Statistical Inference

About This Presentation

Title:

Statistical Inference

Description:

Statistical Inference Making decisions regarding the population base on a sample ... – PowerPoint PPT presentation

Number of Views:187

Avg rating:3.0/5.0

Slides: 166

Provided by: lave9

Category:

more less

Transcript and Presenter's Notes

Title: Statistical Inference

1
Statistical Inference

Making decisions regarding the population base on
a sample

2
Decision Types

Estimation

Deciding on the value of an unknown parameter

Hypothesis Testing

Deciding a statement regarding an unknown
parameter is true of false

Prediction

Deciding the future value of a random variable

All decisions will be based on the values of
statistics

3
Estimation

Definitions

An estimator of an unknown parameter is a sample
statistic used for this purpose

An estimate is the value of the estimator after
the data is collected

The performance of an estimator is assessed by
determining its sampling distribution and
measuring its closeness to the parameter being
estimated

4
Examples of Estimators
5
The Sample Proportion

Let p population proportion of interest or
binomial probability of success.
Let

sample proportion or proportion of successes.
is a normal distribution with
6
(No Transcript)
7
The Sample Mean

Let x1, x2, x3, , xn denote a sample of size n
from a normal distribution with mean m and
standard deviation s.
Let

is a normal distribution with
8
(No Transcript)
9
Confidence Intervals
10
Estimation by Confidence Intervals

Definition

An (100) P confidence interval of an unknown
parameter is a pair of sample statistics (t1 and
t2) having the following properties

Pt1 lt t2 1. That is t1 is always smaller
than t2.

Pthe unknown parameter lies between t1 and t2
P.

the statistics t1 and t2 are random variables

Property 2. states that the probability that the
unknown parameter is bounded by the two
statistics t1 and t2 is P.

11
Critical values for a distribution

The a upper critical value for a any distribution
is the point xa underneath the distribution such
that PX gt xa a

a
xa
12
Critical values for the standard Normal
distribution

PZ gt za a

a
za
13
Critical values for the standard Normal
distribution

PZ gt za a

Confidence Intervals for a proportion p

Let
and
Then t1 to t2 is a (1 a)100 P100 confidence
interval for p
15

Logic

Thus t1 to t2 is a (1 a)100 P100 confidence
interval for p
16
Example

Suppose we are interested in determining the
success rate of a new drug for reducing Blood
Pressure

The new drug is given to n 70 patients with
abnormally high Blood Pressure

Of these patients to X 63 were able to reduce
the abnormally high level of Blood Pressure

The proportion of patients able to reduce the
abnormally high level of Blood Pressure was

17
and za/2 1.960
If P 1 a 0.95 then a/2 .025

Then

and
Thus a 95 confidence interval for p is 0.8297 to
0.9703
18
What is the probability that p is beween 0.8297
and 0.9703?
Is it 95 ?
Answer p (unknown) , 0.8297 and 0.9703 are
numbers. Either p is between 0.8297 and 0.9703
or it is not. The 95 refers to success of
confidence interval procedure prior to the
collection of the data. After the data is
collected it was either successful in capturing p
or it was not.
19
Statistical Inference

Making decisions regarding the population base on
a sample

20
Two Areas of Statistical Inference

Estimation
Hypothesis Testing

21
Estimation

Definitions

An estimator of an unknown parameter is a sample
statistic used for this purpose

An estimate is the value of the estimator after
the data is collected

The performance of an estimator is assessed by
determining its sampling distribution and
measuring its closeness to the parameter being
estimated

22
Confidence Intervals

Estimation of a parameter by a range of values
(an interval)

23
Estimation by Confidence Intervals

Definition

An (100) P confidence interval of an unknown
parameter is a pair of sample statistics (t1 and
t2) having the following properties

Pt1 lt t2 1. That is t1 is always smaller
than t2.

Pthe unknown parameter lies between t1 and t2
P.

the statistics t1 and t2 are random variables

Property 2. states that the probability that the
unknown parameter is bounded by the two
statistics t1 and t2 is P.

