Testing of Hypothesis1

7/31/2019 Testing of Hypothesis1

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Sampling andSampling Distributions


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A statistical population is the aggregate of all the unitspertaining to a study

.i.e. it is the set of all elements about which we wish to

make inferences.

A sample is a subset of a population.

The process of drawing a sample from a large

population is called sampling.

STATISTIC: Characteristic or measure obtained from asampl

e.

PARAMETER: Characteristic or measure obtainedfrom a population.

A sampling distribution is the probability distribution,under repeated sampling of the population, of a givenstatistic.


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Consider a very large population.

Assume we repeatedly take samplesof a given size from the population

and calculate the sample mean for

each sample.Different samples will lead to

different sample means.The distribution of these means is

the sampling distribution of the

sample mean.


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When all of the possible sample means arecomputed, then the following properties are true:

The mean of the sample means will be the meanof the population ().The variance of the sample means will be the

variance of the population divided by the samplesize (2/n).

The standard deviation of the distribution of a samplestatistic is known as thestandard error of the statistic.

The nature of the sampling distribution depends onthe distribution of the population and/or thestatistic being considered and the sample sizeused.


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Testing of Hypothesis


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Hypothesis is an assumption about a population

A few examples are as follows:

1. Mean purchases made by females (1) is more than

or equal to the mean purchases made by males (2)

in a textile stores (1 > 2).

2. Mean age of female shoppers (1) is less than orequal to that of male shoppers (2) in a book

exhibition (1 < 2).

3. Mean monthly income of buyers () in a shop ismore than or equal to Rs 10000\- ( > 10000).

4. The mean stay-over time of customers () in a shop

is at most 45 minutes ( < 45).


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Definitions

Parameter: It is a function of population values.

Statistic: It is a function of sample values.

Null Hypothesis: It is an assumption about the

population parameter which the statement of nochange. It is denoted by H0.

Alternate Hypothesis: It is the statement of

assumption which can be considered to be thealternative to the null hypothesis is called thealternative hypothesis. It is denoted by H1.


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As long as there is no apparent contradiction tothe null hypothesis, we retain this belief. But,

when we find observations contradicting it, thereis a reason to suspect the validity of this nullhypothesis and the problem of testing the nullhypothesis arises.

When we proceed to test H0, we must be awareof the assumption that is expected to be valid if

null hypothesis turns out to be valid if nullhypothesis turns out to be invalid. Thisassumption is known as alternative hypothesis.


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H0: The mean I.Q. of all persons in a city is 105

H1: The mean I.Q. of all persons in the city is 100

(if it is known that the mean I.Q. is 105 or 100 andnothing else)OR

H1: The mean I.Q. of all the persons in the city is lessthan 105

(if it is known that the mean I.Q. is not more than 105)OR

H1: The mean I.Q. of all the persons in the city is morethan 105

(if it is known that the mean I.Q. is not less than 105)ORH1: The mean I.Q. of all the persons is not equal to 105

(if any information is absent)


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The first thing to do when given a claim is to

write the claim mathematically (if possible), and

decide whether the given claim is the null or

alternative hypothesis.

If the given claim contains equality, or a

statement of no change from the given or

accepted condition, then it is the null hypothesis,

otherwise, if it represents change, it is thealternative hypothesis.


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Example

"He's dead, said Dr. X to Captain K.

Mr. S, as the science officer, is put in charge ofstatistically determining the correctness of Xs'statement and deciding the fate of the crew member(to vaporize or try to revive)

His first step is to arrive at the hypothesis to be

tested.Does the statement represent a change in previouscondition?

Yes, there is change, thus it is the alternativehypothesis, H1No, there is no change, therefore is the nullhypothesis, H0


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The correct answer is that there is change.

Dead represents a change from the acceptedstate of alive.

The null hypothesis always represents no

change.

Therefore, the hypotheses are:

H0: Patient is alive. H1: Patient is not alive (dead).


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PROCEDURE IN HYPOTHESIS TESTING

1.Formulate the Hypothesis: Set up a null hypothesis basedon the belief and an appropriate alternate hypothesis.

2. Set up a Suitable Significance Level: The confidence withwhich a null hypothesis is rejected or accepted depends uponthe significance level used for the purpose.

A level of significance say 5% means the risk of making awrong decision is only in 5 out of 100 cases. Level ofsignificance widely used is 5% or 1%. Thus, a 1% level of

significance provides greater confidence to the decision than a5% significance level as the risk of making wrong decision isonly in 1 out of 100 cases. It is denoted by a Greek alphabet

alpha (). Where (1)is the CONFIDENCE LEVEL.


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3. Select Test Criterion: The test criterion is selectedon the basis of sample size. If the sample is large (n 30), the z-test implying normal distribution is used;

whereas if the sample size is small (n < 30), the t-testis more suitable. The most commonly used tests are z,t, F and 2.

A corresponding TEST STATISTIC is calculated.

4. Decision Criterion: The Test Statistic calculated inthe previous step is now classified to fall within theacceptance region or the rejection region at the given

level of significance. Accordingly the null hypothesisis accepted or rejected.

