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TRANSCRIPT
Chapter 9 9-1
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistics for Economics
Department of Economics
Dr. Mohammad ZainalECON 506
Chapter 9Hypothesis Testing:Single Population
Chapter Goals
After completing this chapter, you should be able to:
Formulate null and alternative hypotheses for applications involving a single population mean from a normal distribution a single population proportion (large samples) the variance of a normal distribution
Formulate a decision rule for testing a hypothesis
Know how to use the critical value and p-value approaches to test the null hypothesis (for both mean and proportion problems)
Know what Type I and Type II errors are
Assess the power of a testEcon 506, by Dr. M. Zainal Ch. 9-2
Chapter 9 9-2
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
What is a Hypothesis?
We perform of hypothesis only when we are making a decision about a population parameter based the value of a sample statistic.
Consider an example of a person who has been indicted for committing a crime and is being tried in a court.
Based on the available evidence, the judge will make one of the following decisions: The person is not guilty (null hypothesis, Ho). The person is guilty (alternative hypothesis, H1).
Econ 506, by Dr. M. Zainal Ch. 9-3
What is a Hypothesis?
A null hypothesis is a claim about a population parameter that is assumed to be true until it is declared false.
An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.
The alternative hypothesis is often what the test is attempting to establish.
Econ 506, by Dr. M. Zainal Ch. 9-4
Chapter 9 9-3
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
What is a Hypothesis?
The research hypothesis should be expressed as the alternative hypothesis
Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis.
Econ 506, by Dr. M. Zainal Ch. 9-5
What is a Hypothesis?
In the court example, we start the prosecution by assuming the person is not guilty.
The prosecutors collect all possible evidences to prove the null hypothesis is false (guilty).
Rejection and nonrejection regions
Not enough evidence to declare theperson guilty and, hence, the nullhypothesis is not rejected in this region
Enough evidence to declare the personguilty and, hence, the null hypothesis isrejected in this region
Nonrejection region Rejection region
C
Critical point
Level of evidence0
Econ 506, by Dr. M. Zainal Ch. 9-6
Chapter 9 9-4
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
What is a Hypothesis?
Actual Situation
The person is not guilty
The person is guilty
Court’s decision
The person is not guilty
Correct decisionType II or
error
The person is guiltyType I or
errorCorrect decision
Econ 506, by Dr. M. Zainal Ch. 9-7
What is a Hypothesis?
A hypothesis is a claim (assumption) about a population parameter:
population mean
population proportion
Example: The mean monthly cell phone bill of this city is μ = $42
Example: The proportion of adults in this city with cell phones is p = .68
9.1
Econ 506, by Dr. M. Zainal Ch. 9-8
Chapter 9 9-5
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
The Null Hypothesis, H0
States the assumption (numerical) to be tested
Example: The average number of TV sets in
U.S. Homes is equal to three ( )
Is always about a population parameter, not about a sample statistic
3μ:H0
3μ:H0 3X:H0
Econ 506, by Dr. M. Zainal Ch. 9-9
The Null Hypothesis, H0
Begin with the assumption that the null hypothesis is true
Similar to the notion of innocent untilproven guilty
Refers to the status quo
Always contains “=” , “≤” or “” sign
May or may not be rejected
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-10
Chapter 9 9-6
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis e.g., The average number of TV sets in U.S.
homes is not equal to 3 ( H1: μ ≠ 3 )
Challenges the status quo
Never contains the “=” , “≤” or “” sign
May or may not be supported
Is generally the hypothesis that the researcher is trying to support
Econ 506, by Dr. M. Zainal Ch. 9-11
Population
Claim: thepopulationmean age is 50.(Null Hypothesis:
REJECT
Supposethe samplemean age is 20: X = 20
SampleNull Hypothesis
20 likely if μ = 50?Is
Hypothesis Testing Process
If not likely,
Now select a random sample
H0: μ = 50 )
X
Econ 506, by Dr. M. Zainal Ch. 9-12
Chapter 9 9-7
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Reason for Rejecting H0
Sampling Distribution of X
μ = 50If H0 is true
If it is unlikely that we would get a sample mean of this value ...
