1Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSChapter 7 Hypothesis Testing

2Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Chapter 7Hypothesis Testing

7-1 Overview

7-2 Fundamentals of Hypothesis Testing

7-3 Testing a Claim about a Mean: Large     Samples

7-4 Testing a Claim about a Mean: Small     Samples

7-5 Testing a Claim about a Proportion

7-6 Testing a Claim about a Standard     Deviation

3Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

7-1 Overview

Definition

Hypothesis

in statistics, is a claim or statement about a property of a population

4Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Rare Event Rule for Inferential Statistics

If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

5Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

7-2

Fundamentals of

Hypothesis Testing

6Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

µx = 98.6

Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means

Assuming that = 98.6

Likely sample means

7Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

z = - 1.96

x = 98.48

or

z = 1.96

x = 98.72

or

µx = 98.6

Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means

Assuming that = 98.6

Likely sample means

8Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means Assuming that

= 98.6

z = - 1.96

x = 98.48

or

z = 1.96

x = 98.72

or

Sample data: z = - 6.64

x = 98.20 or

µx = 98.6

Likely sample means

9Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Components of aFormal Hypothesis

Test

10Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Null Hypothesis: H0

Statement about value of    population parameter

Must contain condition of equality

=, , or

Test the Null Hypothesis directly

Reject H0 or fail to reject H0

11Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Alternative Hypothesis: H1

Must be true if H0 is false

, <, >

‘opposite’ of Null

12Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Note about Forming Your Own Claims (Hypotheses)

If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.

13Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Note about Testing the Validity of Someone Else’s Claim

Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality).

14Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Test Statistic

a value computed from the sample data that is used in making the decision about the

rejection of the null hypothesis

15Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Test Statistic

a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis

For large samples, testing claims about population means

z = x - µx

n

16Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical RegionSet of all values of the test statistic that

would cause a rejection of the null hypothesis

17Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical RegionSet of all values of the test statistic that

would cause a rejection of thenull hypothesis

CriticalRegion

18Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical RegionSet of all values of the test statistic that

would cause a rejection of the null hypothesis

CriticalRegion

19Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical RegionSet of all values of the test statistic that

would cause a rejection of the null hypothesis

CriticalRegions

20Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Significance Level denoted by the probability that the test

  statistic will fall in the critical   region when the null hypothesis   is actually true.

common choices are 0.05, 0.01,   and 0.10

21Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical ValueValue or values that separate the critical region

(where we reject the null hypothesis) from the values of the test statistics that do not lead

to a rejection of the null hypothesis

22Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical Value

Critical Value( z score )

Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead

to a rejection of the null hypothesis

23Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Critical Value

Critical Value( z score )

Fail to reject H0Reject H0

Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead

to a rejection of the null hypothesis

24Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Two-tailed,Right-tailed,Left-tailed Tests

The tails in a distribution are the extreme regions bounded

by critical values.

25Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Two-tailed TestH0: µ = 100

H1: µ 100

26Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Two-tailed TestH0: µ = 100

H1: µ 100 is divided equally between

the two tails of the critical region

27Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Two-tailed TestH0: µ = 100

H1: µ 100

Means less than or greater than

is divided equally between the two tails of the critical

region

28Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Two-tailed TestH0: µ = 100

H1: µ 100

Means less than or greater than

100

Values that differ significantly from 100

is divided equally between the two tails of the critical

region

Fail to reject H0Reject H0 Reject H0

29Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Right-tailed TestH0: µ 100

H1: µ > 100

30Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Right-tailed TestH0: µ 100

H1: µ > 100

Points Right

31Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Right-tailed TestH0: µ 100

H1: µ > 100

Values that differ significantly

from 100100

Points Right

Fail to reject H0 Reject H0

32Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Left-tailed Test

H0: µ 100

H1: µ < 100

33Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Left-tailed Test

H0: µ 100

H1: µ < 100Points Left

34Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Left-tailed Test

H0: µ 100

H1: µ < 100

100

Values that differ significantly

from 100

Points Left

Fail to reject H0Reject H0

35Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Conclusions in Hypothesis Testing

always test the null hypothesis

1. Reject the H0

2. Fail to reject the H0

need to formulate correct wording of final conclusion

See Figure 7-4

36Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

FIGURE 7-4 Wording of Final Conclusion

Does theoriginal claim contain

the condition ofequality

Doyou rejectH0?.

Yes

(Original claim

contains equality

and becomes H0)

No(Original claimdoes not containequality and

becomes H1)

Yes

(Reject H0)

“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”

“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”

“The sample datasupports the claim that . . . (original claim).”

“There is not sufficientevidence to support the claimthat. . . (original claim).”

Doyou reject

H0?

Yes

(Reject H0)

No(Fail to

reject H0)

No(Fail to

reject H0)

(This is theonly case inwhich theoriginal claimis rejected).

(This is theonly case inwhich theoriginal claimis supported).

Start

37Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Accept versus Fail to Reject

some texts use “accept the null hypothesis

we are not proving the null hypothesis

sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect)

38Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Type I ErrorThe mistake of rejecting the null hypothesis

when it is true.

(alpha) is used to represent the probability of a type I error

Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6

39Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Type II Errorthe mistake of failing to reject the null

hypothesis when it is false.

ß (beta) is used to represent the probability of a type II error

Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6

40Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Table 7-2 Type I and Type II Errors

True State of Nature

We decide to

reject the

null hypothesis

We fail to

reject the

null hypothesis

The null

hypothesis is

true

The null

hypothesis is

false

Type I error

(rejecting a true

null hypothesis)

Type II error

(rejecting a false

null hypothesis)

Correct

decision

Correct

decision

Decision

41Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Controlling Type I and Type II Errors

For any fixed , an increase in the sample size n will cause a decrease in

For any fixed sample size n , a decrease in will cause an increase in . Conversely, an increase in will cause a decrease in .

To decrease both and , increase the sample size.

42Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Definition

Power of a Hypothesis Test

is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis.