician correct? Set ?
  at 0.05.
• We will use the standard error formula for
  unequal N.
• H0 ?D ?N
• H1 ?D ?N

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This is the formula for standard error when we
have unequal N.
SS2 457.92- (60.4)2 8
SS1 431.22 - (62.2)2
9
1.349
1.9
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Independent t-test (Unequal N)
1.349 1.9 1 1
9 8 - 2
9 8
0.226
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• We can now calculate t.

t (6.91 - 7.55) -2.83 0.226
df N1 N2 - 2 9 8 - 2 15
tcrit0.05 2.131
Since tobs15 -2.83 gt -2.131, we reject the
H0. Depression does have an effect on
sleep. (t15 -2.83, p lt 0.05)
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Assumptions of the t-test

• The t-distribution is used in two sample
  statistics following three assumptions.
• 1) The sampling distribution of the difference
  between the means is normally distributed.
• 2) We can get an unbiased estimate of variance
  using the sample (i.e., S X1 - X2).
• 3) Both samples are drawn from populations with
  equal variances. This is referred to as
  homogeneity of variance.

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The F-distribution

• Sometimes the variances of the populations are
  not equal. We can test for this using the F
  distribution in which we calculate an F ratio.

Where S 2 is the estimate variance of the
population based on the sample and is equal to...
The F-ratio is also useful in studies where we
expect dual effects. That is, the independent
variable causes some subjects scores to
increase while it cause others to decrease.
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• E.g., A researcher believes that drug X will
  increase heart rate in some patients while
  decreasing it in other patients (Hint dual
  effects). He compares the heart rate of 5
  patients given drug X to those who receive no
  drug. The data are provided below. Is he
  correct? Use normal decision rules.

?X1 385 ?X2 366 ?X12 29 829
?X22 26 884
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• H0 ?12 ?22
• H1 ?12 gt ?22

29 829 - (385)2 5
26 884 - (366)2 5
184
92.8
Now calculate estimated variance.
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Because group 1 has the larger estimate of
population variance, it will be in the numerator
of the F-ratio formula, whereas the estimate of
variance for group 2 will be in the denominator.
F 46/ 23.2 1.98
This F ratio must now be compared to a critical
F-ratio. Thus, we need to calculate degrees of
freedom.
There are two separate degrees of freedom for an
F-ratio one for the numerator and one for the
denominator.
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• Use Table D on page 330 to determine Fcrit for df
  4,4 (Note, use Table D1 if performing a two
  tailed test).
• The numbers in the top row of the table represent
  the df of the numerator. The numbers in the
  column on the left represent the df of the
  denominator.
• Note, the numbers in bold (bottom) represent
  Fcrit where ? 0.01, whereas number in normal
  font (top) represent Fcrit where ? 0.05.

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• Go to the column for df 4 in the numerator and
  go down until you get to the fourth row where df
  4 in the denominator.
• Our Fcrit 6.39
• Our calculated F ratio must be greater than Fcrit
  in order to reject the H0 and assume that the
  variances are not homogeneous.
• Since F4,4 1.98 lt Fcrit 6.39, we do not
  reject H0. The variances are homogeneous (F4,4
  1.98, p gt 0.05).

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Control and Experimental Groups

• In some cases in psychology, our independent
  variable is a type of treatment (receiving a
  drug, some type of training or instruction,
  etc.). We then compare a group receiving a
  treatment to a group that receives no treatment.
• E.g., In our previous example, one group received
  drug X, the other received no drug.
• Control Group a sample that receives no
  treatment.
• Experimental Group a sample that receives
  treatment.

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t-test for Correlated Samples

• Whenever we carry out a two sample study in
  psychology, there is a great deal of variability.
  Because of this variability, the scores in the
  two samples overlap.

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• This variability (overlap) may be due to one of
  three factors.
• 1) The independent variable.
• In this case, depressed or nondepressed.
• 2) Random error.
• E.g., variation in procedure from subject to
  subject, external distractions etc.
• 3) Subject differences.
• E.g., some subjects sleep more or less than other
  subjects, and this has nothing to do with
  depression.
• We can minimize variability caused by subject
  differences by using correlated samples.

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t-test for Correlated Samples

• There are two types of correlated sample studies.
• 1) The Before and After Design A single sample
  is measured before and after the introduction of
  some independent variable. The same subjects are
  in both samples.
• 2) The Matched Group Design Each individual in
  one sample is matched with a subject in the other
  sample group. The matching is done so that the
  two individuals are equivalent (or nearly
  equivalent) with respect to a specific variable
  the researcher would like to control.

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• In these cases, we must calculate the difference
  between the scores of the two samples. We use
  these differences in our calculation of t.
• Note, we will also use a slightly different
  estimate of the standard error of the difference
  between the means, and a slightly different
  t-formula.