... then we reject the null
hypothesis that μ = 50.
20
... if in fact this werethe population mean…
X
Econ 506, by Dr. M. Zainal Ch. 9-13
Level of Significance,
Defines the unlikely values of the sample statistic if the null hypothesis is true
Defines rejection region of the sampling distribution
Is designated by , (level of significance)
Typical values are .01, .05, or .10
Is selected by the researcher at the beginning
Provides the critical value(s) of the test
Econ 506, by Dr. M. Zainal Ch. 9-14
Chapter 9 9-8
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Level of Significance and the Rejection Region
H0: μ ≥ 3 H1: μ < 3
0
H0: μ ≤ 3 H1: μ > 3
Representscritical value
Lower-tail test
Level of significance =
0Upper-tail test
Two-tail test
Rejection region is shaded
/2
0
/2H0: μ = 3 H1: μ ≠ 3
Econ 506, by Dr. M. Zainal Ch. 9-15
Errors in Making Decisions
Type I Error
Reject a true null hypothesis
Considered a serious type of error
The probability of Type I Error is Called level of significance of the test
Set by researcher in advance
Econ 506, by Dr. M. Zainal Ch. 9-16
Chapter 9 9-9
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Errors in Making Decisions
Type II Error
Fail to reject a false null hypothesis
The probability of Type II Error is β
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-17
Outcomes and Probabilities
Actual Situation
Decision
Do NotReject
H0
No Error(1 - )
Type II Error( β )
RejectH0
Type I Error( )
Possible Hypothesis Test Outcomes
H0 FalseH0 True
Key:Outcome
(Probability) No Error( 1 - β )
Econ 506, by Dr. M. Zainal Ch. 9-18
Chapter 9 9-10
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Type I & II Error Relationship
Type I and Type II errors can not happen atthe same time
Type I error can only occur if H0 is true
Type II error can only occur if H0 is false
If Type I error probability ( ) , then
Type II error probability ( β )
Econ 506, by Dr. M. Zainal Ch. 9-19
Factors Affecting Type II Error
All else equal, β when the difference between
hypothesized parameter and its true value
β when
β when σ
β when n
Econ 506, by Dr. M. Zainal Ch. 9-20
Chapter 9 9-11
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Formulating Hypotheses
Left-Tailed
Test
Right-Tailed
Test
Two-Tailed
Test
Sign in the null hypothesis Ho
= or = or =
Sign in the alternative hypothesis HA
< >
Critical region In the left tail In the right tail In both tails
Econ 506, by Dr. M. Zainal Ch. 9-21
Formulating Hypotheses
Example 1: A manufacturing company has worked to reduce the noise of one of their tools. They would like to show in its advertising that the tool is quieter. The average of the prior design was 150 decibels.
What is the appropriate hypothesis test?
Econ 506, by Dr. M. Zainal Ch. 9-22
Chapter 9 9-12
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Formulating Hypotheses
What is the appropriate test?
H0: µ ≥ 150 (the tool is not quieter) status quoHA: µ < 150 (the tool is quieter) wants to support
If the null hypothesis is rejected, the company has sufficient evidence to support that the tool is now quieter.
Econ 506, by Dr. M. Zainal Ch. 9-23
Formulating Hypotheses
Example 2: The average annual income of buyers of Ford pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim.
What is the appropriate hypothesis test?
Econ 506, by Dr. M. Zainal Ch. 9-24
Chapter 9 9-13
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Formulating Hypotheses
Example 1: The average annual income of buyers of Ford F150 pickup trucks is claimed to be $65,000 per year. An industry analyst would like to test this claim.
What is the appropriate test?
H0: µ = 65,000 (income is as claimed) status quoHA: µ ≠ 65,000 (income is different than claimed)
The analyst will believe the claim unless sufficient evidence is found to discredit it.
