HYPOTHESIS
TESTING
Purpose

The purpose of hypothesis testing is to help the
researcher or administrator in reaching a
decision concerning a population by examining a
sample from that population
Hypothesis


It is a statement about one or more population
It is usually concerned with the parameter of the
population about which the statement is made
Research Hypothesis

It is the assumption that motivate the research.
It is usually the result of long observation by the
researcher. This hypothesis led directly to the
second type of hypothesis
Statistical Hypothesis

This is stated in a way that can be evaluated by
appropriate statistical technique.
Statistical hypothesis



It is composed of two types:
Null hypothesis( Ho): It is the particular
hypothesis under test, and it is the hypothesis of
“no difference”
Alternative hypothesis (HA): which disagree with
the null hypothesis
Test Statistic


It is a mathematical expression of sample values
which provides a basis for testing a statistical
hypothesis .
The result of this test will determine whether we
will accept the null hypothesis and so the HA
will be rejected , or we reject the null hypothesis
and so the HA will be accepted.
Errors


There are two possible errors to come to the
wrong conclusion:
Type 1 error: rejection of the null hypothesis
when it is true. It is presented by alpha, which is
the level of significance, often the 5%, 1%, and
0.1% (α=0.05, 0.01, and 0.001) levels are chosen
. The selection depends on the particular
problem
P-value

It is the smallest value of α for which the Ho
can be rejected , so it gives a more precise
statement about probability of rejection of Ho
when it is true than the alpha level, so instead of
saying the test statistic is significant or not , we
will mention the exact probability of rejecting
the Ho when it is true
Steps in conducting hypothesis
testing

Hypothesis testing can be presented as NINE
steps:
1.Data

The nature of the data whether it consists of
counts, or measurement will determine the test
statistic to be used
2.Hypotheses


Null Hypothesis (Ho): which is the hypothesis
of no difference, and the alternative hypothesis(
HA )
If we accept the Ho we will say that the data to
be tested does not provide sufficient evidence to
cause rejection. If it is rejected we say that the
data are not compatible with Ho and support
the alternative hypothesis (HA)
3.Test statistic

It uses the data of the sample to reach to a
decision to reject or to accept the null
hypothesis. The general formula for a test
statistic is:
relevant statistic-hypothesized parameter
Test statistic
= ----------------------------------------standard error of the relevant statistic
4.Test statistic--Example
_
x -µ
Z=------------σ/√n
5. Distribution of test statistic

It is the key for statistical inference
6. Decision Rule

It will tell us to reject the null hypothesis if the
test statistic falls in the rejection area, and to
accept the it if it falls in the acceptance region
6. Decision Rule



The critical values (tabulated value) that
discriminate between acceptance and rejection
regions depends on alpha level of significance
If the value of the test statistic falls in the
rejection region area , it is considered statistically
significant
If it falls in the acceptance area it is considered
not statistically significant
6. Decision Rule

Whenever we reject a null hypothesis , there is
always a possibility of type 1 error( rejection of
Ho when it is true). This is why we should
decrease this error to the least possible.
Critical values

The values of the test statistic that separate the
rejection region from the acceptance region
Acceptance region

A set of values of the test statistic leading to
acceptance of the null hypothesis
( values of the test statistic not included in the
critical region)
Rejection region

A set of values of the test statistic leading to
rejection of the null hypothesis
7. Computed test statistic

This should be computed and compared with
the acceptance and rejection regions
8. Statistical decision

It consists of rejecting or not rejecting the Ho .
It is rejected if the computed value of the test
statistic falls in the rejection area , and it is not
rejected if the computed value of the test
statistic falls in the acceptance region
9. Conclusion

If Ho is rejected , we conclude that HA is true.
If Ho is not rejected we conclude that HA may
be true.
Two sided test

