Chapter 3Chapter 3

Hypothesis Hypothesis TestingTesting

Curriculum Object

• Specified the problem based the form of hypothesis

• Student can arrange for hypothesis step

• Analyze a problem bassed for hypothesis

11.3

Introduction…Introduction…

• In addition to estimation, hypothesis testing is a procedure for making inferences about a population.

Parameter

Population

Sample

Statistic

Inference

What is Hypothesis Testing?

• Sample information can be used to obtain point estimates or confidence intervals about population parameters

• Alternatively, sample information can be used to test the validity of conjectures about these parameters

RESumeRESume

• A hypothesis is a statement about a population parameter from one or more populations

• Statistically testable hypotheses are formulated based on theories that are used to make predictions

• A hypothesis test is a procedure that – States the hypothesis to be tested– Uses sample information and formulates a

decision rule– Based on the outcome of the decision rule the

hypothesis is statistically validated or rejected

11.6

Concepts of Hypothesis Testing• There are two hypotheses.

• H0: — the ‘null’ hypothesis

• H1: — the ‘alternative’ or ‘research’ hypothesis

• The null hypothesis (H0) will always state that the parameter equals the value specified in the alternative hypothesis (H1)

pronouncedH “nought”

– The hypothesis that there were no effects is called the NULL HYPOTHESIS.

– The null hypothesis states that in the general population there is no change, no difference, or no relationship.

– In the context of an experiment, H0 predicts that the independent variable (treatment) will have no effect on the dependent variable for the population.

– Form :H0: μA- μB=0 or μA= μB

Null hypothesis (H0)

Alternative hypothesis (H1)– The alternative hypothesis (H1) states

that there is a change, a difference, or a relationship for the general population.

– H1 is a statement of what a statistical hypothesis test is set up to establish.

– Form : H1: μA≠ μB• For example

– H1: the two drugs have different effects, on average.

– H1: the new drug is better than the current drug, on average.

Type I ErrorType I Error

• A type I error is made when the researcher rejected the null hypothesis when it should not have been rejected.

• For example, – H0: there is no difference between the two drugs

on average.• A type I error would occur if we concluded that

the two drugs produced different effects when in fact there was no difference between them.

• The hypothesis test procedure is therefore adjusted so that there is a guaranteed 'low' probability of rejecting the null hypothesis wrongly; this probability is never 0.

• This probability of a type I error can be precisely computed as

P(type I error) = significance level = α

• The exact probability of a type II error is generally unknown.

Type II ErrorType II Error• A type II error is made when the null hypothesis

is accepted when it should have been rejected. • For example, – H0: there is no difference between the two drugs on

average.• A type II error would occur if it was concluded that the

two drugs produced the same effect, i.e. there is no difference between the two drugs on average, when in fact they produced different ones.

• The probability of a type II error is generally unknown, but is symbolised by and written

– P(type II error) = β

RESUMERESUME

The following table gives a summary of possible results of any hypothesis test: – Decision

Reject H0 Don't reject H0

H0 Type I Error Right decision

– Truth

H1 Right decision Type II Error

)()( ErrorIITypePErrorITypeP

Goal: Keep , reasonably small

Example

So, We have 3 kinds of hypothesis :

Steps in aplllying Hypothesis testing :

01

00

01

00

01

00

:

: .

:

: .

:

: .

H

Hc

H

Hb

H

Ha

i. State The Hypothesis

ii. Choose a significance level

Two Tailed Test Two Tailed Test • H0: μ = μo• H1: μ ≠ μo

Rejection Region H0 Rejection Region H0 Acceptance Region H0

½ α ½ α

iii. H0 accepted if: -z1/2α < z < z1/2 α

One Tailed Test (right)• H0: μ = μo• H1: μ > μo Critical region

Rejection Region H0

Acceptance region H0

α

iii. H0 accepted if : z ≤ z α

One tailed (left)• H0: μ = μo• H1: μ < μo

(critical region) Rejection Region H0 Acceptance Region H0

α

iii. H0 accepted if: z ≥ -zα

iv. Calculation :

unknown , 0

0

nsX

Z

n

XZ

5.515.48 xSuppose :

057434.0028717.0028717.090.190.1

90.110/5.2

505.5190.1

10/5.2

505.48

,5.210,n suppose

50with ,5.5150with ,5.48

trueis ,Reject

Error I Type

21

00

ZPZP

zz

XPXP

HHP

P

Implies that 5.76% of all random samples would lead to rejection of the Howhen Ho is true

2643.0

0000.02643.0

52,5.515.48

63.010/5.2

525.51

43.410/5.2

525.48

52,5.515.48

12

2

1

zPzP

XP

z

z

XP

I. Test on the Mean of a normal Distribution, variance knownI. Test on the Mean of a normal Distribution, variance known

• Suppose that we wish to test the hypothesis :

• Test Statistics :

Example