Transcript Statistical Hypotheses Testing

Statistical Hypotheses Testing
Stat 700 Lectures
Hypothesis Testing
Overview of this Lecture
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The problem of hypotheses testing
Elements and logic of hypotheses testing
(hypotheses, decision rule, one- and two-tailed tests,
significance level, Type I and Type II errors, power of
test, implications of the decision, p-values)
Steps in performing a hypotheses test
Large-sample test for the population mean
Two-sample tests for the population means
Large-sample test for the population proportion
Two-sample tests for the population proportions
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The problem of hypotheses testing
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Statement of the Problem:
Given a population (equivalently a distribution) with a
parameter of interest, , (which could be the mean,
variance, standard deviation, proportion, etc.), we
would like to decide/choose between two
complementary statements concerning . These
statements are called statistical hypotheses.
The choice or decision between these hypotheses is
to be based on a sample data taken from the
population of interest.
The ideal goal is to be able to choose the hypothesis
that is true in reality based on the sample data.
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Some Situations where Hypotheses
Testing is Relevant
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Example: A drug manufacturer would like to compare
a newly developed pill for eliminating migraine
headaches relative to a standard drug. Such a
comparison is to be done by comparing the mean
time to cessation of headache after taking the pill. Let
 denote the mean time to headache cessation after
taking the new pill. If 0 is the mean time to
headache cessation for the standard drug, then the
manufacturer would like to decide between:
Statement 1 (Null):  > 0 (new drug is not better)
Statement 2 (Alternative):  < 0 (new drug is better)
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Some Situations …
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Example: A medical researcher would like to compare
the effectiveness of two treatments (for example,
chemotherapy versus radiation-based) for a particular
type of cancer, with the effectiveness being
measured in terms of the five-year survival rate of
patients. If p1 denotes the proportion of patients
surviving 5 years which were treated with
chemotherapy, and p2 is the survival proportion for
those treated with radiation, then the researcher
would like to decide between:
Statement 1: p1 < p2;
Statement 2: p1 > p2.
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Some Situations ...
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Example: The Food and Drug Administration would
like to check that the amount of an active ingredient
of a certain substance in a certain type of medication
is as specified in the label. If  is the mean amount of
this substance, then the FDA would like to decide
between the statements:
Statement 1 (Null):  = 0, where 0 is the specified
amount;
Statement 2 (Alternative):   0.
This is an example of a two-sided hypothesis since it
indicates that either  < 0 or  > 0.
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Elements and Logic of Statistical
Hypotheses Testing
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Consider a population or distribution whose mean is
. To introduce the elements and discuss the logic of
hypotheses testing, we consider the problem of
deciding whether  = 0, where 0 is a pre-specified
value, or   0. This is the type of problem that the
FDA might be interested.
The first step in hypotheses testing, which should be
done before you gather your sample data, is to set up
your statistical hypotheses, which are the null
hypothesis (H0) and the alternative hypothesis (H1).
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The Statistical Hypotheses
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The null hypothesis, H0, is usually the hypothesis that
corresponds to the status quo, the standard, the
desired level/amount, or it represents the statement
of “no difference.”
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The alternative hypothesis, H1, on the other hand, is
the complement of H0, and is typically the statement
that the researcher would like to prove or verify.
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These hypotheses are usually set-up in such a way
that deciding in favor of H1 when in fact H0 is the true
statement will not be a desirable outcome.
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An Analogy to Remember
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Setting the null and alternative hypotheses has an
analog in the justice system where the defendant is
“presumed innocent” until “proven guilty.”
In the court system, the null hypothesis corresponds
to the defendant being innocent (this is the status
quo, the standard, etc.).
The alternative hypothesis, on the other hand, is that
the defendant is guilty.
Note that it is very difficult to reject the null (convict
the defendant), and only “a proof (based on good
evidence) beyond a reasonable doubt” will warrant
rejection of H0.
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The Hypotheses in our Problem
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For the problem we are considering, the appropriate
hypotheses will be:
H0:  = 0
H1:   0.
Another word of caution: It is not proper for a
researcher to set up the hypotheses after seeing the
sample data; however, a data maybe used to
generate a hypotheses, but to test these generated
hypotheses you should gather a new set of sample
data!
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Determine the Type of Sample Data that
will be Gathered
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The second step is to determine what kind of sample
data you will be gathering. Is it a simple random
sample? A stratified sample?
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For the moment we will assume that a simple random