Econ 506, by Dr. M. Zainal Ch. 9-25
Power of the Test
The power of a test is the probability of rejecting a null hypothesis that is false
i.e., Power = P(Reject H0 | H1 is true)
Power of the test increases as the sample size increases
Econ 506, by Dr. M. Zainal Ch. 9-26
Chapter 9 9-14
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Tests for the Mean
Known Unknown
Hypothesis Tests for
Econ 506, by Dr. M. Zainal Ch. 9-27
Test of Hypothesisfor the Mean (σ Known)
Convert sample result ( ) to a z value
The decision rule is:
α0
0 z
n
σμx
z if H Reject
σ Known σ Unknown
Hypothesis Tests for
Consider the test
00 μμ:H
01 μμ:H
(Assume the population is normal)
x
9.2
Econ 506, by Dr. M. Zainal Ch. 9-28
Chapter 9 9-15
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Decision Rule
Reject H0Do not reject H0
zα0
μ0
H0: μ = μ0
H1: μ > μ0
Critical value
Z
α0
0 z
n
σμx
z if H Reject
nσ/Zμx if H Reject α00
n
σzμ α0
Alternate rule:
x
cxEcon 506, by Dr. M. Zainal Ch. 9-29
p-Value Approach to Testing
p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H0 is true
Also called observed level of significance
Smallest value of for which H0 can be rejected
Econ 506, by Dr. M. Zainal Ch. 9-30
Chapter 9 9-16
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
p-Value Approach to Testing
Convert sample result (e.g., ) to test statistic (e.g., z statistic )
Obtain the p-value For an upper
tail test:
Decision rule: compare the p-value to
If p-value < , reject H0
If p-value , do not reject H0
(continued)
x
)μμ | nσ/
μ-x P(z
true) is H that given , nσ/
μ-x P(z value-p
00
00
Econ 506, by Dr. M. Zainal Ch. 9-31
Example: Upper-Tail Z Test for Mean ( Known)
A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume = 10 is known)
H0: μ ≤ 52 the average is not over $52 per month
H1: μ > 52 the average is greater than $52 per month(i.e., sufficient evidence exists to support the manager’s claim)
Form hypothesis test:
Econ 506, by Dr. M. Zainal Ch. 9-32
Chapter 9 9-17
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Suppose that = .10 is chosen for this test
Find the rejection region:
Reject H0Do not reject H0
= .10
1.280
Reject H0
Example: Find Rejection Region(continued)
1.28nσ/
μxz if H Reject 0
0
Econ 506, by Dr. M. Zainal Ch. 9-33
Example: Sample Results
Obtain sample and compute the test statistic
Suppose a sample is taken with the following results: n = 64, x = 53.1 ( = 10 was assumed known)
Using the sample results,
0.88
64
105253.1
n
σμx
z 0
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-34
Chapter 9 9-18
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example: Decision
Reach a decision and interpret the result:
Reject H0Do not reject H0
= .10
1.280
Reject H0
Do not reject H0 since z = 0.88 < 1.28
i.e.: there is not sufficient evidence that themean bill is over $52
z = 0.88
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-35
Example: p-Value Solution
Calculate the p-value and compare to (assuming that μ = 52.0)
Reject H0
= .10
Do not reject H0 1.28
0
Reject H0
Z = .88
(continued)
.1894
.810610.88)P(z
6410/
52.053.1zP
52.0) μ | 53.1xP(
p-value = .1894
Do not reject H0 since p-value = .1894 > = .10
Econ 506, by Dr. M. Zainal Ch. 9-36
Chapter 9 9-19
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
One-Tail Tests
In many cases, the alternative hypothesis focuses on one particular direction
H0: μ ≥ 3
H1: μ < 3
H0: μ ≤ 3
H1: μ > 3
This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3
This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3
Econ 506, by Dr. M. Zainal Ch. 9-37
Upper-Tail Tests
Reject H0Do not reject H0
zα0
μ
H0: μ ≤ 3
H1: μ > 3 There is only one
critical value, since the rejection area is in only one tail
Critical value
Z
x
cx
Econ 506, by Dr. M. Zainal Ch. 9-38
Chapter 9 9-20
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Lower-Tail Tests
Reject H0 Do not reject H0
There is only one critical value, since the rejection area is in only one tail
-z 0
μ
H0: μ ≥ 3
H1: μ < 3
Z
Critical value
x
cx
Econ 506, by Dr. M. Zainal Ch. 9-39
Two-Tail Tests
In some settings, the alternative hypothesis does not specify a unique direction
Do not reject H0 Reject H0Reject H0
There are two critical values, defining the two regions of rejection
/2
0
H0: μ = 3 H1: μ 3
/2
Lower critical value
Uppercritical value
3
z
x
-z/2 +z/2
Econ 506, by Dr. M. Zainal Ch. 9-40
Chapter 9 9-21
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Testing Example
Test the claim that the true mean # of TV sets in US homes is equal to 3.
