Statistical Test of Hypotheses

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Statistical Test of Hypotheses

Professor M. Kabir

Department of Statistics,Jahangirnagar University


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Hypothesis

A hypothesis may be defined is simply as

a statement about one or more

populations. The hypothesis is frequently concerned

with the parameters of the populations

about which the statement is made.


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Hypothesis

A hospital administrator may hypothesize thatan average length of stay of patient admittedto the hospital in five days;

A public health nurse may hypothesize that aparticular educational program will result inimproved communication between nurse andthe patient

A physician may hypothesize that a certaindrug will be effective in 90% of the cases forwhich it is used.


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Hypothesis

By means of hypothesis testing onedetermines whether or not suchstatements are compatible with available

data.

Types of Hypotheses

There are two types of hypotheses

- Research hypotheses

- Statistical hypotheses


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Hypothesis

The research hypothesis is the conjecture or

supposition that motivates the research.

Research hypotheses lead directly to statistical

hypotheses.

Statistical hypotheses are hypotheses are

stated in such a way that they may me

evaluated by appropriate statisticaltechniques.


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Hypothesis

Steps in Hypothesis Testing

Data

Assumptions

Hypotheses

Test Statistic

Distribution of Test Statistic

Decision Rule


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Hypothesis

There are two statistical hypotheses involved inhypotheses testing. These are null hypothesesand alternative hypotheses.

A null hypothesis specifies a hypothesized realvalue, or values for a parameter.

It is denoted by the symbol Ho. The nullhypothesis is sometimes referred to as a

hypothesis of no difference, since it is astatement of agreement with conditionspresumed to be true in the population ofinterest.


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Hypothesis

An alternative hypothesis specifies a real valueor range of values for a parameter that will beconsidered when the null hypothesis is

rejected.The alternative hypothesis is a statement of what

we will believe is true if our sample data causeus to reject the null hypothesis. Usually the

alternative hypothesis and research hypothesisare the same, and in fact the two terms areused interchangeably. We shall designatealternative hypothesis by the symbol Ha.


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Hypothesis

The test statistic is some statistic that may becomputed from the data of the sample. Thetest statistic serves as a decision maker, since

the decision to reject or not to reject the nullhypothesis depends on the magnitude of thetest statistic value

What is rejection region? The rejection region

consists of the set of values of a statistic forwhich the null hypothesis is rejected. Thevalues of the boundaries of the region arecalled the critical values.


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Hypothesis

What is type one error? A type I error occurs

when the null hypothesis is rejected when in

fact it is true. The significance level is the

probability of a type one error when the nullhypothesis is true.

What is type II error? A type II error occurs

when the null hypothesis is not rejected whenit is false


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Hypothesis

The power of a test is the probability of

rejecting the null hypothesis when it is false.

The probability of a type I error is denoted by

, and the probability of a type II error is by

.

The power is defined as

Power= 1probability of type II error Power = 1- .


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Hypothesis

Normal Test when population mean and

variance is known

n

xz

x

/


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Hypothesis

General Formula for Test Statistic

The following is a general formula for a

test statistic that will be applicable inmany of the hypothesis tests discussed

Test statistic = relevant statistic-

hypothesized parameter/ standard errorof the relevant statistic


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Hypothesis

Distribution of Test Statistic

It has been pointed out that the key to

statistical inference is the sampling

distribution.

The distribution of test statistic

for example follows the standard normal

distribution if the null hypothesis is true andthe assumptions are met.

n

xz

/


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Hypothesis

Decision rule:

The decision rule tells us to reject the null

hypothesis if the value of the test statistic that

we compute from our sample is one of the

values in the rejection and to reject the null

hypothesis if the computed value of the test

statistic is one of the values in the non-rejection region.


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Hypothesis

Significance level: The decision as to whichvalues go into the rejection region and whichones go into the non rejection region is made

on the basis of the desired level of significance,designated by .

The term level of significance reflects the factthat hypothesis tests are sometimes called

significance tests, and computed value of thetest statistic that falls in the rejection region issaid to be significant.


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Hypothesis

The level of significance, specifies the area under

the curve of the distribution of the test statistic that is

above the values on the horizontal axis constituting

the rejection region.Types of errors

The error committed when a true null hypothesis is

rejected is called type I error . The type II error is the

error committed when a false null hypothesis is notrejected. The probability of committing type II is

designated by


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Hypothesis

Whenever we reject a null hypothesis there is

always the concomitant risk of committing a

type I error, rejecting a true null hypothesis.

Whenever we fail to reject a null hypothesisthe risk of falling to reject a false null

hypothesis is always present.


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Hypothesis

Statistical Decision

The statistical decision consists of rejecting orof not rejecting the null hypothesis . It is

rejected if the computed value of the teststatistic falls in the rejection region, and is notrejected if the computed value of the teststatistic falls in the non-rejection region.

Conclusion: If Ho is rejected we conclude thatHa is true. If Ho is not rejected we concludethat Ho may be true.


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Hypothesis

The p value is the smallest value of for which the

null hypothesis can be rejected. For Z= -2.12 the p

value is 0.034.

The p value for a hypothesis testing is the probabilityof obtaining when Ho is true, a value of the test

statistic as extreme or more extreme than the one

actually computed.

If p value is less than or equal to , we reject thehypothesis. If p value is greater than , we do not

reject the hypothesis. We accept the hypothesis


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Hypothesis

Hypothesis Accept Ho Reject Ho

Accept Ho Correct Type II error

Reject Ho Type I error Correct


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Steps in Hypothesis Testing

Evaluate data

Review assumption

State hypothesis

Select rest statistics Determine distribution of test statistic

State decision rule

Calculate test statistic

Make statistical decision Do not reject Ho

Reject Ho

Conclude Ho may be true

Calculate Ha is true


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Hypothesis Testing: A Single

Population Mean

We consider the testing of a hypothesis about a

population mean fewer than three different

conditions

When sampling is from a normally distributed

population of values with known variance

When sampling is from a normally distributed

population of values with unknown variance


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Population MeanWhen sampling is from a normally distributed

population and the population variance isknown, the test statistic for testing Ho:

EX. If random sample of size 10 is drawn from anormal population with mean and variance

are respectively 27 and 20 respectively. Canwe conclude the mean age of this population isdifferent from 30 years?

n

x

z

/


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Population MeanEX. If random sample of size 10 is drawn from

a normal population with mean and varianceare respectively 27 and 20 respectively. Can

we conclude the mean age of this population isdifferent from 30 years?

Calculation of test statistic

We have z= (27-30)/ 1.4142 = - 2.12

We reject hypothesis. We conclude thatpopulation mean is different from 30 years.


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Population MeanTesting Ho by means of a confidence interval

The 95% confidence interval of population meanis given by

27 plus-minus 1.96 Square root of 20/1027+ 2.7718, 27-2.7718

The age lies between 29.77 to 24.23 years

Since the interval does not include 30, we say 30is not a candidate for the mean we areestimating and there fore population mean isnot equal =30 and Ho is rejected.


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Population Mean

In general, when testing null hypothesis by

means of a two sided confidence interval, we

reject Ho at the level of significance if the

hypothesized parameter is not contained withthe 100 ( 1-) percent confidence interval.

If the hypothesized parameter is contained

within the interval, Ho cannot be rejected at

the level of significance


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Population Mean

Sampling from a normally distributed

population: Population Variance is unknown

The test statistic for testing Ho: Population

mean= = o is

Statistic is

sample mean- pop population / s/square root of n

Example: Will we be able to conclude that themean BMI for the population is 35 .


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Population Mean

Can we reject the hypothesis that

population mean is equal to 35.

Body Mass Index ( BMI) measurementsfor 14 male subjects are given below.

Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14

BMI 23 25 21 37 39 21 23 24 32 57 23 26 32 45


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Population MeanData: The data consist of BMI measurements on

14 subjects as given above.

Assumptions: The 14 subjects constitute a simple

random sample from a population of similarsubjects. We assume that BMI measurementsin this population are approximately normallydistributed.

Hypotheses: Population mean 35Population is not equal to 35

Test statistic is with d.f is n-1,t= (30.5-35)/2.8434

ns

xt

/

ns

xt

/


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Population MeanThe calculated value of t = -1.58

With 13 degree of the value of t is2.16

Since computed value of t is less than the table

value. We accept the hypothesis. Based on thedata the mean population from which thesample drawn may be 35.

Hypothesis Testing: population standard deviation is

not known

If the population standard deviation is notknown , the usual practice is to use the samplestandard deviation as an estimate. The test

statistic for testing Ho= = o,

ns

xt

/


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Population Mean

then , which when Ho is true , is

distributed approximately as the

standard normal distribution if n is

large.ns

xt

/

ns

xt

/


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Population MeanEx. A study was conducted to describe the

menopausal status , menopausal symptoms,energy expenditure, and aerobic fitness of

healthy midlife women and to determine therelationship among these factors. The meanscore of maximum oxygen uptake for a sample242 was 33.3 with a standard deviation of12.14. The researcher wishes to know if, on thebasis of these data, one may conclude that themean score for a population of such women isgreater than 30

ns

xt

/


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Population Mean

Data Maximum oxygen uptake for 242women with mean = 33.3 and s = 12.14

Assumptions: The data constitute a simplerandom from a population of healthymidlife women similar to those in thesample.

Hypotheses Ho: grater than equal to 30Ha: is greater than 30

ns

xt

/


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Population Mean

Data Maximum oxygen uptake for 242women with mean = 33.3 and s = 12.14

Assumptions: The data constitute a simplerandom from a population of healthymidlife women similar to those in thesample.

Hypotheses Ho: grater than equal to 30Ha: is greater than 30

ns

xt

/


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Population Mean

Given the above test statistic

We have z= ( 33.3-30)/0.7804= 3.3/0.7804

= 4.23We reject the hypothesis since computed

value is greater than table value. We

conclude that the mean score for thesampled population is greater than 30.

ns

xt

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Statistical Test of Hypotheses

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