sample of size n will be obtained, so the data will be
representable by X1, X2, …, Xn, with n > 30.
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Also, determine if you know the population standard
deviation . We assume for the moment that we do.
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The Decision Rule
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The decision rule is the procedure that states when
the null hypothesis, H0, will be rejected on the basis
of the sample data.
To specify the decision rule, one specifies a test
statistic, which is a quantity that is computed from the
sample data, and whose sampling distribution under
H0 is known or can be determined. Such a statistic
measures the agreement of the sample data with the
null hypothesis specification.
For our problem, a logical choice for the test statistic
is:
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The Test Statistic:
X
or equivalently, Z c 
X  0
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n
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The latter is a reasonable choice since it measures
how far the sample mean is from the population
mean under H0. The larger the value of |Zc| the more
it will indicate that H0 is not true.
Furthermore, under H0, by virtue of the Central Limit
Theorem, the sampling distribution of Zc will be
approximately standard normal.
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When to Reject H0 and its Consequences
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Having decided which test statistic to use, the next
step is to specify the precise situation in which to
reject H0. We have said that it is logical to reject H0 if
the absolute value of Zc is large.
But how “large” is “large”?
For the moment, let us specify a critical value,
denoted by C, such that if
 |Zc| > C
then H0 will be rejected.
Before deciding on the value of C, let us examine the
consequences of our decision rule.
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Possible Errors of Decision
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Remember at this stage that either H0 is correct, or
H1 is correct. Thus, there is a “true state of reality,”
but this state is not known to us (otherwise we
wouldn’t be performing a test).
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On the other hand, our decision on whether to reject
H0 will only be based on partial information, which is
the sample data.
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We may therefore represent in a table the possible
combinations of “states of reality” and “decision
based on the sample” as follows:
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States of Reality and Decisions Made
Decision
Made Based
on Sample
Data
According to
Rule
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Do not reject
H0
Reject H0
State of Reality
H0 True
H0 False
Correct
Error in
Decision
Decision
(Type II error)
Error in
Correct
Decision
Decision
(Type I error)
In decision-making, there is therefore the possibility
of committing an error, which could either be an error
of Type I or an error of Type II.
Which of these two types of error is more serious??
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Assessing the Two Types of Errors
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From the table in the preceding slide, we have:
Type I error: committed when H0 is rejected when in
reality it is true.
Type II error: committed when H0 is not rejected
when in reality it is false.
Just like in the court trial alluded to earlier, an error of
Type I is considered to be a more serious type of
error (“convicting an innocent man”).
Therefore, we try to minimize the probability of
committing the Type I error.
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Setting the Probability of a Type I Error
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In trying to minimize, however, the probability of a
Type I error, we encounter an obstacle in that the
probabilities of the Type I and Type II errors are
inversely related. Thus, if we try to make the
probability of a Type I error very, very small, then it
will make the probability of a Type II error quite large.
As a compromise we therefore specify a maximum
tolerable Type I error probability, called the
significance level, and denoted by , and choose the
critical value C such that the probability of a Type I
error is (at most) equal to .
This  is conventionally set to 0.10, 0.05, or 0.01.
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Determining the Critical Value, C
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Let us now determine the critical value C in our test.
Recall that our test will reject H0 if |Zc| > C.
By definition,
P{Type I error} = P{reject H0 | H0 is true} = P{|Zc| > C |
H0 is true}.
But, under H0, Zc is distributed as standard normal,
so if we want P{Type I error} = , then we should
choose the critical value C to be:
C = Z/2, which is the value such that P{Z > Z/2} =
/2.
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The Resulting Decision Rule
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Given a significance level of , for testing the null
hypothesis H0:  = 0 versus the alternative
hypothesis H1:   0, the appropriate test statistic,
under the assumptions that (a)  is known, and (b) n
> 30 is given by:
X  0
Reject H 0 if Z c 
 z .
 n
2
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Data Gathering and Making the Decision
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Having specified the final decision rule, the next step
is to gather the sample data and to compute the
sample mean and the value of Zc.
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If |Zc| > z/2 then H0 is rejected; otherwise, we say
that we “fail to reject H0.”
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Note: If  is not known, then we could replace it in the
formula of Zc by the sample standard deviation S.
The final step is to make the relevant conclusion.
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On the Conclusion that One Could Make
