Download 9.1 Introduction to Hypothesis Testing

Essentials of Business Statistics: Communicating
with Numbers
By Sanjiv Jaggia and Alison Kelly
Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved.
Chapter 9 Learning Objectives
LO 9.1
LO 9.2
LO 9.3
LO 9.4
Define the null hypothesis and the alternative hypothesis.
Distinguish between Type I and Type II errors.
Conduct a hypothesis test using the p-value approach.
Conduct a hypothesis test using the critical value
approach.
LO 9.5 Differentiate between the test statistics for the
population mean.
LO 9.6 Specify the test statistic for the population proportion.
Hypothesis Testing
9-2
9.1 Introduction to Hypothesis Testing
LO 9.1 Define the null hypothesis and the alternative hypothesis.
 Hypothesis tests resolve conflicts between two
competing opinions (hypotheses).
 In a hypothesis test, define


H0, the null hypothesis, the presumed default state
of nature or status quo.
HA, the alternative hypothesis, a contradiction of
the default state of nature or status quo.
Hypothesis Testing
9-3
LO 9.1
9.1 Introduction to Hypothesis
Testing
 In statistics we use sample information to make inferences
regarding the unknown population parameters of interest.
 We conduct hypothesis tests to determine if sample
evidence contradicts H0.
 On the basis of sample information, we either


“Reject the null hypothesis”
 Sample evidence is inconsistent with H0.
“Do not reject the null hypothesis”
 Sample evidence is not inconsistent with H0.
 We do not have enough evidence to “accept” H0.
Hypothesis Testing
9-4
LO 9.1
9.1 Introduction to Hypothesis
Testing
 Defining the Null Hypothesis and Alternative
Hypothesis
 General guidelines:
Null hypothesis, H0, states the status quo.
 Alternative hypothesis, HA, states whatever we wish to
establish (i.e., contests the status quo).
 Use the following signs in hypothesis tests
H0
=
>
<  specify the status quo,

HA
≠
<
>
 contradict H0.
 Note that H0 always contains the “equality.”
Hypothesis Testing
9-5
LO 9.1
9.1 Introduction to Hypothesis
Testing
 One-Tailed versus Two-Tailed Hypothesis Tests


Two-Tailed Test
 Reject H0 on either side of the hypothesized value
of the population parameter.
 For example:
H0: m = m0 versus HA: m ≠ m0
H0: p = p0 versus HA: p ≠ p0
The “≠” symbol in HA indicates that both tail areas of
the distribution will be used to make the decision
regarding the rejection of H0.
Hypothesis Testing
9-6
LO 9.1
9.1 Introduction to Hypothesis
Testing
 One-Tailed versus Two-Tailed Hypothesis Tests


One-Tailed Test
 Reject H0 only on one side of the hypothesized value
of the population parameter.
 For example:
H0: m < m0 versus HA: m > m0 (right-tail test)
H0: m > m0 versus HA: m < m0 (left-tail test)
Note that the inequality in HA determines which tail area
will be used to make the decision regarding the rejection
of H0.
Hypothesis Testing
9-7
LO 9.1
9.1 Introduction to Hypothesis
Testing
 Three Steps to Formulate Hypotheses
1. Identify the relevant population parameter of interest
(e.g., m or p).
H0
HA
Test Type
2. Determine whether
it is a one- or a
two-tailed test.
=
>
<
≠
<
>
Two-tail
One-tail, Left-tail
One-tail, Right-tail
3. Include some form of the equality sign in H0
and use HA to establish a claim.
Hypothesis Testing
9-8
9.1 Introduction to Hypothesis Testing
LO 9.2 Distinguish between Type I and Type II errors.
 Type I and Type II Errors

Type I Error: Committed when we reject H0 when H0 is
actually true.


Type II Error: Committed when we do not reject H0
and H0 is actually false.



Occurs with probability a, which is chosen a priori.
Occurs with probability b. Power of the test = 1b
For a given sample size n, a decrease in a will increase
b and vice versa.
Both a and b decrease as n increases.
Hypothesis Testing
9-9
9.2 Hypothesis Test of the Population
Mean When s Is Known
LO 9.3 Conduct a hypothesis test using the p-value approach.
 Hypothesis testing enables us to determine whether
the sample evidence is inconsistent with what is
hypothesized under the null hypothesis (H0).
 Basic principle: First assume that H0 is true and then
determine if sample evidence contradicts this
assumption.
 Two approaches to hypothesis testing:
 The p-value approach.
 The critical value approach.
Hypothesis Testing
9-10
LO 9.3
9.2 Hypothesis Test of the Population
Mean When s Is Known
 The p-value Approach

