HYPOTHESIS HYPOTHESIS TESTINGTESTING

PurposePurpose

The purpose of hypothesis testing is The purpose of hypothesis testing is to help the researcher or to help the researcher or administrator in reaching a decision administrator in reaching a decision concerning a population by concerning a population by examining a sample from that examining a sample from that populationpopulation

HypothesisHypothesis

It is a statement about one or more It is a statement about one or more populationpopulation

It is usually concerned with the It is usually concerned with the parameter of the population about parameter of the population about which the statement is made which the statement is made

Research HypothesisResearch Hypothesis

It is the assumption that motivate It is the assumption that motivate the research. It is usually the result the research. It is usually the result of long observation by the of long observation by the researcher. This hypothesis led researcher. This hypothesis led directly to the second type of directly to the second type of hypothesishypothesis

Statistical HypothesisStatistical Hypothesis

This is stated in a way that can be This is stated in a way that can be evaluated by appropriate statistical evaluated by appropriate statistical technique. technique.

Statistical hypothesisStatistical hypothesis

It is composed of two types:It is composed of two types: Null hypothesis( Ho): It is the Null hypothesis( Ho): It is the

particular hypothesis under test, and particular hypothesis under test, and it is the hypothesis of “no difference”it is the hypothesis of “no difference”

Alternative hypothesis (HAlternative hypothesis (HAA): which ): which disagree with the null hypothesisdisagree with the null hypothesis

Test StatisticTest Statistic

It is a mathematical expression of It is a mathematical expression of sample values which provides a basis sample values which provides a basis for testing a statistical hypothesis . for testing a statistical hypothesis .

The result of this test will determine The result of this test will determine whether we will accept the null whether we will accept the null hypothesis and so the Hhypothesis and so the HAA will be will be rejected , or we reject the null rejected , or we reject the null hypothesis and so the Hhypothesis and so the HAA will be will be accepted. accepted.

ErrorsErrors

There are two possible errors to There are two possible errors to come to the wrong conclusion:come to the wrong conclusion:

Type 1 error: rejection of the null Type 1 error: rejection of the null hypothesis when it is true. It is hypothesis when it is true. It is presented by alpha, which is the level presented by alpha, which is the level of significance, often the 5%, 1%, and of significance, often the 5%, 1%, and 0.1% (0.1% (αα=0.05, 0.01, and 0.001) levels =0.05, 0.01, and 0.001) levels are chosen . The selection depends are chosen . The selection depends on the particular problem on the particular problem

P-valueP-value

It is the smallest value of It is the smallest value of αα for which for which the Ho can be rejected , so it gives a the Ho can be rejected , so it gives a more precise statement about more precise statement about probability of rejection of Ho when it probability of rejection of Ho when it is true than the alpha level, so is true than the alpha level, so instead of saying the test statistic is instead of saying the test statistic is significant or not , we will mention significant or not , we will mention the exact probability of rejecting the the exact probability of rejecting the Ho when it is trueHo when it is true

Steps in conducting Steps in conducting hypothesis testinghypothesis testing

Hypothesis testing can be presented Hypothesis testing can be presented as as NINENINE steps: steps:

11..DataData

The nature of the data whether it The nature of the data whether it consists of counts, or measurement consists of counts, or measurement will determine the test statistic to be will determine the test statistic to be usedused

22..HypothesesHypotheses

Null Hypothesis (Ho): which is the Null Hypothesis (Ho): which is the hypothesis of no difference, and the hypothesis of no difference, and the alternative hypothesis( Halternative hypothesis( HAA))

If we reject the Ho we will say that If we reject the Ho we will say that the data to be tested does not the data to be tested does not provide sufficient evidence to cause provide sufficient evidence to cause rejection. If it is rejected we say rejection. If it is rejected we say that the data are not compatible that the data are not compatible with Ho and support the alternative with Ho and support the alternative hypothesis (Hhypothesis (HAA))

33..Test statisticTest statistic

It uses the data of the sample to reach It uses the data of the sample to reach to a decision to reject or to accept the to a decision to reject or to accept the null hypothesis. The general formula null hypothesis. The general formula for a test statistic is:for a test statistic is:

relevant statistic-hypothesized relevant statistic-hypothesized parameterparameter

Test statisticTest statistic = = ----------------------------------------------------------------------------------

standard error of the relevant standard error of the relevant statisticstatistic

44..Test statistic--ExampleTest statistic--Example

_ _

x -µx -µ

Z=-------------Z=-------------

σ σ √n√n

55 . .Distribution of test Distribution of test statisticstatistic

It is the key for statistical inferenceIt is the key for statistical inference

66 . .Decision RuleDecision Rule

It will tell us to reject the null It will tell us to reject the null hypothesis if the test statistic falls in hypothesis if the test statistic falls in the rejection area, and to accept the the rejection area, and to accept the it if it falls in the acceptance regionit if it falls in the acceptance region

66 . .Decision RuleDecision Rule

The critical values that discriminate The critical values that discriminate between acceptance and rejection between acceptance and rejection regions depends on alpha level of regions depends on alpha level of significance significance

If the value of the test statistic falls in If the value of the test statistic falls in the rejection region area , it is the rejection region area , it is considered statistically significant considered statistically significant

If it falls in the acceptance area it is If it falls in the acceptance area it is considered not statistically significantconsidered not statistically significant

66 . .Decision RuleDecision Rule

Whenever we reject a null Whenever we reject a null hypothesis , there is always a hypothesis , there is always a possibility of type 1 error( rejection possibility of type 1 error( rejection of Ho when it is true). This is why we of Ho when it is true). This is why we should decrease this error to the should decrease this error to the least possible.least possible.

