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S TATISTICS. E LEMENTARY. Chapter 7 Hypothesis Testing . M ARIO F . T RIOLA. E IGHTH. E DITION. 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

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M ARIO F . T RIOLA

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STATISTICS

ELEMENTARY

Chapter 7 Hypothesis Testing

MARIO F. TRIOLA

EIGHTH

EDITION


Chapter 7 hypothesis testing l.jpg

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

Chapter 7Hypothesis Testing


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Definition

Hypothesis

In statistics, a hypothesis is a claim or statement about a property of a population.


Rare event rule for inferential statistics l.jpg

If, under a given assumption,

theprobability of an observed event

is exceptionally small,

we conclude that

theassumption is probably not correct.

Rare Event Rule for Inferential Statistics


Figure 7 1 central limit theorem l.jpg

The Expected Distribution of Sample Means Assuming that  = 98.6

z= - 1.96

x = 98.48

z = 1.96

x= 98.72

or

or

Figure 7-1 Central Limit Theorem

Likely sample means

or

z= - 6.64

Sample data:x= 98.20

µx = 98.6


Components of a formal hypothesis test l.jpg

Components of aFormal Hypothesis Test


Null hypothesis h 0 l.jpg

Statement about the value of a POPULATION PARAMETER

Must contain condition of EQUALITY: = ,≤ , or ≥

Test the Null Hypothesis directly

RejectH0 or fail to rejectH0

Null Hypothesis: H0


Alternative hypothesis h 1 l.jpg

Must be true if H0 is false

Must contain condition of INEQUALITY:  , < ,or >

‘Opposite’ of Null Hypothesis

Alternative Hypothesis: H1


Test statistic l.jpg

Test Statistic

  • A value computed from the sample data that is used in making the decision about whether to reject the null hypothesis

  • For large samples, when testing claims about population means, the test statistic is a z-score corresponding to the sample mean.


Critical region l.jpg

Set of all values of the test statistic that would cause a rejection of the

null hypothesis

Critical Region

Critical

Region


Critical region11 l.jpg

Set of all values of the test statistic that would cause a rejection of the

null hypothesis

Critical Region

Critical

Region


Critical region12 l.jpg

Set of all values of the test statistic that would cause a rejection of the

null hypothesis

Critical Region

Critical

Regions


Significance level l.jpg

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

Significance Level


Critical value l.jpg

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

Critical Value

Reject H0

Fail to reject H0

Critical Value

( z score )


Two tailed right tailed left tailed tests l.jpg

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

The tails in a distribution are the extreme regions bounded

by critical values.


Two tailed test l.jpg

H0: µ = 100

H1: µ  100

Two-tailed Test

 is divided equally between

the two tails of the critical

region

UNEQUAL means less than or greater than

Reject H0

Fail to reject H0

Reject H0

100

Values that differ significantly from 100


Right tailed test l.jpg

H0: µ  100

H1: µ > 100

Fail to reject H0

Reject H0

Right-tailed Test

Values that

differ significantly

from 100

100


Left tailed test l.jpg

H0: µ  100

H1: µ < 100

Left-tailed Test

Reject H0

Fail to reject H0

Values that

differ significantly

from 100

100


Conclusions in hypothesis testing l.jpg

Always test the NULL hypothesis:

Reject H0

Fail to rejectH0

Be careful to include the correct wording of the final conclusion

Conclusions in Hypothesis Testing

or


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Wording of Final Conclusion

Start

Only case in which original claim is rejected

Claim

contains

equality?

“There is sufficient

evidence to reject

the claim that. . .

(original claim).”

Yes

Yes

Claim becomes H0

Reject

H0?

No

“There is not

sufficient evidence

to reject the claim

that (original claim).”

No

Claim

becomes H1

Only case

in which

original

claim is

supported

“There issufficient evidence to support

the claim that . . .

(original claim).”

Yes

Reject

H0?

No

“There is not

sufficient evidence

to support the claim

that (original claim).”


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Some texts use “accept the null hypothesis”

We are not proving the null hypothesis (can’t PROVE equality)

If the sample evidence is not strong enough to warrant rejection, then the null hypothesis may or may not be true (just as a defendant found NOT GUILTY may or may not be innocent)

“Fail to Reject” versus “Accept”


Type i error l.jpg

Rejecting the null hypothesis when it is true.

(alpha) represents the probability of a type I error

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

Type I Error


Type ii error l.jpg

Failing to reject the null hypothesis when it is false.

β (beta) represents 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 really isn’t 98.6

Type II Error


Type i and type ii errors l.jpg

Type I and Type II Errors

NULL HYPOTHESIS

TRUE

FALSE

Type I error

α

Rejecting a true

null hypothesis

Reject the null

hypothesis

CORRECT

DECISION

Type II error

β

Failing to reject a

false null hypothesis

Fail to reject the

null hypothesis

CORRECT


Controlling type i and type ii errors l.jpg

, , and nare interrelated. If one is kept constant, then an increase in one of the remaining two will cause a decrease in the other.

For any fixed , an increase in the sample size nwill cause a ??????? in 

For any fixed sample size n, a decrease in  will cause a ??????? in .

Conversely, an increase in  will cause a ??????? in  .

To decrease both  and , ??????? the sample size n.

Controlling Type I and Type II Errors


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