24
Confidence Interval for a Proportion

100(1 a) Confidence Interval for the
population proportion

Interpretation For about 100(1 a)P of all
randomly selected samples from the population,
the confidence interval computed in this manner
captures the population proportion.
25
Comment

The usual choices of a are 0.05 and 0.01
In this case the level of confidence, 100(1 -
a), is 95 and 99 respectively
Also the tabled value za/2 is
z0.025 1.960 and
z0.005 2.576 respectively

26
Example

Suppose we are interested in determining the
success rate of a new drug for reducing Blood
Pressure

The new drug is given to n 70 patients with
abnormally high Blood Pressure

Of these patients to X 63 were able to reduce
the abnormally high level of Blood Pressure

The proportion of patients able to reduce the
abnormally high level of Blood Pressure was

27
and za/2 1.960
If P 1 a 0.95 then a/2 .025

Then

and
Thus a 95 confidence interval for p is 0.8297 to
0.9703
28
What is the probability that p is beween 0.8297
and 0.9703?
Is it 95 ?
Answer p (unknown) , 0.8297 and 0.9703 are
numbers. Either p is between 0.8297 and 0.9703
or it is not. The 95 refers to success of
confidence interval procedure prior to the
collection of the data. After the data is
collected it was either successful in capturing p
or it was not.
29
Error Bound
For a (1 a) confidence level, the approximate
margin of error in a sample proportion is
30
Factors that Determine the Error Bound

1. The sample size, n. When sample size
increases, margin of error decreases.

3. The multiplier za/2. Connected to the (1
a) level of confindence of the Error Bound.
The value of za/2 for a 95 level of confidence
is 1.96 This value is changed to change the level
of confidence.
31
Determination of Sample Size

In almost all research situations the researcher
is interested in the question

How large should the sample be?
32
Answer

Depends on

How accurate you want the answer.

Accuracy is specified by

Specifying the magnitude of the error bound

Level of confidence

33
Error Bound

If we have specified the level of confidence then
the value of za/2 will be known.
If we have specified the magnitude of B, it will
also be known

Solving for n we get
34
Summarizing

The sample size that will estimate p with an
Error Bound B and level of confidence P 1 a
is

where
B is the desired Error Bound
za/2 is the a/2 critical value for the standard
normal distribution
p is some preliminary estimate of p.
If you do not have a preliminary estimate of p,
use p 0.50

35
Reason
For p 0.50
n will take on the largest value.
Thus using p 0.50, n may be larger than
required if p is not 0.50. but will give the
desired accuracy or better for all values of p.
36
Example

Suppose that I want to conduct a survey and want
to estimate p proportion of voters who favour a
downtown location for a casino

I know that the approximate value of p is
p 0.50. This is also a good choice for p if
one has no preliminary estimate of its value.
I want the survey to estimate p with an error
bound B 0.01 (1 percentage point)
I want the level of confidence to be 95 (i.e. a
0.05 and za/2 z0.025 1.960
Then

Confidence Intervals
for the mean , m,
of a Normal Population

Confidence Intervals for the mean of a Normal
Population, m

Then t1 to t2 is a (1 a)100 P100 confidence
interval for m
39

Logic

has a Standard Normal distribution
Hence
Thus t1 to t2 is a (1 a)100 P100 confidence
interval for m
40
Example

Suppose we are interested average Bone Mass
Density (BMD) for women aged 70-75

A sample n 100 women aged 70-75 are selected
and BMD is measured for eahc individual in the
sample.

The average BMD for these individuals is

The standard deviation (s) of BMD for these
individuals is

41
If P 1 a 0.95 then a/2 .025
and za/2 1.960

Then

and
Thus a 95 confidence interval for m is 24.10 to
27.16
42
Determination of Sample Size

Again a question to be asked

How large should the sample be?
43
Answer

Depends on

How accurate you want the answer.

Accuracy is specified by

Specifying the magnitude of the error bound

Level of confidence

44
Error Bound

If we have specified the level of confidence then
the value of za/2 will be known.
If we have specified the magnitude of B, it will
also be known

Solving for n we get
45
Summarizing

The sample size that will estimate m with an
Error Bound B and level of confidence P 1 a
is

where
B is the desired Error Bound
za/2 is the a/2 critical value for the standard
normal distribution
s is some preliminary estimate of s.

