### Chapter 9

```Statistics for Managers
Using Microsoft® Excel
5th Edition
Chapter 9
Fundamentals of Hypothesis Testing:
One Sample Tests
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-1
Learning Objectives
In this chapter, you will learn:
 The basic principles of hypothesis testing
 How to use hypothesis testing to test a mean or
proportion
 The assumptions of each hypothesis-testing
procedure, how to evaluate them, and the
consequences if they are seriously violated
 How to avoid the pitfalls involved in hypothesis
testing
 Ethical issues involved in hypothesis testing
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-2
The 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 = \$52
 population proportion
Example: The proportion of adults in this city with
cell phones is π = .68
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-3
The Null Hypothesis, H0
 States the assumption (numerical) to be tested
 Always contains “=” , “≤” or “” sign
Example: The mean number of TV sets in U.S.
Homes is equal to three. H 0 : μ  3
a sample statistic.
H0 : μ  3
H0 : X  3
 May or may not be rejected
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-4
The Alternative Hypothesis, H1
 Is the opposite of the null hypothesis
 e.g., The mean 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 proven
 For one tail tests is generally the hypothesis
that the researcher is trying to prove
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-5
Hypothesis Testing
 We assume the null hypothesis is true
 If the null hypothesis is rejected we have
proven the alternate hypothesis
 If the null hypothesis is not rejected we have
proven nothing as the sample size may have
been to small
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-6
The Hypothesis Testing
Process
 Claim: The population mean age is 50.
 H0: μ = 50,
H1: μ ≠ 50
 Sample the population and find sample mean.
Population
Sample
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-7
The Hypothesis Testing
Process
 Suppose the sample mean age was X = 20.
 This is significantly lower than the claimed mean
population age of 50.
 If the null hypothesis were true, the probability of
getting such a different sample mean would be very
small, so you reject the null hypothesis .
 In other words, getting a sample mean of 20 is so
unlikely if the population mean was 50, you
conclude that the population mean must not be 50.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-8
The Hypothesis Testing
Process
Sampling
Distribution of X
20
If it is unlikely that you
would get a sample
mean of this value ...
μ = 50
If H0 is true
... if in fact this were
the population mean…
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
X
... then you reject
the null hypothesis
that μ = 50.
Chap 9-9
The Test Statistic and
Critical Values
 If the sample mean is close to the assumed
population mean, the null hypothesis is not rejected.
 If the sample mean is far from the assumed
population mean, the null hypothesis is rejected.
 How far is “far enough” to reject H0?
 The level of significance of the test statistic ()
creates a “line in the sand” for decision making.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-10
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 the compliment of the confidence coefficient
 Is selected by the researcher before sampling
 Provides the critical value of the test
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-11
The Test Statistic and
Critical Values
Distribution of the test statistic
Region of
Rejection
Region of
Rejection
 /2
 /2
 /2
Critical Values
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-12
Errors in Decision Making
 Type I Error
 Reject a true null hypothesis
 Considered a serious type of error
 The probability of a Type I Error is 
 Called level of significance of the test
 Set by researcher in advance
 Type II Error
 Failure to reject false null hypothesis
 The probability of a Type II Error is β
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-13
Errors in Decision Making
Possible Jury Trial Outcomes
The Truth
The Verdict
Innocent
Guilty
Innocent
No Error
Type II Error
Guilty
Type I Error
No Error
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-14
Errors in Decision Making
Possible Hypothesis Test Outcomes
Actual Situation
Decision
H0 True
H0 False
Do Not
Reject H0
No Error
Probability 1 - α
Type II Error
Probability β
Reject H0
Type I Error
Probability α
No Error
Probability 1 - β
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-15
Type I & II Error Relationship
 Type I and Type II errors can not happen at
the 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 )
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-16
Level of Significance, α
Claim: The population
mean age is 50.
H0: μ = 50
H1: μ ≠ 50
H0: μ ≤ 50
H1: μ > 50
/2
Represents
critical value
/2
Two-tail test
0

Rejection
region is
Upper-tail test
0
H0: μ ≥ 50
H1: μ < 50
Lower-tail test 
0
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-17
8 Steps in
Hypothesis Testing
1. State the null hypothesis, H0
State the alternative hypotheses, H1
2. Choose the level of significance, α
3. Choose the sample size, n
4. Determine the appropriate test statistic to use
5. Collect the data
6. Compute the p-value for the test statistic from the sample
result
7. Make the statistical decision: Reject H0 if the p-value is
less than alpha
8. Express the conclusion in the context of the problem
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-18
Hypothesis Tests for the Mean
Hypothesis
Tests for 
 Known
Z Test
Normal
Distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
 Unknown
t Test
Student t
Distribution
Chap 9-19
Hypothesis Testing:
σ Unknown
 If the population standard deviation is
unknown, you use the sample standard
 Because of this change, you also use the t
distribution instead of the Z distribution to
test the null hypothesis about the mean.
