Download Test of Hypothesis - Lyle School of Engineering

BOBBY B. LYLE
SCHOOL OF ENGINEERING
EMIS - SYSTEMS ENGINEERING PROGRAM
SMU
EMIS 7370 STAT 5340
Department of Engineering Management, Information and Systems
Probability and Statistics for Scientists and Engineers
Tests of Hypothesis
Basic Concepts & Tests of Proportions
Dr. Jerrell T. Stracener
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Tests of Hypothesis
- Basic Concepts
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Testing Hypotheses
In many situations the reason for gathering and
analyzing data is to provide a basis for deciding
on a course of action. Let us assume that either
of two courses of action is possible: A1 or A2, and
that we would be clear whether one or the other
is the better action, if only we knew the nature of
a certain population - that is, if we knew the
probability distribution of a certain random
variable.
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Testing Hypotheses
The whole population or the distribution of
probability is usually unattainable, therefore, we
are forced to settle for such information as
can be gleaned from a sample and to make our
choice between the two actions on the basis of
the sample.
1. Obtain random sample of size n
2. Apply decision rule to data
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Testing Hypotheses
Statistical Hypothesis - is a statement about a
probability distribution and is usually a statement
about the values of one or more parameters of
the distribution. For example, a company may
want to test the hypothesis that the true average
lifetime of a certain type of TV is at least 500
hours, i.e., that   500.
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Testing Hypotheses
The hypothesis to be tested is called the null
hypothesis and is denoted by H0. To construct a
criterion for testing a given null hypothesis, an
Alternative hypotheses, must be formed denoted by
H1 or HA.
Remark: To test the validity of a statistical
hypothesis the test is conducted, and according
to the test plan the hypothesis is rejected if the
results are improbable under the hypothesis. If
not, the hypothesis is accepted. The test leads
to one of two possible actions: accept H0 or
reject H0 (accept H1)
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Testing Hypotheses
Test Statistic - The statistic upon which a test of
a statistical hypothesis is based.
Critical Region - The range of values of a test
statistic which, for a given test, requires the
rejection of H0.
Remark: Acceptance or rejection of a statistical
hypothesis does not prove nor disprove the
hypothesis! Whenever a statistical hypothesis is
accepted or rejected on the basis of a sample,
there is always the risk of making a wrong
decision. The uncertainty with which a decision
is made is measured in terms of probability.
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Testing Hypotheses
There are two possible decision errors associated
with testing a statistical hypothesis:
A Type I error is made when a true hypothesis is
rejected.
A Type II error is made when a false hypothesis
is accepted.
Decision
Accept H0
Reject H0
(Accept H1)
True Situation
H0 true
H0 false
correct
Type II error
Type I error
correct
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Testing Hypotheses
The decision risks are measured in terms of
probability.
 = P(Type I error)
= P(reject H0|H0 is true)
= Producers risk
 = P(Type II error)
= P(accept H0|H1 is true)
= Consumers risk
Remark: 100% is commonly referred to as the
significance level of a test.
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Testing Hypotheses
Note: For fixed n,  increases as  decreases, and
vice versa, as n increases, both  and  decrease.
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Important Properties
• The Type I error and Type II error are related.
A decrease in the probability of one generally
results in an increase in the probability of the
other.
• The size of the critical region, and therefore
the probability of committing a Type I error,
can always be reduced by adjusting the critical
value(s).
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Important Properties
• An increase in the sample size n will reduce 
and  simultaneously.
• If the null hypothesis is false,  is a maximum
when the true value of a parameter approaches
the hypothesized value. The greater the distance
between the true value and the hypothesized
value, the smaller  will be.
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P-Value
• A p-value is the lowest level of significance at
which the observed value of the test statistic is
significant.
• Reject H0 if the computed p-value is less than or
equal to the desired level of significance .
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Power Function
Before applying a test procedure, i.e., a decision
rule, we need to analyze its discriminating power,
i.e., how good the test is. A function called the
power function enables us to make this analysis.
Power Function
= PF()
= PR()
= P(rejecting H0|true parameter value)
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Power Function
A plot of the power function vs the test parameter
value is called the power curve and 1 - power curve
is the OC curve.
ideal power curve
PR()
1