If the rejection area is divided into the two tails
the test is called two-sided test ,
One sided test
If the rejection region is only in one tail it is
called one-sided test
 The decision will depend on the nature of the
research question being asked by the researcher
Single population mean , known
population variance
_
x -µ
Z=------------σ/√n
Single population mean with
unknown population variance
_
x -µ
t =------------s/√n
Difference between two populations
means with known variances
_ _
(X1 –X2) – (µ1-µ2)
Z=------------------------------√ σ21 /n1 + σ22 /n2
Difference between two population
mean with unknown and unequal
variances
_ _
(X1 –X2) – (µ1-µ2)
t=------------------------------2
2
√ s 1 /n1 + s 2 /n2
Difference between two population
mean with unknown but assumed
equal variances
_ _
(X1 –X2) – (µ1-µ2)
t=------------------------------Sp√ 1 /n1 + 1 /n2
Paired t-test
_
d -µd
t =------------Sd /√n
Single population proportion
˜
P -P
Z=------------√P(1-P)n
Difference between two population
proportions
˜ ˜
(P1-P2) –(P1-P2)
Z=----------------------------------------√P1(1-P1)/n1 + P2(1-P2)/n2
Example

A certain breed of rats shows a mean weight
gain of 65 gm, during the first 3 months of life.
16 of these rats were fed a new diet from birth
until age of 3 months. The mean was 60.75 gm.
If the population variance is 10 gm , is there a
reason to believe at the 5% level of significance
that the new diet causes a change in the average
amount of weight gained



Ho : µ =65
HA: µ ≠ 65
Z 1-α/2
value)
_
x -µ
Answer
α=0.05
Z=1.96 (critical
60.75-65
Z=---------- = ----------- = -5.38
σ /√n
√10/ √16
Sine the calculated values falls in the rejection
region , we reject the Ho, and accept the HA

In the above example , if the population
variance is unknown, and the sample Sd is 3.84
Answer
t
1- α/2
df =n-1
=± 2.1315
_
x -µ
60.75-65
t =-------------=------------= - 4.1315
s /√n
3.84/ √16
Sine the calculated values falls in the rejection region , we
reject the Ho, and accept the HA
Question
In a study two types of dental cements were
used to hold a crown on tooth cast. The amount
of force in foot pounds required to pull each
cemented crown from the cast was reported :
_
X
n
σ
----------------------------------------------------------Cement 1: 45
50
4.1
Cement 2:
42
50
3.4


Test the hypothesis that µ1= µ2 at α=0.05
If σ1 and σ2 are unknown but assumed to be
equal test the same hypothesis if :
S1=6.2 S2= 5.2

If σ1 and σ2 are unknown and unequal test the
same hypothesis
Question

In a dental clinic it is hypothesized that 90% of
all 4-years old children give no evidence of
dental caries. In a study of 100 children 82 gave
no such evidence , would you accept the quoted
value of the 90% ? Use α=0.05
Question

Two communities were sampled to learn their positions
concerning fluoridation prior to campaigns being
launched. The results were :
n1= 110
n2= 75
˜
˜
˜
P1=0.52
P2=0.55
Do these two communities have equal proportions
Patient No.
Nurse 1
Nurse 2
1
147.9
143.0
2
150.9
151.5
3
150.9
152.1
4
158.1
158.1
5
151.2
151.1
6
160.2
160.5
7
157.8
158.0
8
150.1
150.0
9
142.1
142.5
10
159.9
160.0
11
141.9
142.0
12
140.8
141.0
13
147.7
148.0
14
143.6
144.0
15
139.9
141.0
Patient No. Nurse 1
Nurse 2
1
147.9
143.0
2
150.9
151.5
3
150.9
152.1
4
158.1
158.1
5
151.2
151.1
6
160.2
160.5
7
157.8
158.0
8
150.1
150.0
9
142.1
142.5
10
159.9
160.0
11
141.9
142.0
12
140.8
141.0
13
147.7
148.0
14
143.6
144.0
15
139.9
141.0
d
d2