(Assume σ = 0.8)
State the appropriate null and alternativehypotheses
H0: μ = 3 , H1: μ ≠ 3 (This is a two tailed test)
Specify the desired level of significance
Suppose that = .05 is chosen for this test
Choose a sample size
Suppose a sample of size n = 100 is selected
Econ 506, by Dr. M. Zainal Ch. 9-41
Hypothesis Testing Example
2.0.08
.16
100
0.832.84
n
σμX
z 0
Determine the appropriate technique σ is known so this is a z test
Set up the critical values
For = .05 the critical z values are ±1.96
Collect the data and compute the test statistic
Suppose the sample results are
n = 100, x = 2.84 (σ = 0.8 is assumed known)
So the test statistic is:
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-42
Chapter 9 9-22
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Testing Example
Is the test statistic in the rejection region?
Reject H0 Do not reject H0
= .05/2
-z = -1.96 0
Reject H0 if z < -1.96 or z > 1.96; otherwise do not reject H0
(continued)
= .05/2
Reject H0
+z = +1.96
Here, z = -2.0 < -1.96, so the test statistic is in the rejection region
Econ 506, by Dr. M. Zainal Ch. 9-43
Hypothesis Testing Example
Reach a decision and interpret the result
-2.0
Since z = -2.0 < -1.96, we reject the null hypothesisand conclude that there is sufficient evidence that the mean number of TVs in US homes is not equal to 3
(continued)
Reject H0 Do not reject H0
= .05/2
-z = -1.96 0
= .05/2
Reject H0
+z = +1.96
Econ 506, by Dr. M. Zainal Ch. 9-44
Chapter 9 9-23
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example: p-Value
Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is = 3.0?
.0228
/2 = .025
-1.96 0
-2.0
.02282.0)P(z
.02282.0)P(z
Z1.96
2.0
x = 2.84 is translated to a z score of z = -2.0
p-value
= .0228 + .0228 = .0456
.0228
/2 = .025
Econ 506, by Dr. M. Zainal Ch. 9-45
Example: p-Value
Compare the p-value with
If p-value < , reject H0
If p-value , do not reject H0
Here: p-value = .0456 = .05
Since .0456 < .05, we reject the null hypothesis
(continued)
.0228
/2 = .025
-1.96 0
-2.0
Z1.96
2.0
.0228
/2 = .025
Econ 506, by Dr. M. Zainal Ch. 9-46
Chapter 9 9-24
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
t Test of Hypothesis for the Mean (σ Unknown)
Convert sample result ( ) to a t test statistic
σ Known σ Unknown
Hypothesis Tests for
x
The decision rule is:
α , 1-n0
0 t
n
sμx
t if H Reject
Consider the test
00 μμ:H
01 μμ:H
(Assume the population is normal)
9.3
Econ 506, by Dr. M. Zainal Ch. 9-47
t Test of Hypothesis for the Mean (σ Unknown)
For a two-tailed test:
The decision rule is:
α/2 , 1-n0
α/2 , 1-n0
0 t
n
sμx
t if or t
n
sμx
t if H Reject
Consider the test
00 μμ:H
01 μμ:H
(Assume the population is normal, and the population variance is unknown)
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-48
Chapter 9 9-25
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example: Two-Tail Test( Unknown)
The average cost of a hotel room in Chicago is said to be $168 per night. A random sample of 25 hotels resulted in x = $172.50 and
s = $15.40. Test at the
= 0.05 level.(Assume the population distribution is normal)
H0: μ = 168 H1: μ 168
Econ 506, by Dr. M. Zainal Ch. 9-49
Example Solution: Two-Tail Test
= 0.05
n = 25
is unknown, so use a t statistic
Critical Value:
t24 , .025 = ± 2.0639 Do not reject H0: not sufficient evidence that true mean cost is different than $168
Reject H0Reject H0
/2=.025
-t n-1,α/2
Do not reject H0
0
/2=.025
-2.0639 2.0639
1.46
25
15.40168172.50
n
sμx
t 1n
1.46
H0: μ = 168 H1: μ 168
t n-1,α/2