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The final step in performing a statistical test of
hypotheses is to make the conclusion relevant to the
particular study, that is, not to simply say that “H0 is
rejected” or “H0 is not rejected.”
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When H0 is rejected, then either that a correct
decision has been made, or an error of Type I has
been committed. But since we have controlled the
probability of committing a Type I error (set to ,
which we could tolerate), then we can conclude in
this case that H0 is not true, and hence that H1 is
correct.
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On Conclusions … continued
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On the other hand, if we did not reject H0, then either
we are making the correct decision, or we are making
a Type II error.
However, since we did not control for the Type II error
probability (when we set the Type I error probability to
be , we “closed our eyes to the probability of a Type
II error”), if we do not reject H0, we cannot conclude
that H0 is true. Rather, we could only say that we
“failed to reject H0 on the basis of the available data.”
This is the basis of the saying that: “you can never
prove a theory, you can only disprove it.”
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Recapitulation: Steps in Hypotheses
Testing
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Step 1: Formulate your null and alternative
hypotheses.
Step 2: Determine the type of sample you will be
getting with regards to sample size, knowledge of the
standard deviation, etc.
Step 3: Specify your level of significance.
Step 4: State precisely your decision rule.
Step 5: Gather your sample data and compute the
test statistic.
Step 6: Decide and make final conclusions.
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The p-Value Approach
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Another approach to making the decision in
hypotheses testing is to compute the p-value
associated with the observed value of the test
statistic.
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By definition, the p-value is the probability of getting
the observed value or more extreme values of the
test statistic under H0.
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In our situation, the p-value would then be:
p-value = P{|Z| > |zc|} where zc is the observed value
of the test statistic.
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Deciding Based on the p-Value
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If the p-value exceed 0.10, then H0 is not rejected
and we say that the result is not significant.
If the p-value is between 0.10 and 0.05, we usually
say that the result is almost significant or tending
towards significance.
If the p-value is between 0.05 and 0.01, we reject H0
and conclude that the result is significant.
If the p-value is less than 0.01 then H0 is rejected and
conclude that the result is highly significant.
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On the Sensitivity of a Test
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Ideally, we would like our test procedure to always
produce the correct decision. However, this is not
possible if the decision is based only on sample data.
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To measure the sensitivity of a test under the
alternative hypothesis, we can compute its power,
which is the probability of rejecting H0 under the
alternative hypothesis.
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That is, Power of Test at 1 = P{reject H0 |  = 1}.
This function could be plotted and can be used to
determine the appropriate sample size.
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Some Concrete Problems
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Situation: The mean yield of corn in the US is about
120 bushels per acre. A survey of 40 farmers this
year gives a sample mean yield of 123.8 bushels per
acre. We want to know whether this is good evidence
that the national mean this year is not 120 bushels
per acre. Assume that the farmers surveyed are an
SRS from the population of all commercial corn
growers and that the standard deviation of the yield in
this population is  = 10 bushels per acre. Test H0: 
= 120 versus H1:   120 at 5% level of significance.
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Solution: Because H1 is a two-sided hypothesis and
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Solution … continued
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Level of significance is  = 0.05, then the appropriate
decision rule is:
Reject H0 if |Zc| > z.025 = 1.96, where the test statistic
is Zc = (Xbar -0)/(/n1/2).
From the given information, the value of this test
statistic is Zc = (123.8 - 120)/[10/401/2] = 2.4033.
Since this value is larger than the critical value of
1.96, then our decision is to reject H0 at 5%
significance level.
We can therefore conclude at the 5% level that the
mean yield of corn for this year is different from the
usual mean yield of 120 bushels per acre.
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P-value Approach Illustrated
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Recall that the p-value is the probability, under H0, of
getting the observed value of the test statistic or more
extreme values. For our problem, we therefore have:
p-value = P{|Z| > 2.4033} = 0.0162.
Based on this value we could reject H0 at the 5%
level, but not at the 1% level.
Another interpretation of the p-value of 0.0162 is that
it is the smallest level of significance at which H0 can
be rejected.
Let us also examine the power of our test.
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Power of the Test
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Let us denote by (1) the power of the test when the
value of the true value of the mean  is 1. Thus,
 ( 1 )  P{reject H 0 |   1}
 P{| Z c | 1.96 |   1}
 P{Z c  1.96 |   1} 
P{Z c  1.96 |   1}.
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Power … continued
Since Z c 
X  0