46
Notes

n increases as B, the desired Error Bound,
decreases
Larger sample size required for higher level of
accuracy
n increases as the level of confidence, (1 a),
increases
za/2 increases as a/2 becomes closer to zero.
Larger sample size required for higher level of
confidence
n increases as the standard deviation, s, of the
population increases.
If the population is more variable then a larger
sample size required

47
Summary

The sample size n depends on
Desired level of accuracy
Desired level of confidence
Variability of the population

48
Example

Suppose that one is interested in estimating the
average number of grams of fat (m) in one
kilogram of lean beef hamburger

This will be estimated by
randomly selecting one kilogram samples, then
Measuring the fat content for each sample.
Preliminary estimates of m and s indicate
that m and s are approximately 220 and 40
respectively.

I want the study to estimate m with an error
bound 5
and
a level of confidence to be 95 (i.e. a 0.05
and za/2 z0.025 1.960)

49
Solution
Hence n 246 one kilogram samples are required
to estimate m within B 5 gms with a 95 level
of confidence.
50
Statistical Inference

Making decisions regarding the population base on
a sample

51
Decision Types

Estimation

Deciding on the value of an unknown parameter

Hypothesis Testing

Deciding a statement regarding an unknown
parameter is true of false

Prediction

Deciding the future value of a random variable

All decisions will be based on the values of
statistics

52
Estimation

Definitions

An estimator of an unknown parameter is a sample
statistic used for this purpose

An estimate is the value of the estimator after
the data is collected

The performance of an estimator is assessed by
determining its sampling distribution and
measuring its closeness to the parameter being
estimated

53
Comments

When you use a single statistic to estimate a
parameter it is called a point estimator
The estimate is a single value
The accuracy of this estimate cannot be
determined from this value
A better way to estimate is with a confidence
interval.
The width of this interval gives information on
its accuracy

54
Estimation by Confidence Intervals

Definition

An (100) P confidence interval of an unknown
parameter is a pair of sample statistics (t1 and
t2) having the following properties

Pt1 lt t2 1. That is t1 is always smaller
than t2.

Pthe unknown parameter lies between t1 and t2
P.

the statistics t1 and t2 are random variables

Property 2. states that the probability that the
unknown parameter is bounded by the two
statistics t1 and t2 is P.

55
Confidence Intervals
Summary
56
Confidence Interval for a Proportion
57
Determination of Sample Size

The sample size that will estimate p with an
Error Bound B and level of confidence P 1 a
is

where
B is the desired Error Bound
za/2 is the a/2 critical value for the standard
normal distribution
p is some preliminary estimate of p.

Confidence Intervals for the mean of a Normal
Population, m

59
Determination of Sample Size

The sample size that will estimate m with an
Error Bound B and level of confidence P 1 a
is

where
B is the desired Error Bound
za/2 is the a/2 critical value for the standard
normal distribution
s is some preliminary estimate of s.

60
Hypothesis Testing

An important area of statistical inference

61
Definition

Hypothesis (H)
Statement about the parameters of the population
In hypothesis testing there are two hypotheses of
interest.
The null hypothesis (H0)
The alternative hypothesis (HA)

Either
null hypothesis (H0) is true or
the alternative hypothesis (HA) is true.
But not both
We say that are mutually exclusive and
exhaustive.

One has to make a decision
to either to accept null hypothesis (equivalent
to rejecting HA)
or
to reject null hypothesis (equivalent to
accepting HA)

There are two possible errors that can be made.
Rejecting the null hypothesis when it is true.
(type I error)
accepting the null hypothesis when it is false
(type II error)

An analogy a jury trial
The two possible decisions are
Declare the accused innocent.
Declare the accused guilty.

The null hypothesis (H0) the accused is
innocent
The alternative hypothesis (HA) the accused is
guilty

The two possible errors that can be made
Declaring an innocent person guilty.
(type I error)
Declaring a guilty person innocent.
(type II error)
Note in this case one type of error may be
considered more serious

68
Decision Table showing types of Error
H0 is True
H0 is False

Correct Decision
Type II Error
Accept H0
Correct Decision
Type I Error
Reject H0
69

To define a statistical Test we
Choose a statistic (called the test statistic)
Divide the range of possible values for the test
statistic into two parts
The Acceptance Region
The Critical Region

To perform a statistical Test we
Collect the data.
Compute the value of the test statistic.
Make the Decision
If the value of the test statistic is in the
Acceptance Region we decide to accept H0 .
If the value of the test statistic is in the
Critical Region we decide to reject H0 .