 All other steps, concepts, and conclusions are
the same as the  known test.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-20
Hypothesis Testing:
σ Unknown
 The t test statistic with n-1 degrees of
freedom is:
t n -1 
Xμ
Hypothesized
S
n
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-21
Hypothesis Testing:
σ Unknown Problem
The mean 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.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-22
Hypothesis Testing Problem
8 steps
 1.
State the appropriate null and alternative hypotheses
H0: μ = 168
H1: μ ≠ 168 (This is a two tailed test)
 2.
Specify the desired level of significance
 = .05 is chosen for this test
Choose a sample size
sample of size n = 25 was selected
Determine the appropriate Test
σ is unknown so this is a t test
 3.
 4.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-23
Hypothesis Testing Problem
(continued)
 5.
Collect the data
The sample results are
n = 25,
 6.
X= \$172.50 S = \$15.40
So the test statistic is:
t 
X μ
S
n

172.50  168
15.40

4.50
 1.4610
3.08
25
The p value for n=25, =.05, t=1.4610 is .1570
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-24
Hypothesis Testing Problem
(continued)
 7.
Is the test statistic in the rejection region?
Reject H0 if p is < alpha; otherwise
do not reject H0
The p-value .1570 is not < alpha .05,
we do not reject the null hypothesis
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-25
Hypothesis Testing Problem
(continued)

8. Express the conclusion in the context of the problem
Since The p-value .1570 is > alpha .05,
we have failed to reject the null hypothesis
Thereby not proving the alternate hypothesis
Conclusion: There is not sufficient evidence to reject the
claim that the mean cost of a hotel room in NYC is \$168
If we had rejected the null hypothesis the conclusion would
have been: There is sufficient evidence to reject the claim that
the mean cost of a hotel room in NYC is \$168
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-26
Hypothesis Testing:
σ Unknown
 Recall that you assume that the sample
statistic comes from a random sample from a
normal distribution.
 If the sample size is small (< 30), you should
use a box-and-whisker plot or a normal
probability plot to assess whether the
assumption of normality is valid.
 If the sample size is large, the central limit
theorem applies and the sampling
distribution of the mean will be normal.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-27
One Tail Tests
 In many cases, the alternative hypothesis
focuses on a 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
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-28
Hypothesis Testing:
Connection to Confidence Intervals
 For X = 172.5, S = 15.40 and n = 25, the 95%
confidence interval is:
172.5 - (2.0639)
15.4
to
172.5  (2.0639)
25
15.4
25
166.14 ≤ μ ≤ 178.86
 Since this interval contains the hypothesized
mean (168), you do not reject the null
hypothesis at  = .05
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-29
Hypothesis Testing
Proportions
 Involves categorical variables
 Two possible outcomes
 “Success” (possesses a certain characteristic)
 “Failure” (does not possesses that
characteristic)
 Fraction or proportion of the population in
the “success” category is denoted by π
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-30
Hypothesis Testing
Proportions
 Sample proportion in the success category is denoted by p
p
X
n

number
of successes
in sample
sample size
 When both nπ and n(1-π) are at least 5, p can be
approximated by a normal distribution with mean and
standard deviation
 (1   )
σp 
μp  
n
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-31
Hypothesis Testing
Proportions
 The sampling distribution of p is
approximately normal, so the test statistic is
a Z value:
Z 
p 
 (1   )
n
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-32
Hypothesis Testing
Proportions Example
A marketing company claims that it receives
8% responses from its mailing. To test this
claim, a random sample of 500 were
surveyed with 30 responses. Test at the  =
.05 significance level.
First, check:
n π = (500)(.08) = 40
n(1-π) = (500)(.92) = 460
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-33
Hypothesis Testing
Proportions Example
H0: π = .08
H1: π ≠ .08
α = .05
n = 500, p = 30/500 =.06
Z
p 
 (1   )
n

.06  .08
.08(1  .08)
  1.648
500
p-value for -1.648 is .0497
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-34
Hypothesis Testing
Proportions Example
Test Statistic:
p-value for -1.648 is .0497
Decision:
Reject H0 at  = .05
Conclusion:
There is sufficient evidence to
reject the company’s claim of
8% response rate.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-35
Using PHStat
Options
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-36
Sample PHStat Output
Input
Output
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-37
Potential Pitfalls and
Ethical Considerations
 Use randomly collected data to reduce selection





biases
Do not use human subjects without informed
consent
Choose the level of significance, α, before data
collection
Do not employ “data snooping” to choose between
one-tail and two-tail test, or to determine the level of
significance
Do not practice “data cleansing” to hide observations
that do not support a stated hypothesis
Report all pertinent findings
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-38
Chapter Summary
In this chapter, we have
 Mentioned Z Test for the mean (σ known)
 Discussed the p–value approaches to hypothesis
testing
 Discussed one-tail and two-tail tests
 Performed t test for the mean (σ unknown)
 Performed Z test for the proportion
 Discussed pitfalls and ethical issues
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.
Chap 9-39