Comparing Populations

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Comparing Populations

• Proportions and means

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Comparing proportions

• Situation
• We have two populations (1 and 2)
• Let p1 denote the probability (proportion) of
  success in population 1.
• Let p2 denote the probability (proportion) of
  success in population 2.
• Objective is to compare the two population
  proportions

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We want to test either
or
or
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The test statistic
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Where
A sample of n1 is selected from population 1
resulting in x1 successes
A sample of n2 is selected from population 2
resulting in x2 successes
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Estimating a difference proportions using
confidence intervals

• Situation
• We have two populations (1 and 2)
• Let p1 denote the probability (proportion) of
  success in population 1.
• Let p2 denote the probability (proportion) of
  success in population 2.
• Objective is to estimate the difference in the
  two population proportions d p1 p2.

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Confidence Interval for d p1 p2 100P 100(1
a)
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Example

• Estimating the increase in the mortality rate for
  pipe smokers higher over that for non-smokers d
  p2 p1

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Comparing Means

• Situation
• We have two normal populations (1 and 2)
• Let m1 and s1 denote the mean and standard
  deviation of population 1.
• Let m2 and s2 denote the mean and standard
  deviation of population 1.
• Let x1, x2, x3 , , xn denote a sample from a
  normal population 1.
• Let y1, y2, y3 , , ym denote a sample from a
  normal population 2.
• Objective is to compare the two population means

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We want to test either
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Consider the test statistic
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If

• will have a standard Normal distribution
• This will also be true for the approximation
  (obtained by replacing s1 by sx and s2 by sy) if
  the sample sizes n and m are large (greater than
  30)

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Note
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Example

• A study was interested in determining if an
  exercise program had some effect on reduction of
  Blood Pressure in subjects with abnormally high
  blood pressure.
• For this purpose a sample of n 500 patients
  with abnormally high blood pressure were required
  to adhere to the exercise regime.
• A second sample m 400 of patients with
  abnormally high blood pressure were not required
  to adhere to the exercise regime.
• After a period of one year the reduction in blood
  pressure was measured for each patient in the
  study.

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We want to test
The exercise group did not have a higher average
reduction in blood pressure
The exercise group did have a higher average
reduction in blood pressure
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The test statistic
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Suppose the data has been collected and
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The test statistic
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We reject H0 if
True hence we reject H0.
Conclusion There is a significant (a 0.05)
effect due to the exercise regime on the
reduction in Blood pressure
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Estimating a difference means using confidence
intervals

• Situation
• We have two populations (1 and 2)
• Let m1 denote the mean of population 1.
• Let m2 denote the mean of population 2.
• Objective is to estimate the difference in the
  two population proportions d m1 m2.

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Confidence Interval for d m1 m2 100P 100(1
a)
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Example

• Estimating the increase in the average reduction
  in Blood pressure due to the exercise regime d
  m1 m2

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Sample size determination

• When comparing two or more populations

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Estimating a difference proportions using
confidence intervals

• Situation
• We have two populations (1 and 2)
• Let p1 denote the probability (proportion) of
  success in population 1.
• Let p2 denote the probability (proportion) of
  success in population 2.
• Objective is to estimate the difference in the
  two population proportions d p1 p2.

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Confidence Interval for d p1 p2 100P 100(1
a)
where
Note B is determined by

• The sample sizes n1 and n2.The level of
  confidence 1 a.The probability of success in
  both populations, p1 and p2.