Example
We are interested in determining if a coin is
fair.
i.e. H0 p probability of tossing a head ½.
To test this we will toss the coin n 10 times.
The test statistic is x the number of heads.
This statistic will have a binomial distribution
with p ½ and n 10 if the null hypothesis is
true.

72
Sampling distribution of x when H0 is true
73

Note
We would expect the test statistic x to be around
5 if H0 p ½ is true.
Acceptance Region 3, 4, 5, 6, 7.
Critical Region 0, 1, 2, 8, 9, 10.
The reason for the choice of the Acceptance
region
Contains the values that we would expect for x if
the null hypothesis is true.

Definitions For any statistical testing
procedure define
a PRejecting the null hypothesis when it is
true P type I error
b Paccepting the null hypothesis when it is
false P type II error

In the last example
a P type I error p(0) p(1) p(2) p(8)
p(9) p(10) 0.109, where p(x) are binomial
probabilities with p ½ and n 10 .
b P type II error p(3) p(4) p(5) p(6)
p(7), where p(x) are binomial probabilities
with p (not equal to ½) and n 10. Note these
will depend on the value of p.

76
Table Probability of a Type II error, b vs. p
Note the magnitude of b increases as p gets
closer to ½.
77

Comments
You can control a P type I error and b P
type II error by widening or narrowing the
acceptance region. .
Widening the acceptance region decreases a P
type I error but increases b P type II
error.
Narrowing the acceptance region increases a P
type I error but decreases b P type II
error.

Example Widening the Acceptance Region
Suppose the Acceptance Region includes in
addition to its previous values 2 and 8 then a
P type I error p(0) p(1) p(9) p(10)
0.021, where again p(x) are binomial
probabilities with p ½ and n 10 .
b P type II error p(2) p(3) p(4) p(5)
p(6) p(7) p(8). Tabled values of are given
on the next page.

79
Table Probability of a Type II error, b vs. p
Note Compare these values with the previous
definition of the Acceptance Region. They have
increased,
80

Example Narrowing the Acceptance Region
Suppose the original Acceptance Region excludes
the values 3 and 7. That is the Acceptance Region
is 4,5,6. Then a P type I error p(0)
p(1) p(2) p(3) p(7) p(8) p(9) p(10)
0.344.
b P type II error p(4) p(5) p(6) .
Tabled values of are given on the next page.

81
Table Probability of a Type II error, b vs. p
Note Compare these values with the otiginal
definition of the Acceptance Region. They have
decreased,
82
Acceptance Region 4,5,6.
Acceptance Region 2,3,4,5,6,7,8.
Acceptance Region 3,4,5,6,7.
a 0.344
a 0.109
a 0.021
83
Hypothesis Testing

An important area of statistical inference

84
Definition

Hypothesis (H)
Statement about the parameters of the population
In hypothesis testing there are two hypotheses of
interest.
The null hypothesis (H0)
The alternative hypothesis (HA)

Either
null hypothesis (H0) is true or
the alternative hypothesis (HA) is true.
But not both
We say that are mutually exclusive and
exhaustive.

86
Decision Table showing types of Error
H0 is True
H0 is False

Correct Decision
Type II Error
Accept H0
Correct Decision
Type I Error
Reject H0
87

The Approach in Statistical Testing is
Set up the Acceptance Region so that a is close
to some predetermine value (the usual values are
0.05 or 0.01)
The predetermine value of a (0.05 or 0.01) is
called the significance level of the test.
The significance level of the test is a Ptest
makes a type I error

Determining the Critical Region
The Critical Region should consist of values of
the test statistic that indicate that HA is true.
(hence H0 should be rejected).
The size of the Critical Region is determined so
that the probability of making a type I error, a,
is at some pre-determined level. (usually 0.05 or
0.01). This value is called the significance
level of the test.
Significance level Ptest makes type I error