X  1

n
n

  0
  0
Z
when   1.


n
n
T herefore,
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 ( 1 )  P Z  1.96 
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
n ( 1   0 ) 
n ( 1   0 ) 

P
Z

1
.
96



.



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
Substituting 0 = 120,  = 10, and n = 40 into the
above expression, we can then calculate the value of
(1) for different values of 1.
The values of 1 and (1) could then be plotted. This
plot is given in the next slide.
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Plot of the Power Function
Graph of the Power Function of the Test in our Example
1.0
0.9
0.8
Power
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
110
120
130
True Value of the Mean
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Problems ...
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Situation: The Survey of Study Habits and Attitudes
(SSHA) is a psychological test that measures the
motivation, attitude toward school, and study habits
of students. Scores range from 0 to 200. The mean
score for US college students is about 115, and the
standard deviation is about 30. A teacher who
suspects that older students have better attitudes
toward school gives the SSHA to 20 students who
are at least 30 years of age. Their mean score is
135.2. Assume that  = 30. Perform a test of H0:  =
115 versus H1:  > 115 using the p-value approach.
Solution: To be done in class.
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Some Comments on Assumptions
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The testing procedure we developed here required
two assumptions:
(a) sample size is at least 30;
(b) population standard deviation is known.
Assumption (b) is not crucial since  could be
replaced by S in the formula for Zc.
When assumption (a) is not satisfied, then we need
to be able to assume that the population is normal
and we need to know the population standard
deviation.
If  is not known, we will need to use the tdistribution, which will be discussed next week.
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Concrete Problems for Testing Two
Means
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Question of Interest: Does cocaine use by pregnant
women cause their babies to have low birth weight?
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Hypothesis:
– H0: Mean birth weight of babies of cocaine users
is greater than or equal to the mean birth weight of
babies from non-cocaine users. Symbolically, 1 >
2.
– H1: 1 < 2.
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Data of the Study
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Data Gathering Performed: Birth weights (measured
in grams) of babies of women who tested positive for
cocaine/crack during a drug-screening test were
compared with the birth weights for women who
either tested negative or were not tested, a group
called “other.” Below is the summary statistics for the
two samples.
Group
Sample Size
Sample Means
Positive Test
Other
134
5974
2733
3118
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Sample
Standard
Deviation
599
672
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Problems … continued
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Study Question: Is the mean hemoglobin level
among breast-fed babies higher than those fed with
standard baby formula without iron supplements?
What are the appropriate hypotheses?
Situation: A study of iron deficiency among infants
compared the samples of infants following different
feeding regimens. One group contained breast-fed
infants, while the children in another group were fed a
standard baby formula without any iron supplements.
A summary of the blood hemoglobin levels at 12
months of age is presented in the following table.
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Summary of the Data from Study
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Group
Sample Size
Sample Means
Breast-Fed
Formula
23
19
13.3
12.4
Population
Standard
Deviation
1.7
1.8
The appropriate test will be done in class.
What conclusions could be made?
What assumptions are needed for the test to be
valid?
What if the standard deviations that were provided
were actually the sample standard deviations?
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Tests of a Population Proportion
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Situation: A peony plant with red petals was crossed
with another plant having streaky petals. A geneticist
states that 75% of the offspring resulting from this
cross will have red flowers. To test this claim, 100
seeds from this cross were collected and germinated
and 58 plants had red petals.
What hypotheses are being tested?
Does the observed data contradict the geneticist’s
claim?
The test will be done in class.
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Testing Differences of Two Population
Proportions
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Situation: A clinical trial examined the effectiveness
of aspirin in the treatment of cerebral ischemia
(stroke). Patients were randomized into treatment
and control groups. The study was double-blind in
the sense that neither the patients nor physicians
who evaluated the patients knew which patients
received aspirin and which received the placebo
tablet.
After 6 months of treatment, the attending physicians
evaluated each patient’s progress as either favorable
or unfavorable.
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Continued ...
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Of the 78 patients in the aspirin group, 63 had
favorable outcomes; 43 of the 77 control (placebo)
patients had favorable outcomes.
Source: William S. Fields, et al (1977), “Controlled
trial of aspirin in cerebral ischemia,” Stroke, 8, 301315.
What hypotheses are being tested?
The hypotheses test will be performed in class.
What conclusions could be made based on this data?
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Another Problem
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Situation: Gastric freezing was once a recommended
treatment for ulcers in the upper intestine. A
randomized comparative experiment found that 28 of
the 82 patients who were subjected to gastric
freezing improved, while 30 of the 78 patients in the
control group improved.
Based on this information, test for the hypothesis of
“no difference” for the two populations.
By the way, what will be the relevant populations in
this study?
The test will be done in class.
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