Econ 506, by Dr. M. Zainal Ch. 9-50
Chapter 9 9-26
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Tests of the Population Proportion
Involves categorical variables
Two possible outcomes
“Success” (a certain characteristic is present)
“Failure” (the characteristic is not present)
Fraction or proportion of the population in the “success” category is denoted by P
Assume sample size is large
9.4
Econ 506, by Dr. M. Zainal Ch. 9-51
Proportions
Sample proportion in the success category is denoted by
When nP(1 – P) > 5, can be approximated by a normal distribution with mean and standard deviation
sizesample
sampleinsuccessesofnumber
n
xp ˆ
Pμ pn
P)P(1σ
p
(continued)
p
p
Econ 506, by Dr. M. Zainal Ch. 9-52
Chapter 9 9-27
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Tests for Proportions
The sampling distribution of is approximately normal, so the test statistic is a z value:
n)P(1P
Ppz
00
0
ˆ
nP(1 – P) > 5
Hypothesis Tests for P
Not discussed in this chapter
p
nP(1 – P) < 5
Econ 506, by Dr. M. Zainal Ch. 9-53
Example: Z Test for Proportion
A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the = .05 significance level.
Check: Our approximation for P is
= 25/500 = .05
nP(1 - P) = (500)(.05)(.95)= 23.75 > 5
p
Econ 506, by Dr. M. Zainal Ch. 9-54
Chapter 9 9-28
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Z Test for Proportion: Solution
= .05
n = 500, = .05
Reject H0 at = .05
H0: P = .08 H1: P .08
Critical Values: ± 1.96
Test Statistic:
Decision:
Conclusion:
z0
Reject Reject
.025.025
1.96
-2.47
There is sufficient evidence to reject the company’s claim of 8% response rate.
2.47
500.08).08(1
.08.05
n)P(1P
Ppz
00
0
ˆ
-1.96
p
Econ 506, by Dr. M. Zainal Ch. 9-55
p-Value Solution
Calculate the p-value and compare to (For a two sided test the p-value is always two sided)
Do not reject H0Reject H0Reject H0
/2 = .025
1.960
Z = -2.47
(continued)
0.01362(.0068)
2.47)P(Z2.47)P(Z
p-value = .0136:
Reject H0 since p-value = .0136 < = .05
Z = 2.47
-1.96
/2 = .025
.0068.0068
Econ 506, by Dr. M. Zainal Ch. 9-56
Chapter 9 9-29
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Power of the Test
Recall the possible hypothesis test outcomes:Actual Situation
Decision
Do Not Reject H0
No error(1 - )
Type II Error( β )
Reject H0Type I Error
( )
H0 FalseH0 TrueKey:
Outcome(Probability)
No Error( 1 - β )
β denotes the probability of Type II Error
1 – β is defined as the power of the test
Power = 1 – β = the probability that a false null hypothesis is rejected
9.5
Econ 506, by Dr. M. Zainal Ch. 9-57
Type II Error
or
The decision rule is:
α0
0 zn/σ
μxz if H Reject
00 μμ:H
01 μμ:H
Assume the population is normal and the population variance is known. Consider the test
nσ/Zμxx if H Reject α0c0
If the null hypothesis is false and the true mean is μ*, then the probability of type II error is
n/σ
*μxzPμ*)μ|xxP(β c
c
Econ 506, by Dr. M. Zainal Ch. 9-58
Chapter 9 9-30
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Type II Error Example
Type II error is the probability of failing to reject a false H0
Reject H0: μ 52
Do not reject H0 : μ 52
5250
Suppose we fail to reject H0: μ 52when in fact the true mean is μ* = 50
cx
Econ 506, by Dr. M. Zainal Ch. 9-59
Type II Error Example
Suppose we do not reject H0: μ 52 when in fact the true mean is μ* = 50
Reject H0: μ 52
Do not reject H0 : μ 52
5250
This is the true distribution of x if μ = 50
This is the range of x where H0 is not rejected
(continued)
cxEcon 506, by Dr. M. Zainal Ch. 9-60
Chapter 9 9-31