Critical valuesCritical values

The values of the test statistic that The values of the test statistic that separate the rejection region from separate the rejection region from the acceptance regionthe acceptance region

Acceptance regionAcceptance region

A set of values of the test statistic A set of values of the test statistic leading to acceptance of the null leading to acceptance of the null hypothesis hypothesis

( values of the test statistic not ( values of the test statistic not included in the critical region)included in the critical region)

Rejection regionRejection region

A set of values of the test statistic A set of values of the test statistic leading to rejection of the null leading to rejection of the null hypothesishypothesis

77 . .Computed test statisticComputed test statistic

This should be computed and This should be computed and compared with the acceptance and compared with the acceptance and rejection regionsrejection regions

88 . .Statistical decisionStatistical decision

It consists of rejecting or not It consists of rejecting or not rejecting the Ho . It is rejected if the rejecting the Ho . It is rejected if the computed value of the test statistic computed value of the test statistic falls in the rejection area , and it is falls in the rejection area , and it is not rejected if the computed value of not rejected if the computed value of the test statistic falls in the the test statistic falls in the acceptance regionacceptance region

99 . .ConclusionConclusion

If Ho is rejected , we conclude that If Ho is rejected , we conclude that HHAA is true. If Ho is not rejected we is true. If Ho is not rejected we conclude that Ho may be true.conclude that Ho may be true.

Two sided testTwo sided test

If the rejection area is divided into If the rejection area is divided into the two tails the test is called two-the two tails the test is called two-sided test , sided test ,

One sided testOne sided test

If the rejection region is only in one If the rejection region is only in one tail it is called one-sided testtail it is called one-sided test

The decision will depend on the The decision will depend on the nature of the research question nature of the research question being asked by the researcher being asked by the researcher

Single population mean , Single population mean , known population varianceknown population variance

_ _ x -µx -µ Z=-------------Z=------------- σ σ √n√n

Single population mean Single population mean with unknown population with unknown population

variancevariance

_ _ x -µx -µ t =-------------t =------------- ss √n√n

Difference between two Difference between two populations means with populations means with

known variancesknown variances

_ _ _ _

(X(X11 –X –X22) – (µ) – (µ11-µ-µ22))

Z=-------------------------------Z=-------------------------------

√ √ σσ2211 /n/n11 + + σσ22

11 /n/n22

Difference between two Difference between two population mean with population mean with unknown and unequal unknown and unequal

variancesvariances

_ _ _ _ (X(X11 –X –X22) – (µ) – (µ11-µ-µ22)) tt=-------------------------------=------------------------------- √ √ ss22

11 /n/n11 + s + s2211 /n/n22

Difference between two Difference between two population mean with population mean with unknown but assumed unknown but assumed

equal variancesequal variances

_ _ _ _ (X(X11 –X –X22) – (µ) – (µ11-µ-µ22)) tt=-------------------------------=------------------------------- Sp√ 1Sp√ 1 /n/n11 + 1 + 1 /n/n22

Paired t-testPaired t-test

_ _

d -µd -µdd

t =-------------t =-------------

SSdd √n√n

Single population Single population proportionproportion

˜̃

P -PP -P

Z=-------------Z=-------------

√√P(1-P)nP(1-P)n

Difference between two Difference between two population proportionspopulation proportions

˜ ˜˜ ˜ (P(P11-P-P22) –(P) –(P11-P-P22)) Z=-----------------------------------------Z=-----------------------------------------

√√PP11(1-P(1-P11)/n)/n1 1 + P+ P22(1-P(1-P22)/n)/n22

ExampleExample

A certain breed of rats shows a mean A certain breed of rats shows a mean weight gain of 65 gm, during the first 3 weight gain of 65 gm, during the first 3 months of life. 16 of these rats were fed months of life. 16 of these rats were fed a new diet from birth until age of 3 a new diet from birth until age of 3 months. The mean was 60.75 gm. If the months. The mean was 60.75 gm. If the population variance is 10 gm , is there a population variance is 10 gm , is there a reason to believe at the 5% level of reason to believe at the 5% level of significance that the new diet causes a significance that the new diet causes a change in the average amount of weight change in the average amount of weight gainedgained

AnswerAnswer Ho=65Ho=65 HHAA≠ 65≠ 65 Z 1-Z 1-αα/2 /2 αα=0.05 =0.05 Z=1.96Z=1.96

(critical value)(critical value) _ _ x -µ 60.75-65x -µ 60.75-65

Z=---------- = ----------- = Z=---------- = ----------- = -5.38-5.38 σ σ √n √10/ √16√n √10/ √16Sine the calculated values falls in the Sine the calculated values falls in the

rejection region , we reject the Ho, rejection region , we reject the Ho, and accept the Hand accept the HAA

In the above example , if the In the above example , if the population variance is unknown, and population variance is unknown, and the sample Sd is 3.84the sample Sd is 3.84

AnswerAnswer

t t 1- 1- αα/2 /2 =± 2.1315=± 2.1315 df =n-1df =n-1

_ _ x -µ 60.75-65x -µ 60.75-65 t =-------------=------------= - 4.1315t =-------------=------------= - 4.1315 ss √n 3.84/ √16√n 3.84/ √16Sine the calculated values falls in the rejection Sine the calculated values falls in the rejection

region , we reject the Ho, and accept the Hregion , we reject the Ho, and accept the HAA