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

16.1 Concepts of Statistical TestsA manager is evaluating software to filter SPAM e-mails cost $15,000.. 16.1 Concepts of Statistical TestsNull and Alternative Hypotheses  Statistical

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Statistical Tests

Chapter 16

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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?

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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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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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One- and Two-Sided Tests

value of a parameter larger (or smaller) than a

specified value

specific value for the population parameter

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Type I and II Errors

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Type I and II Errors

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Other Tests

plots and control charts all use tests of

hypotheses

for association is that there is no association

between two variables shown in the scatterplot

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Sampling Distribution

of the statistic that estimates the parameter

specified in the null and alternative hypotheses

sample that differs from H 0 by as much as this

one if H 0 is true?

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16.2 Testing the Proportion

is approximately normal with mean p = 0.20

and SE( ) = 0.04 (note that the hypothesized

pˆ pˆ

pˆ

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What is the chance of making a Type I error?

Possible sampling distributions for .

Chance of a Type I error shown in shaded area.pˆ

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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

statistic

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z–Test for SPAM Software Example

= -2.25

n p

p

p z

/ ) 1

(

ˆ

0 0

) 20

0 1

( 20

0

20

0 11

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p–Value for SPAM Software Example

Interpret the p-value as a weight of evidence against H0; small values mean that H0 is not plausible.

012

0 )

25

2 (

)

P

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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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Type II Error

to miss meaningful deviations from the null

hypothesis and produce a Type II error

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Summary

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Checklist

sample from the relevant population

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4M Example 16.1: DO ENOUGH

HOUSEHOLDS WATCH?

Motivation

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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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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0 06

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4M Example 16.1: DO ENOUGH

HOUSEHOLDS WATCH?

Message

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

Use s from the sample as an estimate of σ to

calculate the estimated standard error of

X

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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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Null and Alternative Hypotheses

in the Denver area

H0: µ ≤ µ0 = $500

Ha: µ > µ0 = $500

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t - Statistic

n s

x t

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Example: Denver Rental Properties

 The firm obtained rents for a sample of size n=45;

the average rent was $647 with s = $299.

t = 3.298 with 44 df; p-value = 0.00097

Reject H0 ; mean rent exceeds break-even value.

45/

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Finding the p-Value in the t-Table

t = 3.298 is larger than any value in the row

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Summary

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Checklist

sample from the relevant population

normally distributed, a normal model can be used

to approximate the sampling distribution of if n

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4M Example 16.2: COMPARING

RETURNS ON INVESTMENTS

Motivation

Does stock in IBM return more, on average,

than Bills? From 1980 through 2005,

T-Bills returned 5% each month.

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x t

0805

0

0050

0 0106

.



t

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earnings than comparable investments in US

Treasury Bills during this period

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16.4 Other Properties of Tests

Significance versus Importance

have made an important or meaningful discovery

test With enough data, a trivial difference from

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16.4 Other Properties of Tests

Confidence Interval or Test?

parameter values that are compatible with the

observed data

hypothesized value for a parameter

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Best Practices

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Best Practices (Continued)

population

test

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substantive importance

that the null hypothesis is true

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