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Statistics for
Business and Economics
8th Edition
Chapter 9
Hypothesis Testing:
Single Population
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-1
9.1

What is a Hypothesis?
A hypothesis is a claim
(assumption) about a
population parameter:

population mean
Example: The mean monthly cell phone bill
of this city is μ = $42

population proportion
Example: The proportion of adults in this
city with cell phones is p = .68
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-2
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 ( H : μ  3 )
0
A hypothesis about a parameter that will be maintained unless
there is strong evidence against the null hypothesis.

Is always about a population parameter, not about a
sample statistic
H0 : μ  3
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
H0 : X  3
Ch. 9-3
The Null Hypothesis, H0
(continued)




Begin with the assumption that the null
hypothesis is true
 Similar to the notion of innocent until
proven guilty
Always contains “=” , “≤” or “” sign
May or may not be rejected
A null hypothesis is a claim (or statement)
about a population parameter that is assumed to
be true until it is declared false.
Ch. 9-4
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 )
Never contains the “=” , “≤” or “” sign
May or may not be supported
If we reject the null hypothesis, then the second
hypothesis, named the “alternative hypothesis” will be
accepted.
Is generally the hypothesis that the researcher is trying to
support
An alternative hypothesis is a claim about a population
parameter that will be true if the null hypothesis is false.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-5
Hypothesis Testing Process
Claim: the
population
mean age is 50.
(Null Hypothesis:
H0: μ = 50 )
Population
Is X 20 likely if μ = 50?
If not likely,
REJECT
Null Hypothesis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Suppose
the sample
mean age
is 20: X = 20
Now select a
random sample
Sample
Ch. 9-6
Rejection and Nonrejection Regions
Figure 9.1 Nonrejection and rejection regions for
the court case.
Four Possible Outcomes for a Test of
Hypothesis (Two Types of Errors)
Actual Situation
Court's Decision
The person is not
guilty
The person is guilty
The person is not
guilty
Correct Decision
Type 2 error
The person is guilty
Type 1 error
Correct Decision
Table: Four possible outcomes for a court case.
Two Types of Errors
Definition
A Type I error occurs when a true null hypothesis is
rejected. The value of α represents the probability of
committing this type of error; that is,
α = P(H0 is rejected | H0 is true)
The value of α represents the significance level of the
test.
A Type II error occurs when a false null hypotheses is not
rejected. The value of β represents the probability of
committing a Type II error; that is,
β = P (H0 is not rejected | H0 is false)
The value of 1 – β is called the power of the test. It
represents the probability of not making a Type II error.
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-10
Level of Significance
and the Rejection Region
Level of significance =
H0: μ = 3
H1: μ ≠ 3

/2
/2
H0: μ ≤ 3
H1: μ > 3

Upper-tail test
H0: μ ≥ 3
H1: μ < 3
Rejection
region is
shaded
0
Two-tail test
0

Lower-tail test
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
0
Represents
critical value
Definition
A two-tailed test
has rejection regions
in both tails, a lefttailed test has the
rejection region in
the left tail, and a
right-tailed test has
the rejection region
in the right tail of the
distribution curve.
Ch. 9-11
Signs in H0 and H1 and Tails of a Test
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Hypothesis Tests for the Mean
Hypothesis
Tests for 
 Known
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 Unknown
Ch. 9-13
Test of Hypothesis
for the Mean (σ Known)
9.2

Convert sample result ( x ) to a z value
Hypothesis
Tests for 
σ Known
σ Unknown
Consider the test
H0 : μ  μ0
The decision rule is:
x  μ0
Reject H0 if z 
 zα
σ
(Assume the population is normal)
n
H1 : μ  μ 0
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-14
Decision Rule
x  μ0
Reject H0 if z 
 zα
σ
n
H0: μ = μ0
H1: μ > μ0
Alternate rule:

Reject H 0 if x  μ 0  Z α σ/ n
Z
x
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Do not reject H0
0
μ0
zα
μ0  z α
Reject H0
σ
n
Critical value x c
Ch. 9-15
Example 1: 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)
Form hypothesis test:
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)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-16
Example: Find Rejection Region
(continued)

Suppose that  = .10 is chosen for this test
Find the rejection region:
Reject H0
 = .10
Do not reject H0
0
1.28
Reject H0
x  μ0
Reject H 0 if z 
 1.28
σ/ n
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-17
Example: Sample Results
(continued)
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,
x  μ0
53.1  52
z 