The value of the test statistic for the hypothesis test of
the population mean m when the population standard
deviation s is known is computed as
z

x  m0
s
n
where m0 is the hypothesized mean value.
p-value: the likelihood of obtaining a sample mean that is
at least as extreme as the one derived from the given
sample, under the assumption that the null hypothesis is
true.
Hypothesis Testing
9-11
LO 9.3
9.2 Hypothesis Test of the Population
Mean When s Is Known
 The p-value Approach
 Under the assumption that m = m0, the p-value is the
likelihood of observing a sample mean that is at least as
extreme as the one derived from the given sample.
 The calculation of the p-value depends on the
specification of the alternative hypothesis:
 Decision rule: Reject H0 if p-value < a.
Hypothesis Testing
9-12
LO 9.3
9.2 Hypothesis Test of the Population
Mean When s Is Known
 Four Step Procedure Using the p-value
Approach
 Step 1.
Specify the null and the alternative
hypotheses.
 Step 2. Specify the significance level.
 Step 3. Calculate the value of the test statistic and
the p-value.
 Step 4. State the conclusion and interpret the
results.
Hypothesis Testing
9-13
9.2 Hypothesis Test of the Population
Mean When s Is Known
LO 9.4 Conduct a hypothesis test using the critical value approach.
 The Critical Value Approach
Rejection region: a region of values such that if
the test statistic falls into this region, then we
reject H0.
 The location of this region is determined
by HA.
 Critical value: a point that separates the rejection
region from the nonrejection region.

Hypothesis Testing
9-14
LO 9.4
9.2 Hypothesis Test of the Population
Mean When s Is Known
 The Critical Value Approach



The critical value approach specifies a region such that if the
value of the test statistic falls into the region, the null
hypothesis is rejected.
The critical value depends on the alternative hypothesis.
Decision Rule: Reject H0 if:
z > za for a right-tailed test
z < za for a left-tailed test
z > za/2 or z < za/2 for a two-tailed test
Hypothesis Testing
9-15
LO 9.4
9.2 Hypothesis Test of the Population
Mean When s Is Known
 The Critical Value Approach
 Determining
the critical value(s) depending on
the specification of the competing hypotheses.
Reject H0 if
z > za/2 or z < za/2
Reject H0 if z < za
Reject H0 if z > za
Hypothesis Testing
9-16
LO 9.4
9.2 Hypothesis Test of the Population
Mean When s Is Known
 Four Step Procedure Using the Critical Value
Approach
Step 1. Specify the null and the alternative
hypotheses.
 Step 2. Specify the significance level and find the
critical value(s) for the test statistic.
 Step 3. Calculate the value of the test statistic.
 Step 4. State the conclusion and interpret the
results.

Hypothesis Testing
9-17
LO 9.4
9.2 Hypothesis Test of the Population
Mean When s Is Known
 Confidence Intervals and Two-Tailed
Hypothesis Tests


Given the significance level a, we can use the
sample data to construct a 100(1  a)% confidence
interval for the population mean m.
Decision Rule
 Reject H0 if the confidence interval does not contain
the value of the hypothesized mean m0.
 Do not reject H0 if the confidence interval does
contain the value of the hypothesized mean m0.
Hypothesis Testing
9-18
LO 9.4
9.2 Hypothesis Test of the Population
Mean When s Is Known
 Implementing a Two-Tailed Test Using a Confidence
Interval

The general specification for a 100(1  a)% confidence
interval of the population mean m when the population
standard deviation s is known is computed as
x  za /2 s

n
or
 x  za / 2 s

n , x  za / 2 s
Decision rule: Reject H0 if m0  x  za /2 s
n
or if m0  x  za /2 s
n
Hypothesis Testing
n

9-19
9.3 Hypothesis Test of the Population
Mean When s Is Unknown
LO 9.5 Differentiate between the test statistics for the population
mean.
 Test Statistic for m When s is Unknown


When the population standard deviation s is
unknown, the test statistic for testing the
population mean m is assumed to follow the tdf
distribution with (n  1) degrees of freedom (df).
The value of tdf is computed as:
x  m0
tdf 
Hypothesis Testing
s
n
9-20
9.4 Hypothesis Test of the Population
Proportion
LO 9.6 Specify the test statistic for the population proportion.
 Test Statistic for p.

P can be approximated by a normal distribution if np
> 5 and n(1p) > 5.
 Test statistic for the hypothesis test of the population
proportion p is assumed to follow the z distribution:
where p  x n
p  p0
z
p0 1  p0  n and p0 is the hypothesized
value of the population
proportion.
Hypothesis Testing
9-21