To find the Critical Region
Find the sampling distribution of the test
statistic when is H0 true.
Locate the Critical Region in the tails (either
left or right or both) of the sampling
distribution of the test statistic when is H0
true.
Whether you locate the critical region in the
left tail or right tail or both tails depends on
which values indicate HA is true.
The tails chosen values indicating HA.

the size of the Critical Region is chosen so that
the area over the critical region and under the
sampling distribution of the test statistic when
is H0 true is the desired level of a Ptype I
error

Sampling distribution of test statistic when H0
is true
Critical Region - Area a
91
The z-test for Proportions

Testing the probability of success in a binomial
experiment

92
Situation

A success-failure experiment has been repeated n
times
The probability of success p is unknown. We want
to test
H0 p p0 (some specified value of p)
Against
HA

93
The Data

The success-failure experiment has been repeated
n times
The number of successes x is observed.
Obviously if this proportion is close to p0 the
Null Hypothesis should be accepted otherwise the
null Hypothesis should be rejected.

94
The Test Statistic

To decide to accept or reject the Null Hypothesis
(H0) we will use the test statistic

If H0 is true we should expect the test statistic
z to be close to zero.

If H0 is true we should expect the test statistic
z to have a standard normal distribution.

If HA is true we should expect the test statistic
z to be different from zero.

The sampling distribution of z when H0 is true
The Standard Normal distribution

Accept H0
96

The Acceptance region

Accept H0
97

Acceptance Region
Accept H0 if

Critical Region
Reject H0 if

With this Choice

98
Summary

To Test for a binomial probability p
H0 p p0 (some specified value of p)
Against
HA
we

Decide on a PType I Error the significance
level of the test (usual choices 0.05 or 0.01)

Collect the data

Compute the test statistic

Make the Decision

Accept H0 if

Reject H0 if

100
Example

In the last provincial election the proportion of
the voters who voted for the Liberal party was
0.08 (8 )

The party is interested in determining if that
percentage has changed

A sample of n 800 voters are surveyed

101

We want to test
H0 p 0.08 (8)
Against
HA

102
Summary

Decide on a PType I Error the significance
level of the test
Choose (a 0.05)

Collect the data
The number in the sample that support the liberal
party is x 92

103

Compute the test statistic

Make the Decision

Accept H0 if

Reject H0 if

104

Since the test statistic is in the Critical
region we decide to Reject H0

Conclude that H0 p 0.08 (8) is false

There is a significant difference (a 5) in the
proportion of the voters supporting the liberal
party in this election than in the last election

105
The two-tailed z-test for Proportions

Testing the probability of success in a binomial
experiment

106
Situation

A success-failure experiment has been repeated n
times
The probability of success p is unknown. We want
to test
H0 p p0 (some specified value of p)
Against
HA

107
The Test Statistic

To decide to accept or reject the Null Hypothesis
(H0) we will use the test statistic

108

Acceptance Region
Accept H0 if

Critical Region
Reject H0 if

With this Choice

109

The Acceptance region

Accept H0
110
The one tailed z-test

A success-failure experiment has been repeated n
times
The probability of success p is unknown. We want
to test
H0 (some specified value of p)
Against
HA
The alternative hypothesis is in this case called
a one-sided alternative

111
The Test Statistic

To decide to accept or reject the Null Hypothesis
(H0) we will use the test statistic

If H0 is true we should expect the test statistic
z to be close to zero or negative

If p p0 we should expect the test statistic z
to have a standard normal distribution.

If HA is true we should expect the test statistic
z to be a positive number.

112

The sampling distribution of z when p p0
The Standard Normal distribution

Reject H0
Accept H0
113

The Acceptance and Critical region

Reject H0
Accept H0
114

Acceptance Region
Accept H0 if

Critical Region
Reject H0 if

The Critical Region is called one-tailed
With this Choice

115
Example

A new surgical procedure is developed for
correcting heart defects infants before the age
of one month.