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Type II Error Example
Suppose we do not reject H0: μ 52 when in fact the true mean is μ* = 50
Reject H0: μ 52
Do not reject H0 : μ 52
5250
β
Here, β = P( x ) if μ* = 50
(continued)
cx
cx
Econ 506, by Dr. M. Zainal Ch. 9-61
Calculating β
Suppose n = 64 , σ = 6 , and = .05
Reject H0: μ 52
Do not reject H0 : μ 52
5250
So β = P( x 50.766 ) if μ* = 50
50.76664
61.64552
n
σzμx α0c
(for H0 : μ 52)
50.766
cxEcon 506, by Dr. M. Zainal Ch. 9-62
Chapter 9 9-32
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Calculating β
Suppose n = 64 , σ = 6 , and = .05
Reject H0: μ 52
Do not reject H0 : μ 52
.1539.3461.51.02)P(z
646
5050.766zP50)μ*|50.766xP(
5250
(continued)
Probability of type II error:
β = .1539
cxEcon 506, by Dr. M. Zainal Ch. 9-63
Power of the Test Example
If the true mean is μ* = 50,
The probability of Type II Error = β = 0.1539
The power of the test = 1 – β = 1 – 0.1539 = 0.8461
Actual Situation
Decision
Do Not Reject H0
No error1 - = 0.95
Type II Errorβ = 0.1539
Reject H0Type I Error = 0.05
H0 FalseH0 TrueKey:
Outcome(Probability)
No Error1 - β = 0.8461
(The value of β and the power will be different for each μ*)
Econ 506, by Dr. M. Zainal Ch. 9-64
Chapter 9 9-33
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Tests of one Population Variance
If the population is normally distributed,
2
22
1n σ
1)s(n
has a chi-square distribution with (n – 1) degrees of freedom
Goal: Test hypotheses about thepopulation variance, σ2
9.6
Econ 506, by Dr. M. Zainal Ch. 9-65
Hypothesis Tests of one Population Variance
The test statistic for hypothesis tests about one population variance is
20
22
1n σ
1)s(nχ
(continued)
Econ 506, by Dr. M. Zainal Ch. 9-66
Chapter 9 9-34
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Decision Rules: Variance
Population variance
Lower-tail test:
H0: σ2 σ02
H1: σ2 < σ02
Upper-tail test:
H0: σ2 ≤ σ02
H1: σ2 > σ02
Two-tail test:
H0: σ2 = σ02
H1: σ2 ≠ σ02
/2 /2
Reject H0 if Reject H0 if Reject H0 if
or
2, 1n χ
2,1 1n χ
2,1 1n 2/χ
2, 1n 2/χ
2,1 1n
21n χχ
2, 1n
21n χχ
2, 1n
21n 2/ χχ
2,1 1n
21n 2/ χχ
Econ 506, by Dr. M. Zainal Ch. 9-67
Proportions
The sample proportion of successes is denoted by p :
When both nπ and n(1- π) are at least 5, pis approximately normally distributed with mean and standard deviation
(continued)
sizesample
sampleinsuccessesofnumber
n
xp
πμp n
π)π(1σp
Econ 506, by Dr. M. Zainal Ch. 9-68
Chapter 9 9-35
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Hypothesis Tests for Variances
Hypothesis Testsfor Variances
Tests for a SinglePopulation Variance
Tests for TwoPopulation Variances
Chi-Square test statistic F test statistic
Econ 506, by Dr. M. Zainal Ch. 9-69
Single Population
Hypothesis Tests for Variances
Tests for a SinglePopulation Variance
Chi-Square test statistic
H0: σ2 = σ02
HA: σ2 ≠ σ02
H0: σ2 σ02
HA: σ2 < σ02
H0: σ2 ≤ σ02
HA: σ2 > σ02
*Two tailed test
Lower tail test
Upper tail test
Econ 506, by Dr. M. Zainal Ch. 9-70
Chapter 9 9-36
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Chi-Square Test Statistic
Hypothesis Tests for Variances
Tests for a SinglePopulation Variance
Chi-Square test statistic *
The chi-squared test statistic for a Single Population Variance is:
2
22
σ
1)s(n
where
2 = standardized chi-square variable
n = sample size
s2 = sample variance
σ2 = hypothesized variance
Econ 506, by Dr. M. Zainal Ch. 9-71
The Chi-square Distribution
The chi-square distribution is a family of distributions, depending on degrees of freedom (Like the t distribution).