0
H0
H1
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Power of a Test
• The power of a test can be computed as 1 - 
• Often different types of tests are compared by
contrasting power properties
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Power Function
The power function of a statistical test of hypothesis is
the probability of rejecting H0 as a function of the true
value of the parameter being tested, say , i.e.,
PF() = PR()
= P(reject H0|)
= P(test statistic falls in CR|)
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Operating Characteristic Function
•The operating characteristic function of a statistical
test of hypothesis is the probability of accepting
H0 as a function of the true value of the parameter
being tested, say , i.e.,
OC() = PA()
= P(accept H0|)
= P(test statistic falls in CA|)
• Note that OC()=1-PF()
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Summary Procedures for Hypothesis Testing
1. State the null hypothesis H0 that  = 0
2. Choose an appropriate alternative hypothesis
H1 from one of the alternatives  < 0,  > 0, or
  0
3. Choose a significance level of size .
4. Select the appropriate test statistic and establish
the critical region. (If the decision is to be based on
a P-value, it is not necessary to state the critical
region.)
5. Compute the value of the test statistic from the
sample data
6. Carry out the test of hypothesis and make a
Decision. Reject H0 if the test statistic has a
value in the critical region (or if the computed
P-value is less than or equal to the desired
significance level ); otherwise, do not reject H0
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Tests of Proportions
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Tests of Proportions
Let X1, X2, . . ., Xn be a random sample of size n
from B(n, p).
Case 1: small sample sizes
To test the Null Hypothesis
H0: p = p0, a specified value, against the
appropriate Alternative Hypothesis
1. HA: p < p0 ,
or
2. HA: p > p0 ,
or
3. HA: p  p0 ,
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Tests of Proportions
at the 100% Level of Significance, calculate
the value of the test statistic using X ~ B(n, p = p0).
Find the number of successes and compute the
appropriate P-Value, depending upon the alternative
hypothesis and reject H0 if P  , where
1. P = P(X  x|p = p0),
or
or
2. P = P(X  x|p = p0),
3. P = 2P(X  x|p = p0) if x < np0, or
P = 2P(X  x|p = p0) if x > np0,
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Tests of Proportions
Case 2: large sample sizes with p not extremely
close to 0 or 1.
To test the Null Hypothesis
H0: p = p0, a specified value, against the
appropriate Alternative Hypothesis
1. HA: p < p0 ,
or
2. HA: p > p0 ,
or
3. HA: p  p0 ,
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Tests of Proportions
Calculate the value of the test statistic
x  np 0
Z
np 0 q 0
and reject H0 if
1.
z  z ,
2.
z  z ,
3.
z  z α
or
or
2
or
z  zα ,
2
depending on the alternative hypothesis.
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Example
A missile manufacturer claims that the probability of success
for missile MX is 0.80. To demonstrate its reliability 25
missiles are fired. If 15 are successful, do you agree with
the manufacturers claim?
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Solution
We want to test
H0: p=0.8
vs
H1: p0.8
At the 10% level of significance (Since a value was given, I
selected 10% to illustrate the process)
The test statistic is X, the observed number of successes,
and X~B(25,p)
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Solution
We have to determine values L and U where
n x
α
n x


PX  L | p 0     p 0 1  p 0  
2
x 0  x 
L
and
n x
α
n x
PX  U | p 0     p 0 1  p 0  
2
x U  x 
n
Note that is we cannot find integer values of L and U, select those values
of L and U which make the value of each of the summations as large as
possible without exceeding /2. Now
 25 
x
25 x
S L    0.8 0.2
 0.05
x 0  x 
and
L
 25 
x
25 x
SU    0.8 0.2
 0.05
x U  x 
25
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Solution
Here,
L=16 for SL=0.0468
U=23 for Su=0.0274
We reject H0 if X<L or if X>U. Since X=15, we reject H0
and conclude that at the 7.42% level of significance that
the manufacturer’s claim is incorrect.
Note that L=17 and U=22, result in
SL= 0.109 and Su=0.0982
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Solution
Using case 2, the value of the test statistic is
x  np0
15  25  0.8
Z

 2.5
np0 q0
25  0.8  0.2
The critical region is Z< -1.64 or Z > 1.64.
Since -2.5 is less than -1.64, reject H0 and conclude that the
manufacturer’s claim is incorrect.
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Example
A manufacturing company has submitted a claim that 90% of
items produced by a certain process are non-defective. An
improvement in the process is being considered that they
feel will lower the proportion of defectives below the current
10%. In an experiment 100 items are produced with the new
process and 5 are defective. Is this evidence sufficient to
conclude that the method has been improved? Use a 0.05
level of significance.
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Solution
Follow the six-step procedure:
1. H0: p=0.9
2. H1:p>0.9
3. =0.05
4. Critical region: Z>1.645
5. Computations: x=95, n=100, np0=(100)(0.90)=90, and
95  90
z
 1.67
(100)(0.9)(0.1)
6. Conclusion: Reject H0 and conclude that the
improvement has reduced the proportion of defects.
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