6 TESTING OF HYPOTHESIS1

8/7/2019 6 TESTING OF HYPOTHESIS1

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Sampling andSampling andSampling DistributionsSampling Distributions

Sampling andSampling andSampling DistributionsSampling Distributions


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

y.

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

A sample is a subset of a population.

The process of drawing a sample from a largepopulation 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 thevariance of the population divided by the samplesize (2/n).The standard deviation of the distribution of a sample

statistic is known as the standard 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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A population comprises of four numbers:

3, 5, 7 and 9

(a) List all possible samples of size 2 thatcan be drawn from the population withoutreplacement.

(b) Show that the mean of the samplingdistribution of sample means is equal to thepopulation mean.

(c) Calculate the standard deviation of the

sampling distribution of sample means andhence, show that it is less than thepopulation standard deviation.


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Testing of HypothesisTesting of HypothesisTesting of HypothesisTesting 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 or

equal 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 ofassumption 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 null

hypothesis 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)OR

H1: 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), anddecide 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 the

alternative 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 betested.

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 based

on 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 u

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 true

nature of things.

Possible states of nature (Based on H0)

Crew member is alive (H0 true /H1 false )

Crew member 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 whether

your decision is correct or incorrect.Possible decisions (Based on H0) / conclusions(Based on claim)

Reject H0 if sufficient evidence to say patient

is dead, is available

Fail to Reject H0 if sufficient evidence to

say patient is dead, is not available


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Statistically speaking State at e

ecisi n e alse

e ect Patient is alive,

Sufficient evidenceof death

Patient is dead,

Sufficient evidenceof death

ail te ect

Patient is alive,

Insufficient evidence

of death

Patient is dead,

Insufficient evidence

of death


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State of Nature

Decision Crew member alive Crew member dead

Vaporize the

crew member

Error Right decision

Try to revive

crew member

Right decision Error


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Following table gives the

possibilities that exist in reality.

Null Hypothesis H0 is

True

Not 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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Level of significance

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 I

Error) is more serious errorthan (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, a test is so designed that

P(Type I Error) <

then is called level of significance

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


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DECISION CRITERION

In p-value of the teststatistic is less than thelevel 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

Testing mean Testing variance


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Testing mean

Singlesample

Sample

size 30

Z test

Sample

size


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Testing

variance

Single sample

Chi Square test

Two samples

F test


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For testing association between two variablesChi-Square test for Independence of

Attributes is used.

Expected frequencies are calculated using the

following formula:

E =

O= Observed frequenciesN

CTRTv


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For fitting a distribution to a given data

Chi-Square test for Goodness of Fit is used

Expected frequencies are calculated

depending upon the distribution.


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Thank You

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6 TESTING OF HYPOTHESIS1