The chi-square distribution curve starts at the origin and lies entirely to the right of the vertical axis.
The chi-square distribution assumes nonnegative values only, and these are denoted by the symbol 2 (read as “chi-square”).
Econ 506, by Dr. M. Zainal Ch. 9-72
Chapter 9 9-37
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
The Chi-square Distribution
The shape of a specific chi-square distribution depends on the number of degrees of freedom.
peak of a 2 distribution curve with 1 or 2 degrees of freedom occurs at zero and for a curve with 3 or more degrees of freedom at (df−2).
0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28
d.f. = 1 d.f. = 5 d.f. = 15
2 22
Econ 506, by Dr. M. Zainal Ch. 9-73
Finding the Critical Value
The critical value, , is found from the
chi-square table
Do not reject H0 Reject H0
2
2
2
H0: σ2 ≤ σ02
HA: σ2 > σ02
Upper tail test:
Econ 506, by Dr. M. Zainal Ch. 9-74
Chapter 9 9-38
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Finding the Critical Value
Econ 506, by Dr. M. Zainal Ch. 9-75
Finding the Critical Value
Example: Find the value of 2 for 7 degrees of freedom and anarea of .10 in the right tail of the chi-square distribution curve.
Econ 506, by Dr. M. Zainal Ch. 9-76
Chapter 9 9-39
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Finding the Critical Value
Example: Find the value of 2 for 9 degrees of freedom and anarea of .05 in the left tail of the chi-square distribution curve.
Econ 506, by Dr. M. Zainal Ch. 9-77
Example
A commercial freezer must hold the selected temperature with little variation. Specifications call for a standard deviation of no more than 4 degrees (or variance of 16 degrees2). A sample of 16 freezers is tested and
yields a sample variance
of s2 = 24. Test to see
whether the standard
deviation specification
is exceeded. Use
= .05
Econ 506, by Dr. M. Zainal Ch. 9-78
Chapter 9 9-40
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Finding the Critical Value
Use the chi-square table to find the critical value:
Do not reject H0 Reject H0
= .05
2
2
2
= 24.9958
= 24.9958 ( = .05 and 16 – 1 = 15 d.f.)
22.516
1)24(16
σ
1)s(n2
22
The test statistic is:
Since 22.5 < 24.9958, do not reject H0
There is not significant evidence at the = .05 levelthat the standard deviation specification is exceeded
Econ 506, by Dr. M. Zainal Ch. 9-79
Lower Tail or Two Tailed Chi-square Tests
H0: σ2 = σ02
HA: σ2 ≠ σ02
H0: σ2 σ02
HA: σ2 < σ02
2/2
Do not reject H0Reject
21-
2
Do not reject H0
Reject
/2
21-/2
2
/2
Reject
Lower tail test: Two tail test:
(2U)(2
L)
Econ 506, by Dr. M. Zainal Ch. 9-80
Chapter 9 9-41
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Confidence Interval Estimatefor σ2
2/2
/2
21-/2
2
/2
(2U)(2
L)
The confidence interval estimate for σ2 is
2L
22
2U
2 1)s(nσ
1)s(n
χχ
Where 2L and 2
U are from the 2 distribution with n -1 degrees
of freedom
Econ 506, by Dr. M. Zainal Ch. 9-81
Example
A sample of 16 freezers yields a sample variance of s2 = 24.
Form a 95% confidence interval for the population variance.
Econ 506, by Dr. M. Zainal Ch. 9-82
Chapter 9 9-42
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Use the chi-square table to find 2L and 2
U :
6.2621
( = .05 and 16 – 1 = 15 d.f.)