 0.88
σ
10
n
64
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-18
Example: Decision
(continued)
Reach a decision and interpret the result:
Reject H0
 = .10
Do not reject H0
1.28
0
z = 0.88
Reject H0
Do not reject H0 since z = 0.88 < 1.28
(So, Accept H0)
i.e.: there is not sufficient evidence that the
mean bill is over $52,
OR, the average bill is not over $52 per month
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-19
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 an upper-tail test since the
alternative hypothesis is focused on
the upper tail above the mean of 3
This is a lower-tail test since the
alternative hypothesis is focused on
the lower tail below the mean of 3
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-20
Upper-Tail Tests

There is only one
critical value, since
the rejection area is
in only one tail
H0: μ ≤ 3
H1: μ > 3

Do not reject H0
Z
0
x
μ
zα
Reject H0
Critical value x c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-21
Lower-Tail Tests
H0: μ ≥ 3

There is only one
critical value, since
the rejection area is
in only one tail
H1: μ < 3

Reject H0
-z
Do not reject H0
0
Z
μ
x
Critical value x c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 9-22
Two-Tail Tests


In some settings, the
alternative hypothesis does
not specify a unique direction
There are two
critical values,
defining the two
regions of
rejection
H0: μ = 3
H1: μ  3
/2
/2
x
3
Reject H0
Do not reject H0
-z/2
Lower
critical value
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
0
Reject H0
+z/2
z
Upper
critical value
Ch. 9-23
Example 9.1 (B.p-336)
Evaluating a new production process
(hypothesis test: Upper tail Z test)



The production manager of Northern Windows Inc. has
asked you to evaluate a proposed new procedure for
producing its Regal line of double-hung windows. The
present process has a mean production of 80 units per
hour with a population standard deviation of σ = 8. the
manager indicates that she does not want to change a
new procedure unless there is strong evidence that the
mean production level is higher with the new process.
Assume n=25 and α=0.05. Also assume that the sample
mean was 83.
What decision would you recommend based on
hypothesis testing?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-24
Example 9.2



Lower Tail Test
The production manager of Twin Forks ball bearing has
asked your assistance in evaluating a modified ball
bearing production process. When the process is
operating properly the process produces ball bearings
whose weights are normally distributed with a
population mean of 5 ounces and a population standard
deviation of 0.1 ounce. The sample mean was 4.962
and n= 16.
A new raw material supplier was used for a recent
production run, and the manager wants to know if that
change has resulted in lowering of the mean weight of
bearings.
What will be your conclusion for a lower tail test?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-25
Example 9.3 Two


tailed test
The production manager of Circuits Unlimited
has asked for your assistance in analyzing a
production process. The process involves
drilling holes whose diameters are normally
distributed with population mean 2 inches and
population standard deviation 0.06 inch. A
random sample of nine measurements had a
sample mean of 1.95 inches.
Use a significance level of 0.05 to determine if
the observed sample mean is unusual and
suggests that the machine should be adjusted.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-26
t Test of Hypothesis for the Mean
(σ Unknown)

Convert sample result ( x ) to a t test statistic
Hypothesis
Tests for 
σ Known
σ Unknown
Consider the test
H0 : μ  μ0
The decision rule is:
x  μ0
Reject H0 if t 
 t n-1, α
H1 : μ  μ0
s
n
(Assume the population is normal)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-27
t Test of Hypothesis for the Mean
(σ Unknown)
(continued)

For a two-tailed test:
Consider the test
H0 : μ  μ0
H1 : μ  μ0
(Assume the population is normal,
and the population variance is
unknown)
The decision rule is:
Reject H0 if t 
x  μ0
x  μ0
 t n-1, α/2 or if t 
 t n-1, α/2
s
s
n
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-28
Example 2: Two-Tail Test
( Unknown)
The average cost of a
hotel room in New York
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.
H0: μ = 168
H1: μ  168
(Assume the population distribution is normal)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-29
Example Solution:
Two-Tail Test
H0: μ = 168
H1: μ  168
  = 0.05
/2=.025
Reject H0
-t n-1,α/2
-2.0639
 n = 25
  is unknown, so
use a t statistic
t n1 
 Critical Value:
t24 , .025 = ± 2.0639
/2=.025
Do not reject H0
Reject H0
0
1.46
t n-1,α/2
2.0639
x μ
172.50  168

