Statistics for business decision making and analysis robert stine and foster chapter 16

Chapter 16

Statistical Tests

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16.1 Concepts of Statistical Tests
A manager is evaluating software to filter
SPAM e-mails (cost $15,000). To make it
profitable, the software must reduce SPAM
to less than 20%. Should the manager buy
the software?



Use a statistical test to answer this question
Consider the plausibility of a specific claim (claims are
called hypotheses)
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16.1 Concepts of Statistical Tests
Null and Alternative Hypotheses


Statistical hypothesis: claim about a parameter of a
population.



Null hypothesis (H0): specifies a default course of action,
preserves the status quo.



Alternative hypothesis (Ha): contradicts the assertion of
the null hypothesis.
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16.1 Concepts of Statistical Tests
SPAM Software Example
Let p = email that slips past the filter
H0: p ≥ 0.20
Ha: p < 0.20
These hypotheses lead to a one-sided test.
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16.1 Concepts of Statistical Tests
One- and Two-Sided Tests


One-sided test: the null hypothesis allows any value of a
parameter larger (or smaller) than a specified value.



Two-sided test: the null hypothesis asserts a specific
value for the population parameter.

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16.1 Concepts of Statistical Tests
Type I and II Errors


Reject H0 incorrectly
(buying software that will not be cost effective)



Retain H0 incorrectly
(not buying software that would have been cost effective)

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16.1 Concepts of Statistical Tests
Type I and II Errors

 indicates a correct decision
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16.1 Concepts of Statistical Tests
Other Tests


Visual inspection for association, normal quantile plots
and control charts all use tests of hypotheses.



For example, the null hypothesis in a visual test for
association is that there is no association between two
variables shown in the scatterplot.

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16.1 Concepts of Statistical Tests
Sampling Distribution


Statistical tests rely on the sampling distribution of the
statistic that estimates the parameter specified in the null
and alternative hypotheses.



Key question: What is the chance of getting a sample
that differs from H0 by as much as this one if H0 is true?

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16.2 Testing the Proportion
SPAM Software Example
=p
ˆ0.11.



Based on n = 100,



Assuming H0 is true, the sampling distribution of
ˆ is approximately normal with mean p = 0.20 and SE( )
p
= 0.04 (note that the hypothesized value p0 = 0.20 is used
ˆ
p
to calculate SE).

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16.2 Testing the Proportion
SPAM Software Example
What is the chance of making a Type I error?

Possible sampling distributions for .
ˆ
p
Chance of a Type I error shown in shaded area.
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16.2 Testing the Proportion
z–Test and p-Value


p-Value: the largest chance of a Type I error if H0 is
rejected based on the observed test statistic.



z-Test: test of H0 based on a count of the standard errors
separating H0 from the test statistic.

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16.2 Testing the Proportion
z–Test for SPAM Software Example

z=

ˆ − p0
p
p 0(1 − p 0) / n

z=

0.11 − 0.20
0.20(1 − 0.20) / 100

= -2.25
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16.2 Testing the Proportion
p–Value for SPAM Software Example

P ( Z ≤ z ) = P ( Z ≤ −2.25) ≈ 0.012
Interpret the p-value as a weight of evidence
against H0; small values mean that H0 is not
plausible.
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16.2 Testing the Proportion
α-Value



α-Value: threshold that sets the maximum tolerance for a
Type I error.



Statistically significant: data contradict the null hypothesis
and lead us to reject H0 (p-value < α).



The p-value in the SPAM example is less than the typical
α of 0.05; should buy the software.
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16.2 Testing the Proportion
Type II Error


Power: probability that a test rejects H0.



If a test has little power when H0 is false, it is likely to miss
meaningful deviations from the null hypothesis and
produce a Type II error.

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16.2 Testing the Proportion
Summary

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16.2 Testing the Proportion
Checklist


SRS condition: the sample is a simple random sample
from the relevant population.



Sample size condition (for proportion): both np0 and n(1 p0 ) are larger than 10.

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4M Example 16.1: DO ENOUGH
Motivation
HOUSEHOLDS
WATCH?

The Burger King ad featuring Coq Roq won critical
acclaim. In a sample of 2,500 homes, MediaCheck
found that only 6% saw the ad. An ad must be viewed
by 5% or more of households to be effective. Based
on these sample results, should the local sponsor run
this ad?

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4M Example 16.1: DO ENOUGH
Mehod
HOUSEHOLDS
WATCH?
Set up the null and alternative hypotheses.
H0: p ≤ 0.05
Ha: p > 0.05
Use α = 0.05. Note that p is the population proportion who watch
this ad. Both SRS and sample size conditions are met.

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4M Example 16.1: DO ENOUGH
Mechanics
HOUSEHOLDS
WATCH?
Perform a one-sided z-test for a proportion.

z=

0.06 − 0.05
0.05(1 − 0.05) / 2,500

z = 2.3 with p-value of 0.011
Reject H0.

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4M Example 16.1: DO ENOUGH
Message
HOUSEHOLDS
WATCH?
The results are statistically significant. We can
conclude that more than 5% of households
watch this ad. The Burger King Coq Roq ad is
cost effective and should be run.

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16.3 Testing the Mean
Similar to Tests of Proportions
ˆ .
withp

X



The hypothesis test of µ replaces



Unlike the test of proportions, σ is not specified. Use s
from the sample as an estimate of σ to calculate the
estimated standard error of .

X

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16.3 Testing the Mean
Example: Denver Rental Properties
A firm is considering expanding into the Denver area. In
order to cover costs, the firm needs rents in this area to
average more than $500 per month. Are Denver rents
high enough to justify the expansion?

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