Previously the procedure was used on infants that
were older than one month and the success rate
was 91

A study is conducted to determine if the success
rate of the new procedure is greater than 91 (n
200)

116

We want to test
H0
Against
HA

117
Summary

Decide on a PType I Error the significance
level of the test
Choose (a 0.05)

Collect the data
The number of successful operations in the sample
of 200 cases is x 187

118

Compute the test statistic

Make the Decision

Accept H0 if

Reject H0 if

119

Since the test statistic is in the Acceptance
region we decide to Accept H0

There is a no significant (a 5) increase in
the success rate of the new procedure over the
older procedure

120
Comments

When the decision is made to accept H0 is made
one should not conclude that we have proven H0.
This is because when setting up the test we have
not controlled b Ptype II error Paccepting
H0 when H0 is FALSE
Whenever H0 is accepted there is a possibility
that a type II error has been made.

121
In the last example

The conclusion that there is a no significant (a
5) increase in the success rate of the new
procedure over the older procedure should be
interpreted

We have been unable to proof that the new
procedure is better than the old procedure

122
Some other comments

When does one use a two-tailed test?
When does one use a one tailed test?
Answer This depends on the alternative
hypothesis HA.
Critical Region values that indicate HA
Thus if only the upper tail indicates HA, the
test is one tailed.
If both tails indicate HA, the test is two
tailed.

123
Also

The alternative hypothesis HA usually corresponds
to the research hypothesis (the hypothesis that
the researcher is trying to prove)

The new procedure is better
The drug is effective in reducing levels of
cholesterol.
There has a change in political opinion from the
time the survey was taken till the present time
(time of current survey).

124
The z-test for the Mean of a Normal Population

We want to test, m, denote the mean of a normal
population

125
Situation

A sample of n observations are collected from a
Normal distribution
The mean of the Normal distribution, m, is
unknown. We want to test
H0 m m0 (some specified value of m)
Against
HA

126
The Data

Let x1, x2, x3 , , xn denote a sample from a
normal population with mean m and standard
deviation s.
Let
we want to test if the mean, m, is equal to some
given value m0.
Obviously if the sample mean is close to m0 the
Null Hypothesis should be accepted otherwise the
null Hypothesis should be rejected.

127
The Test Statistic

To decide to accept or reject the Null Hypothesis
(H0) we will use the test statistic

If H0 is true we should expect the test statistic
z to be close to zero.

If H0 is true we should expect the test statistic
z to have a standard normal distribution.

If HA is true we should expect the test statistic
z to be different from zero.

128

The sampling distribution of z when H0 is true
The Standard Normal distribution

Accept H0
129

The Acceptance region

Accept H0
130

Acceptance Region
Accept H0 if

Critical Region
Reject H0 if

With this Choice

131
Summary

To Test for mean m, of a normal population
H0 m m0 (some specified value of m)
Against
HA

Decide on a PType I Error the significance
level of the test (usual choices 0.05 or 0.01)

132

Collect the data

Compute the test statistic

Make the Decision

Accept H0 if

Reject H0 if

133
Example

A manufacturer Glucosamine capsules claims that
each capsule contains on the average

500 mg of glucosamine

To test this claim n 40 capsules were selected
and amount of glucosamine (X) measured in each
capsule.
Summary statistics
134

We want to test

Manufacturers claim is correct
against
Manufacturers claim is not correct
135
The Test Statistic
136
The Critical Region and Acceptance Region
Using a 0.05
za/2 z0.025 1.960
We accept H0 if -1.960 z 1.960
reject H0 if z lt -1.960 or z gt 1.960
137
The Decision
Since z -2.75 lt -1.960 We reject H0
Conclude the manufacturerss claim is incorrect
138
Hypothesis Testing
A review of the concepts
139

In hypotheses testing there are two hypotheses
The Null Hypothesis (H0)
The Alternative Hypothesis (HA)
The alternative hypothesis is usually the
research hypothesis - the hypothesis that the
researcher is trying to prove.
The null hypothesis is the hypothesis that the
research hypothesis is not true.

140

A statistical Test is defined by
Choosing a statistic (called the test statistic)
Dividing the range of possible values for the
test statistic into two parts
The Acceptance Region
The Critical Region

141

To perform a statistical Test we
Collect the data.
Compute the value of the test statistic.
Make the Decision
If the value of the test statistic is in the
Acceptance Region we decide to accept H0 .
If the value of the test statistic is in the
Critical Region we decide to reject H0 .

142

You can compare a statistical test to a meter

Value of test statistic
Acceptance Region
Critical Region
Critical Region
Critical Region is the red zone of the meter
143
Value of test statistic
Acceptance Region
Critical Region
Critical Region
Accept H0
144
Acceptance Region
Value of test statistic
Critical Region
Critical Region
Reject H0
145
Acceptance Region
Critical Region
Sometimes the critical region is located on one
side. These tests are called one tailed tests.
146

Whether you use a one tailed test or a two tailed
test depends on

The hypotheses being tested (H0 and HA).
The test statistic.

147
If only large positive values of the test
statistic indicate HA then the critical region
should be located in the positive tail. (1 tailed
test)
If only large negative values of the test
statistic indicate HA then the critical region
should be located in the negative tail. (1 tailed
test)
If both large positive and large negative values
of the test statistic indicate HA then the
critical region should be located both the
positive and negative tail. (2 tailed test)
148
Usually 1 tailed tests are appropriate if HA is
one-sided.
Two tailed tests are appropriate if HA is two
-sided. But not always
149

Once the test statistic is determined, to set up
the critical region we have to find the sampling
distribution of the test statistic when H0 is true

This describes the behaviour of the test
statistic when H0 is true
150

We then locate the critical region in the tails
of the sampling distribution of the test
statistic when H0 is true

a /2
a /2
The size of the critical region is chosen so that
the area over the critical region is a.
151

This ensures that the Ptype I error
Prejecting H0 when true a

a /2
a /2
152

To find Ptype II error P accepting H0 when
false b, we need to find the sampling
distribution of the test statistic when H0 is
false

sampling distribution of the test statistic when
H0 is false
sampling distribution of the test statistic when
H0 is true
b
a /2
a /2
153
The p-value approach to Hypothesis Testing
154
In hypothesis testing we need

A test statistic
A Critical and Acceptance region for the test
statistic

The Critical Region is set up under the sampling
distribution of the test statistic. Area a
(0.05 or 0.01) above the critical region. The
critical region may be one tailed or two tailed
155

The Critical region

a/2
a/2
Accept H0
156
In test is carried out by

Computing the value of the test statistic
Making the decision
Reject if the value is in the Critical region and
Accept if the value is in the Acceptance region.

157
The value of the test statistic may be in the
Acceptance region but close to being in the
Critical region, or The it may be in the Critical
region but close to being in the Acceptance
region.
To measure this we compute the p-value.
158
Definition Once the test statistic has been
computed form the data the p-value is defined to
be
p-value Pthe test statistic is as or more
extreme than the observed value of the test
statistic
more extreme means giving stronger evidence to
rejecting H0
159
Example Suppose we are using the z test for
the mean m of a normal population and a 0.05.
Z0.025 1.960
Thus the critical region is to reject H0 if Z lt
-1.960 or Z gt 1.960 . Suppose the z 2.3, then
we reject H0
p-value Pthe test statistic is as or more
extreme than the observed value of the test
statistic P z gt 2.3 Pz lt -2.3
0.0107 0.0107 0.0214
160
Graph
p - value
-2.3
2.3
161
If the value of z 1.2, then we accept H0
p-value Pthe test statistic is as or more
extreme than the observed value of the test
statistic P z gt 1.2 Pz lt -1.2
0.1151 0.1151 0.2302
23.02 chance that the test statistic is as or
more extreme than 1.2. Fairly high, hence 1.2 is
not very extreme
162
Graph
p - value
1.2
-1.2
163
Properties of the p -value

If the p-value is small (lt0.05 or 0.01) H0 should
be rejected.
The p-value measures the plausibility of H0.
If the test is two tailed the p-value should be
two tailed.
If the test is one tailed the p-value should be
one tailed.
It is customary to report p-values when reporting
the results. This gives the reader some idea of
the strength of the evidence for rejecting H0

164
Summary

A common way to report statistical tests is to
compute the p-value.
If the p-value is small ( lt 0.05 or lt 0.01) then
H0 is rejected.
If the p-value is extremely small this gives a
strong indication that HA is true.
If the p-value is marginally above the threshold
0.05 then we cannot reject H0 but there would be
a suspicion that H0 is false.

165
Next topic Students t - test

Write a Comment

User Comments (0)