 1.46
s
15.40
n
25
Do not reject H0: not sufficient evidence that
true mean cost is different than $168
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-30
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-31
Proportions
(continued)

Sample proportion in the success category is
denoted by p̂


ˆp  x  number of successes in sample
n
sample size
When nP(1 – P) > 9, p̂ can be approximated
by a normal distribution with mean and
standard deviation

P(1 P)
μp̂  P
σ p̂ 
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-32
Hypothesis Tests for Proportions

The sampling
distribution of p̂ is
Hypothesis
approximately
Tests for P
normal, so the test
statistic is a z
nP(1 – P) < 9
nP(1 – P) > 9
value:
z
pˆ  P0
P0 (1 P0 )
n
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Not discussed
in this chapter
Chap 10-33
Example 3: 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
p̂ = 25/500 = .05
nP(1 - P) = (500)(.05)(.95)
= 23.75 > 9
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chap 10-34
Z Test for Proportion: Solution
Test Statistic:
H0: P = .08
H1: P  .08
 = .05
n = 500,
p̂
pˆ  P0
.05  .08
z

 2.47
P0 (1 P0 )
.08(1  .08)
500
n
= .05
Decision:
Critical Values: ±1.96
Reject
Reject
Reject H0 at  = .05
Conclusion:
.025
.025
-1.96
0
1.96
z
-2.47
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
There is sufficient
evidence to reject the
company’s claim of 8%
response rate.
Chap 10-35
Two Procedures
Two procedures to make tests of hypothesis
1. The p-value approach
2. The critical-value approach
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
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
Decision rule: compare the p-value to 

If p-value <  , reject H0

If p-value   , do not reject H0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 10-37
HYPOTHESIS TESTS ABOUT μ: σ
KNOWN
Definition
Assuming that the null hypothesis is true, the p-value can
be defined as the probability that a sample statistic (such
as the sample mean) is at least as far away from the
hypothesized value in the direction of the alternative
hypothesis as the one obtained from the sample data
under consideration.
Note that the p–value is the smallest significance level at
which the null hypothesis is rejected.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Figure 9.5 The p–value for a right-tailed
test.
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Copyright © 2010 John Wiley & Sons. All right reserved
Figure 9.6 The p–value for a two-tailed
test.
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Copyright © 2010 John Wiley & Sons. All right reserved
Calculating the z Value for x
When using the normal distribution, the value of z
for x for a test of hypothesis about μ is computed
as follows:
z
x 
x

where  x 
n
The value of z calculated for x using this formula is
also called the observed value of z.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Steps to Perform a Test of Hypothesis
Using the p–Value Approach
1.
2.
3.
4.
State the null and alternative hypothesis.
Select the distribution to use.
Calculate the p–value.
Make a decision.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Example 9-1 (p–Value Approach)
At Canon Food Corporation, it used to take an average of 90 minutes
for new workers to learn a food processing job. Recently the
company installed a new food processing machine. The supervisor at
the company wants to find if the mean time taken by new workers to
learn the food processing procedure on this new machine is different
from 90 minutes.
A sample of 20 workers showed that it took, on average, 85 minutes
for them to learn the food processing procedure on the new machine.
It is known that the learning times for all new workers are normally
distributed with a population standard deviation of 7 minutes.
Find the p–value for the test that the mean learning time for the food
processing procedure on the new machine is different from 90
minutes. What will your conclusion be if α = .01?
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Example 9-1: Solution
H1: μ ≠ 90
 Step 2: The population standard deviation σ is
known, the sample size is small (n < 30), but the
population distribution is normal. We will use the
normal distribution to find the p–value and make
the test.
 Step 1: H0: μ = 90
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Example 9-1: Solution
 Step 3:

7
x 

 1.56524758 min utes
n
20
x 
85  90
z

 3.19
x
1.56524758
see 3.19 from z table, we find .9993 so 1-.9993= 0.0007
p-value = 2(.0007) = .0014
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Figure 9: The p-value for a two-tailed
test.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Example 9-1: Solution
 Step 4: Because α = .01 is greater than the p-
value of .0014, we reject the null hypothesis at
this significance level.
Therefore, we conclude that the mean time for
learning the food processing procedure on the
new machine is different from 90 minutes.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved