Introduction to Biostatistics. This lecture was given as a part of the Introduction to Epidemiology & Community Medicine Course given for third-year medical students.

2. Hypothesis Testing
&
Chi-Square Statistic

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Definitions
In statistics, a hypothesis is a claim or
statement about a property of a population.
A hypothesis test (or test of significance) is a
standard procedure for testing a claim about a
property of a population.

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Main Objective
The main objective of this chapter is to
develop the ability to conduct.

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Examples of Hypotheses that can be Tested
• Genetics: The Genetics & IVF Institute claims
that its XSORT method allows couples to increase
the probability of having a baby girl.
• Business: A newspaper headline makes the
claim that most workers get their jobs through
networking.
• Medicine: Medical researchers claim that when
people with colds are treated with echinacea, the
treatment has no effect.

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Examples of Hypotheses that can be Tested
• Aircraft Safety: The Federal Aviation
Administration claims that the mean weight of an
airline passenger (including carry-on baggage) is
greater than 185 lb, which it was 20 years ago.
• Quality Control: When new equipment is used
to manufacture aircraft altimeters, the new
altimeters are better because the variation in the
errors is reduced so that the readings are more
consistent. (In many industries, the quality of
goods and services can often be improved by
reducing variation.)

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Null Hypothesis:
H0
• The null hypothesis (denoted by H0) is
a statement that the value of a
population parameter (such as
proportion, mean, or standard
deviation) is equal to some claimed
value. (µ1 equal µ2)
• We test the null hypothesis directly.
• Either reject H0 or fail to reject H0.

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Alternative Hypothesis:
H1
• The alternative hypothesis (denoted
by H1 or Ha or HA) is the statement that
the parameter has a value that
somehow differs from the null
hypothesis.
• The symbolic form of the alternative
hypothesis must use one of these
symbols: , <, >.

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HYPOTHISES
One tail
H0:µ1≤µ2
H1:µ1>µ2
H0:µ1≥µ2
H1:µ1‹µ2
Two tails
H0:µ1=µ2
H1:µ1=µ2

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Note about Forming Your
Own Claims (Hypotheses)
If you are conducting a study and want
to use a hypothesis test to support
your claim, the claim must be worded
so that it becomes the alternative
hypothesis.

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Note about Identifying
H0 and H1
Figure 8-2

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Example:
Consider the claim that the mean weight of
airline passengers (including carry-on
baggage) is at most 195 lb (the current value
used by the Federal Aviation Administration).
Follow the three-step procedure outlined in
Figure 8-2 to identify the null hypothesis and
the alternative hypothesis.

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Example:
Step 1: Express the given claim in symbolic
form. The claim that the mean is at
most 195 lb is expressed in symbolic
form as  ≤ 195 lb.
Step 2: If  ≤ 195 lb is false, then  > 195 lb
must be true.

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Example:
Step 3: Of the two symbolic expressions
 ≤ 195 lb and  > 195 lb, we see that
 > 195 lb does not contain equality,
so we let the alternative hypothesis
H1 be  > 195 lb. Also, the null
hypothesis must be a statement that
the mean equals 195 lb, so we let H0
be  = 195 lb.
Note that the original claim that the mean is at
most 195 lb is neither the alternative hypothesis
nor the null hypothesis. (However, we would be
able to address the original claim upon
completion of a hypothesis test.)

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Conclusions
in Hypothesis Testing
We always test the null hypothesis. The
initial conclusion will always be one of
the following:
1. Reject the null hypothesis.
2. Fail to reject (Accept) the null
hypothesis .

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Type I Error
• A Type I error is the mistake of
rejecting the null hypothesis when it
is actually true.
• The symbol  (alpha) is used to
represent the probability of a type I
error.

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Type II Error
• A Type II error is the mistake of failing
to reject the null hypothesis when it is
actually false.
• The symbol  (beta) is used to
represent the probability of a type II
error.

18. Chi-Square Statistic

19. Cares of this type of statistical tests to test whether the
sample was selected views of a society with a certain
probability distribution or a particular theory.
This test is used when data is in the form of a nominal or
occurrences and is intended to reconcile the quality
here study how similar occurrences of the sample,
which is usually called loops note Observed with the
expected frequencies Expected variable under study in
the original community.
It uses Chi-Square test statistical comparison between the
observed and expected . If the sample is representative
of the population in frequency and identical with the
value of And cares of this type of statistical tests to test
whether the sample was selected views of a society
with a certain probability distribution or a particular
theory.

20. This test is used when data is in the form of a nominal or occurrences and is
intended to reconcile the quality here study how similar occurrences of
the sample, which is usually called loops note Observed with the expected
frequencies Expected variable under study in the original community.
It uses Chi-Square test statistical comparison between the observed and
expected . If the sample is representative of the population in frequency
and identical with the value of And cares of this type of statistical tests to
test whether the sample was selected views of a society with a certain
probability distribution or a particular theory.
This test is used when data is in the form of a nominal or occurrences and is
intended to reconcile the quality here study how similar occurrences of
the sample, which is usually called loops note Observed with the expected
frequencies Expected variable under study in the original community.
It uses Chi-Square test statistical comparison between the observed and
expected . If the sample is representative of the population in frequency
and identical with the value of Chi-Square is usually zero and increase this
value to be more than zero whenever there is a difference between the
occurrences of the sample (note) and between the occurrences of the
theoretical distribution of the community (expected).

21. Statistical hypotheses:
H0: Views group, which have been selected track the distribution of a particular or
specific probabilistic theory.
HA: Views group, which has been selected for inconsistent with this distribution or a
particular theory.
Test tally: is usually zero and increase this value to be more than zero whenever there
is a difference between the occurrences of the sample (note) and between the
occurrences of the theoretical distribution of the community (expected).
Statistical hypotheses:
H0: Views group, which have been selected track the distribution of a particular or
specific probabilistic theory.
HA: Views group, which has been selected for inconsistent with this distribution or a
particular theory.
Test tally: is usually zero and increase this value to be more than zero whenever there
is a difference between the occurrences of the sample (note) and between the
occurrences of the theoretical distribution of the community (expected).
Statistical hypotheses:
H0: Views group, which have been selected track the distribution of a particular or
specific probabilistic theory.
HA: Views group, which has been selected for inconsistent with this distribution or a
particular theory.

22. Where Oi the observed frequency .
Where Ei the expected frequency.
ii npE   i iOn

23. Areas of rejection and acceptance
)(2
v
V is degrees of freedom
is The number of views

25. the decision:
1- We accept the Null hypothesis if (the value of
the Chi-Square Calculated is smaller than the
value of the Chi-Square tabular) that is:

26. 2- We reject the Null hypothesis if (the value of
the Chi-Square Calculated is greater than the
value of the Chi-Square tabular) that is:

27. Example:
One of the researchers selected a sample of size
n = 800 people from a city, and their distribution
by blood type was as follows:
blood typeABABO
Number of people (frequency observed)200150100350
Is this distribution consistent with the distribution of other city members whose
blood distribution was divided according to the following percentages:
blood typeABABO
Percentage of people25%15%15%45%
Use a significant level α= 0.05

28. Answer
Statistical hypotheses:
Ho: There is no difference in the blood groups between the distribution of
the
observed and expected distribution.
H1: There is difference in the blood groups between the distribution of the
observed and expected distribution .
ii npE 
360)45.0(800
120)15.0(800
120)15.0(800
200)25.0(800
44
33
22
11




npE
npE
npE
npE

29. blood typeABABO
Number of people (frequency observed)200150100350
The expected frequency200120120360
11.11
360
)360350(
120
)120100(
120
)120150(
200
)200200()( 2222
1
2
2
0 




 








 
k
i i
ii
E
EO

v=4-1=3
)(2
v
815.7)3(2
05.0  
815.7)3(2
05.0  
)(22
0 v 
Decision
We can reject the null Hypothesis then the type blood in The two cities is
different .