5. Conclusion: On the basis of the decision theconclusion is stated.


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ERRORS IN DECISION MAKING

The problem of testing of a hypothesis isactually a problem of deciding whether toaccept or to reject the null hypothesis H0, infavor of alternate hypothesis H1.

The decision of rejecting or accepting of thenull hypothesis is taken on the basis of

observations made only on a sample of unitsselected from the population. This decisioncannot be always correct. When this decisionis not correct, an error is said to occur.


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States of nature are something that you, as a

decision maker has no control over.

Either it is, or it isn't. This represents the truenature of things.

Possible states of nature (Based on H0)

Patient is alive (H0 true - H1 false )

Patient is dead (H0 false - H1 true)


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Decisions are something that you have controlover.

You may make a correct decision or an incorrectdecision.

It depends on the state of nature as to whetheryour decision is correct or in error.

Possible decisions (Based on H0) / conclusions(Based on claim)

Reject H0 / "Sufficient evidence to say patient

is dead"

Fail to Reject H0 / "Insufficient evidence to

say patient is dead"


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Statistically speakingState of NatureDecision H0 True H0 False

Reject H0 Patient is alive,

Sufficient evidence

of death

Patient is dead,

Sufficient evidence

of deathFail to

reject H0

Patient is alive,

Insufficient evidence

of death

Patient is dead,

Insufficient evidence

of death


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In English...State of Nature

Decision H0 True H0 False

Reject H0 Vaporize a live

person

Vaporize a dead

personFail to

reject H0

Try to revive a

person

Try to revive a

person


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Following table gives the

possibilities that exist in reality.

Null Hypothesis H0 is

TrueNot True

Decision

Reject H0 Type I Error No Error

Do not reject H0 No Error Type II Error


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Type I Error

Reject H0, when H0 is True

Type II ErrorDo Not Reject H0, when H0 is Not True

Which of the two errors is more serious?

Type I or Type II?


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Critical Region: The setting up of a decision criterion

involves partitioning the set of possible values of the

test statistic into two subsets; one of which is attributedto the decision: Reject H0 and the other to the decision:

Do not reject (Accept) H0.

For example, Reject H0 if x < 50

And Accept H0 if x 50

Critical Region for a test is the region whichcorresponds to the subset of sample space for which the

hypothesis H0 is to be rejected if a sample point falls in

the region.


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Level of significance, Test of Significance and Power of a

test

To design a good test we would like to arrive at adecision criterion in such a way that none of the twoerrors, (Type I Error and Type II Error) occur.

But when P(Type I Error) 0, P(Type II Error) 1& when P(Type II Error) 0, P(Type I Error) 1

Hence, no test can be perfect. We therefore design atest such that one of the two probabilities is restrictedto a small value (0 < < 1 and is closer to 0) andthen minimize probability of the other error.


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The error in rejecting H0, when it is true (Type IError) is more serious error than (Type II Error),therefore an upper limit is put on P(Type I Error)

and P(Type II Error) is simultaneously minimized.This upper limit is known as level of significance.

Thus, if a test is so designed that

P(Type I Error) <

then is called level of significance and the test sodesigned is called a test of significance.

Hence, = Max. P(Type I Error).

And P(Type II Error is not committed)= 1P(Type II Error)

measures strength of a test and it is known as powerfunction or the power of the test.


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Tails of a testThe rejection region in hypothesis testing can be on both sides ofthe curve with the non-rejection region in between the two

rejection regions.A hypothesis test with two rejection regions is called a two-tailtest and a test with one rejection region is called a one-tail test.

The one rejection region can be either on the right or the left sideof the curve.

If the rejection region is on the right side of the curve, the test isknown as the right-tail test.

When the test has a rejection region on the left side, then it isknown as the left-tail test.

To find out whether a particular test is one-tail or two-tail and if itis one-tail, is it left-tail or right tail test, we use the sign in thealternative hypothesis. If the alternative hypothesis has a sign, itis a two-tail test.


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Two-Tail Test To test the hypothesis that the average

monthly income of households in a certain town is Rs.

5000/-;

H0: = 5000

H1: 5000

= 5000

Non-rejection

regionRejection

region

Rejection

region

Area = /2 Area = /2


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b) H0: = 10H1: > 10

= 10

Rejection

region

Area = Non-rejection region


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Signs in the tails of a test

Two-tail

Test

Left-tail

Test

Right-tail

Test

1.Sign in the null hypothesis

H0= = or = or

2.Sign in the alternate

hypothesis H1

< >

3. Rejection regionIn both

tails

In the

left tail

In the

right tail


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DECISION CRITERIONIn p-value of the test statistic is lessthan the level of significance , reject

H0.


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Distributions used intesting of hypothesis

In order to test different parameters, for

different sample sizes and comparisons of

such parameters for multiple populations,

different statistical distributions are used.


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Testing of Hypotheses

One SampleTests

Testfor

Mean

Test forProportion

Two SampleTests

Testfor

Mean

Test forProportion


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Testing ofHypothesis

Large SampleTests (n > 30)

Use z-test

Small SampleTests ( n < 30)

Use t-test

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Testing of Hypothesis1