Powerpoint Topik 8

7/28/2019 Powerpoint Topik 8

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TOPIC 8

Hypothesis Testing

Introduction

Hypothesis testing is a procedure based

on sample data and probability theory to

statistical evidence in favor of a certain

belief about a parameter.

6 Steps of Hypothesis Testing Procedure

S.1 State null & alternative hypotheses

S.2 Select the significance level

S.3 Formulate a decision rule

S.4 Compute the test statistic

S.5 Make a conclusion of the test

S.6 Make the related decision

S.1 State null & alternative hypotheses

A pair of competing statistical hypothesesconsists of

0 the value of a population parameter.

Alternative Hypothesis (H1): A statementthat is accepted if the sample dataprovide evidence that the null hypothesisis false.

Threepossiblestatistical

one-tail test(directional test)

H0 : 0H1 : > 0

H0 : 0H1 : > 0

H0 : 0 H0 : 0

Threepossiblestatistical

ypo esesto test a

populationmean

H1 : < 0 H1 : < 0

two-tail test(Nondirectional test)

H0 : = 0H1 : 0

H0 : = 0H1 : 0

ypo esesto test a

populationproportion

EXAMPLE 1:

For each of the following pairs of null andalternative hypotheses, determine whether thepair would be appropriate for a hypothesis test. Ifa air is deemed ina ro riate ex lain wh .

a) H0 : > 90 ; H1 : 90

b) H0 : 75 ; H1 : > 85

c) H0 : = 58 ; H1 : 58

d) H0 : 0.48 ; H1 : > 0.52

e) H0 : p 0.65 ; H1 : > 0.65

x x


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EXAMPLE 2:

For each of the following statements, formulateappropriate null and alternative hypotheses.

a) The average college student spends no morethan RM500 per semester at the universitys

.

b) The proportion adult drinks 2 cups of coffeeper day is more than 0.62.

c) The average SAT score for entering freshmen isat least 1200.

d) The proportion candidate passing on thequalifying exam differs from 0.33.

S.2 Select the significance level

significance level () measure the prob.of rejecting H0 when it is true

in practice, significance level is set at 1%(0.01), 5% (0.05) and 10% (0.10)

S.3 Formulate a decision rule

Decision Rule : A statement of the specificconditions under which the H0 is rejected.

r t ca a ue :

- the dividing point between the regionwhere the H0 is rejected and the regionwhere it is not rejected

Critical Value :

- it is determined by the type of alternativehypothesis, sampling distribution of thetest statistic and level of significance ()

Rejection Region : The rejection region is a

range of values such that if the test statisticfalls into that range, the H0 is rejected infavor of the H1.

One-tail Test: Level of Significance, =0.01(right tail) Sampling distribution of the

test statistic Z

Decision Rule :

2.3263Critical Value: Z

RejectionRegion

( = 0.01)

e ec 0Z > 2.3263

One-tail Test: Level of Significance, =0.10(left tail) Sampling distribution of the

test statistic Z

Decision Rule :

-1.2816Critical Value: -Z

RejectionRegion

( = 0.10)

Reject H0 ifZ < -1.2816


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Two-tail Test: Level of Significance, =0.05Sampling distribution of thetest statistic Z

RejectionRegion

RejectionRegion

-1.96 1.96Critical Value: -Z/2 Critical Value: Z/2

(/2 = 0.025)

Decision Rule: Reject H0 if Z > 1.96 or Z < -1.96

(/2 = 0.025)

Decision Rule for Z-Test and t-Test

AlternativeHypothesis

Reject H0 if

Z -Test t -Test

, n-1

< Z < -Z t < -t , n-1

Z > Z/2 orZ < -Z/2

t > t/2 , n-1 or

t < -t/2, n-1

Test Statistic

A value, determined from sample

S.4 Compute the test statistic

n orma on, use o e erm ne w e eror not to reject the H0.

Type of Inference Test Statistic

Testing the when2 known n

xZ

=

Testing the when

2

unknown

Testing the

ns

xt

=

n

)1(

pZ

=

S.5 Make a conclusion of the test

If the test statistic falls in the rejectionregion, H0 is rejected.

t e test stat st c oes not a n t erejection region, H0 is not rejected.

Failure to reject a null hypothesis DOESNOT constitute proof that it is true.

Therefore, we never say that we acceptthe null hypothesis

S.6 Make the related decision

If H0 is rejected, we conclude that thereis enough statistical evidence to infer

.

If H0 is not rejected, we conclude thatthere is not enough statistical evidence toinfer that the H1 is true.


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2 types of errors may occur when decidingwhether to reject H0 based on the stat. value.

a) Type I error ():

Types of Errors

e ec 0 w en s rue.

b) Type II error ():

Do not reject H0 when it is false.

The Type I error can be directly controlled.It is actually the level of significance.

Do not reject H0 Reject H0

H0 is true Correct decision Type I error()

H is false T e II error Correct decision() (1 )

Power of a test (1-):

It represents the probability of rejecting thenull hypothesis when it is false.

EXAMPLE 3:

Following a major earthquake, the city engineermust determine whether the stadium isstructurally sound for an upcoming athletic event.If the null h othesis is the stadium isstructurally sound, and the alternative hypothesis

is the stadium is not structurally sound, whichtype of error (Type I or Type II) would theengineer least like to commit?

Testing the Population Mean when thePopulation Variance is Known

EXAMPLE 4:

When a machine is set properly, it produces nailshaving a mean length of 1.325 inches, with astandard deviation of 0.0396 inches. For a sample

of 80 nails, the mean length is 1.3229 inches.Using the 0.05 level of significance, examinewhether the machine is adjusted properly.

Testing Hypotheses and ConfidenceInterval Estimators

Confidence interval estimator can be usedto conduct the two-sided alternative testso ypot eses at t e eve o s gn cance

If the hypothesized value under nullhypothesis falls into the (1)100%confidence interval estimate, H0 is notrejected


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Refer to EXAMPLE 4:

The 95% confidence interval estimate of isLCL = 1.3142, UCL = 1.3316

We have 95% confident that the population meaneng s somew ere e ween . an .inches. Since = 1.325 falls within the intervalestimate, we conclude at 5% that the mean lengthdo not differ from 1.325 inches.

The conclusion is the same as that reachedat the level of significance

Testing the Population Mean when the

Population Variance is UnknownEXAMPLE 5:

cash sales is no more than $80, but an InternalRevenue Service agent believes the dealer isuntruthful. Observing a sample of 20 cashcustomers, the agent finds the mean purchase tobe $85, with a standard deviation of $21.

Assuming population is approximately normaldistributed, and using the 0.05 level ofsignificance, is the agents suspicion confirmed?

Testing the Population Proportion

EXAMPLE 6:

A simple random sample of 300 items is,

reveals that 4% of the sampled are defective.At the 0.10 level of significance, can weconclude that less than 7% of the items in theshipment are defective?


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Making Decision Using p-value Approach

The p-value of a test is the lowest level ofsignificance at which the null hypothesiscan be rejected.

The calculation of p-value depends on thetype of alternative hypothesis, and thesampling distribution of the test statistics

Decision Rule: If the p-value is smallerthan the significance level (), H0 isrejected.

Calculation of the p-value

p-value

Example 4:(two tail test)

2 P(Z > |Z*|) = 2 P(Z > 0.47)= 0.6384

Example 5:(right tail test)

P(t > t*) = P(t > 1.065)= 0.1502 (from Excel)

Example 6:(left tail test)

P(Z < Z*) = P(Z < -2.04)= 0.0207

Z* or t* is the value of the test statistic

Making Decision Using p-value

Example 4:

Example 6:

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