Example(continued)
2.025
/2=.025
2.975
/2=.025
(2U)(2
L)27.4884 57.489σ13.096
6.2621
1)24(16σ
27.4884
1)24(16
1)s(nσ
1)s(n
2
2
2L
22
2U
2
χχ
We are 95% confident that the population variance is between 13.096 and 57.489 degrees2. (Taking the square root, we are 95% confident that the population standard deviation is between 3.619 and 7.582 degrees.)
Econ 506, by Dr. M. Zainal Ch. 9-83
Problems
Example: Write the null and alternative hypotheses for each of
the following examples. Determine if each is a case of a two-
tailed, a left-tailed, or a right-tailed test.
o To test if the mean number of hours spent working per week
by college students who hold jobs is different from 20 hours.
o To test whether or not a bank's ATM is out of service for an
average of more than 10 hours per month.
o To test if the mean length of experience of airport security
guards is different from three years.
o To test if the mean credit card debt of college seniors is less
than $1000.
Econ 506, by Dr. M. Zainal Ch. 9-84
Chapter 9 9-43
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Problems
Econ 506, by Dr. M. Zainal Ch. 9-85
Problems
The management of Priority Health Club claims that its
members lose an average of 10 pounds or more within the first
month after joining the club. A consumer agency that wants to
check this claim took a random sample of 36 members of this
health club and found that they lost an average of 9.2 pounds
within the first month of membership with a standard deviation
of 2.4 pounds.
o Find the p-value for this test.
o What will your decision be if = .01? What if = .05?
Econ 506, by Dr. M. Zainal Ch. 9-86
Chapter 9 9-44
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Problems
Data:
Hypothesis:
Test statistic:
(a) P-value =
(b)
- Since the value of is smaller than the p-value, then we decide not to reject at %1 level of significance.
- Since the value of is greater than the p-value, then we decide to reject at %5 level of significance.
36, 9.2, 2.4n x s
0 1: 10 . : 10H vs H
0 9.2 102
2.4 / 6/c
xZ
s n
( 2) 0.02275P Z
0.01
0.05
Econ 506, by Dr. M. Zainal Ch. 9-87
Problems
Canon Inc. has introduced a copying machine that featurestwo-color copying capability in a compact system copier. Theaverage speed of the standard compact system copier is 27copies per minute. Suppose the company wants to testwhether the new copier has the same average speed as itsstandard compact copier. Conduct a test at an = 0.05 levelof significance.
Ho : = 27
HA : 27
n = 24
Econ 506, by Dr. M. Zainal Ch. 9-88
Chapter 9 9-45
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Problems
For = 0.05 and df = 24 - 1 = 23, the critical values of t are ± 2.069
The rule is:
Do not reject Ho if: [-2.069 t 2.069]
The test statistic is:
Lower Rejection Region
Upper Rejection Region
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
.025 .025
.95
Nonrejection Region
t 2.069-2.069
The t Distribution
7.4
24.6 277.424
⇒
2.41.51
1.59
Econ 506, by Dr. M. Zainal Ch. 9-89
Problems
Direct Mailing Company sells computers and computer parts by
mail. The company claims that at least 90% of all orders are
mailed within 72 hours after they are received. The quality
control department at the company often takes samples to
check if this claim is valid. A recently taken sample of 150
orders showed that 129 of them were mailed within 72 hours.
Do you think the company's claim is true? Use a 2.5%
significance level.
Econ 506, by Dr. M. Zainal Ch. 9-90
Chapter 9 9-46
Statistics for Business and Economics, 7/e Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Problems
Econ 506, by Dr. M. Zainal Ch. 9-91
Chapter Summary
Addressed hypothesis testing methodology
Performed Z Test for the mean (σ known)
Discussed critical value and p-value approaches to hypothesis testing
Performed one-tail and two-tail tests
Performed t test for the mean (σ unknown)
Performed Z test for the proportion
Discussed type II error and power of the test
Performed a hypothesis test for the variance (χ2)
Econ 506, by Dr. M. Zainal Ch. 9-92