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Note if B, a, p1 and p2 are given
then
and
Note there are many solutions for n1 and n2.
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Special solutions - case 1 n1 n2 n.
then
and
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Special solutions - case 2 Choose n1 and n2 to
minimize N n1 n2 total sample size
Note
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hence
if
or
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Also
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Summary The sample sizes required, n1 and n2,
to estimate p1 p2 within an error bound B with
level of confidence 1 a are
if the objectives are to minimize the total
sample size N n1 n2 .
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Special solutions - case 3 Choose n1 and n2 to
minimize C C0 c1 n1 c2 n2 total cost of
the study
Note
C0 fixed (set-up) costs c1 cost per unit in
population 1 c2 cost per unit in population 2
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hence
if
or
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Also
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Summary The sample sizes required, n1 and n2,
to estimate p1 p2 within an error bound B with
level of confidence 1 a are
Summary The sample sizes required, n1 and n2,
to estimate p1 p2 within an error bound B with
level of confidence 1 a are
if the objectives are to minimize the total
cost C C0 c1 n1 c2 n2 .
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Example It is known that approximately 4 of
individuals aged 70-80 with high cholesterol
suffer a heart attack or stroke within a 10 year
period. One is interested in determining if this
rate is decreased for individuals who receive a
new medication
A study is proposed in which n1 individuals will
receive the new medication while n2 will receive
a placebo in a double blind study. double blind
study both patient and physician administering
the treatment are unaware of the treatment (drug
or placebo)
What should the sample sizes be in each group if
we want to estimate the difference in the rate of
heart attack or stroke within 0.5 with a 99
level of confidence and minimize the total
cost C C0 c1 n1 c2 n2 . Assume that the
cost for the medication is 100 times that of the
cost of administering a placebo
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The sample sizes required are
Where za/2 z0.005 2.576 B 0.005 p1 ? p2 ?
0.04 and
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hence
and
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Estimating a difference means using confidence
intervals

• Situation
• We have two populations (1 and 2)
• Let m1 denote the mean of population 1.
• Let m2 denote the mean of population 2.
• Objective is to estimate the difference in the
  two population proportions d m1 m2.

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Confidence Interval for d m1 m2 100P
100(1 a)
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The sample sizes required, n1 and n2, to estimate
m1 m2 within an error bound B with level of
confidence 1 a are
Equal sample sizes
Minimizing the total sample size N n1 n2 .
Minimizing the total cost C C0 c1n1 c2n2 .
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Comparing Means small samples

• Situation
• We have two normal populations (1 and 2)
• Let m1 and s1 denote the mean and standard
  deviation of population 1.
• Let m2 and s2 denote the mean and standard
  deviation of population 1.
• Let x1, x2, x3 , , xn denote a sample from a
  normal population 1.
• Let y1, y2, y3 , , ym denote a sample from a
  normal population 2.
• Objective is to compare the two population means

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We want to test either
or
or
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Consider the test statistic
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If the sample sizes (m and n) are large the
statistic
will have approximately a standard normal
distribution
This will not be the case if sample sizes (m and
n) are small
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The t test for comparing means small samples

• Situation
• We have two normal populations (1 and 2)
• Let m1 and s denote the mean and standard
  deviation of population 1.
• Let m2 and s denote the mean and standard
  deviation of population 1.
• Note we assume that the standard deviation for
  each population is the same.
• s1 s2 s

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Let
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The pooled estimate of s.
Note both sx and sy are estimators of s.
These can be combined to form a single estimator
of s, sPooled.
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The test statistic
If m1 m2 this statistic has a t distribution
with n m 2 degrees of freedom
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are critical points under the t distribution with
degrees of freedom n m 2.
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Example

• A study was interested in determining if
  administration of a drug reduces cancerous tumor
  size.
• For this purpose n m 9 test animals are
  implanted with a cancerous tumor.
• n 3 are selected at random and administered the
  drug.
• The remaining m 6 are left untreated.
• Final tumour sizes are measured at the end of the
  test period

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We want to test
The treated group did not have a lower average
final tumour size.
vs
The exercize group did have a lower average final
tumour size.
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The test statistic
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Suppose the data has been collected and
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The test statistic
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We reject H0 if
with d.f. n m 2 7
Hence we accept H0.
Conclusion The drug treatment does not result in
a significant (a 0.05) smaller final tumour
size,