RM PREZENTATION

8/8/2019 RM PREZENTATION

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HYPOTHESISTESTS

PRESENTED BYPRIYA CHOPRA

097619


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A hypothesis is anassumption about thepopulation parameter.

A parameter is acharacteristic of thepopulation, like itsmean or variance.

The parameter must beidentified beforeanalysis.

I assume the mean GP A of this class is 3.5!

1984-1994 T/Maker Co.

W hat is a Hypothesis?


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States the A ssumption (numerical) to be tested

e.g. The grade point average of our class is at

least 3.0 ( H

0: Q u 3.0)Begin with the assumption that the null

hypothesis is TRUE.

The Null Hypothesis, H 0

Refers to the Status QuoA lways contains the = signThe Null Hypothesis may or may not be rejected.


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Is the opposite of the null hypothesise.g. The grade point average of our class is

less than 3.0 ( H 1 : Q < 3.0)C hallenges the Status Quo.Never contains the = signThe A lternative Hypothesis may or may

not be accepted.It is generally the hypothesis that is believed tobe true by the researcher.

The A lternative Hypothesis, H 1


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L evel of Significance, E and

the Rejection Region H

0: Q u 3 H 1 : Q < 3

0

0

0

H 0: Q e 3 H 1 : Q > 3

H 0: Q ! 3

H 1 : Q { 3

E

E

E /2

C riticalValue(s)

RejectionRegions


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Decision Errors

Educated guesses can be wrong.Def: Drawing a false conclusion from an hypothesis test Never know for sure if a difference is due just to sampling error

or if it truly reflects a treatment effect.

Two Types Type I: Falsely rejecting null

Seeing something thats not there. False positive.

Type II: Falsely retaining null M issing something that is there. False negative.


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Decision Errors Example

Di d the tra ining work or i s th i s group of people just more talented than usual?You implement a training program to improve job performance,and then compare the performance of trainees to average

performance. Youd make a mistake if you.

a. Conclude participants dont differ from average, but inreality the training DOES improve performance.

Type II error

b. Conclude participants do better than average, but in realitythe training does NOT improve performance.

Type I error


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H 0 is true H 0 Is not true

H 0 Is rejected Type I error No error

H0 Is notrejected No error Type II error


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HYPOTHESIS TEST

A statistical hypothesis test is a method of makingstatistical decisions using experimental data.

Critical tests of this kind may be called tests of significance, and when such tests are available we maydiscover whether a second sample is or is not

significantly different from the first.


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Two-Tailed Test: A test of a statistical hypothesis, wherethe region of rejection is on both sides of the samplingdistribution, is called a two-tailed test .

For the same example :

H0: = 50,

against

H1: not equal to 50


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Z TEST

z- test is based on the normal probability distribution andis used for judging the significance of several statisticalmeasures, particularly the mean.

The relevant test statistic, z , is worked out and compared

with its probable value (to be read from table showingarea under normal curve) at a specified level of significance for judging the significance of the measureconcerned.

U sed even when binomial distribution or t -distributionis applicable on the presumption that such a distributiontends to approximate normal distribution as n becomeslarger.


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U sed for comparing the mean of a sample to somehypothesised mean for the population in case of largesample, or when population variance is known.

z -test is also used for judging he significance of

difference between means of two independent samples incase of large samples, or when population variance isknown.

z -test is also used for comparing the sample proportion

to a theoretical value of population proportion or for judging the difference in proportions of two independentsamples when n happens to be large.


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EXA M PLE

A company manufacturing tyres finds that the tyre life isnormally distributed with a mean life of 40000 km and

the standard deviation of 3000 km. It is believed that thechange in production process will result in a better

product and the company has developed a new tyre. Asample of 64 new tyres has been selected. The companyhas found that the mean life of these new tyres is 41200km. Can it be concluded that the new tyre is significantly

better than the old one?


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SOL U TIONWe would like to test whether the mean life of the tyres has increased beyond 40000

km. Null hypothesis and alternative hypothesis are as under:

H0: = 40000 km

H1: > 40000 km

Significance level is taken as 0.05

Computation

OR

= 412000-400000/(3000/8)= 3.2

Decision At 0.05 level of significance, the critical value of z is -+ 1.64. Thecomputed value of z = 3.2 lies in the rejection region.

Therefore we reject the null hypothesis that mean is equal to 40000km and accept thatthe mean value is greater than 40000 km.


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T TEST

V ery similar to z-scores Provides way of judging how extreme a sample mean is A bunch of t-scores form a t-distribution

Done when is unknownU

sed for hypothesis testing: Ex: You wonder if college students really get 8 hours of sleepHo: = 8 (College students do get eight hours of sleep)Ha: { 8 (College students dont get eight hours of sleep)

t-distribution provides foundation for t-test

can do by hand with table can do on SPSS

K ey difference: t-test done when is unknown


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Review: Different M easures of Stand. Dev.

nn

x x

x

!

2

2

W

Have all thescores in apopulation

E.g., SAT scores (ETS has

every single score).

Have onlyscores in asample , wantto estimatevariability in

population

E.g., hours of sleepstudents in this class sleptlast night

(N eed to adjust becauseweve only got sampledata.)

1

22

! n n

x x

s x

Calculate differently based on available information


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Different M easures of Sampling Error

If x is known, do z-testU se x to get measure of sampling error indistribution.

If x is not known, do t-testU se x to get measure of sampling error indistribution.

x

x x

x z

son

W

Q

W W

!

!

...

x

x x

s x

t

son

s s

...

Q!

!


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t-distributions vs. z-distributions


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Comparing Frequency & Sampling Distributions (T1)

F requency D-z Sampling D z Sampling D - t

Havex s xbars xbars

Compare

Amt. of V ariab. +

M eas. of V ariab.

FormulaW

Q!

x z

x

x z

W

Q!

Q x Q x Q x

xW x s

x s x

t

Q!

xW


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EX AM P L E:

A biscuit manufacturer wishes to compare the performance of two biscuit production lines, A and B. Thelines produce packets of biscuits with a nominal weight of 300 grams. Two random samples of 15 packets each fromthe two lines are weighed (in grams). The sample data issummarised as follows:

Line A mean x1 = 309.8; standard deviation SI = 3.5

Line B mean X2 = 305.2; standard deviation S2= 7.0

Carry out a t-test on this sample data to test whether thetwo production lines produce packets of different weights.


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SOL U TIONNull Hypothesis: Two production lines produce packetsof same weights.

A lternative Hypothesis: Two production lines produce packets of different weights.

The value of the test statistic

The table shows that the probability of getting a t-value of 2.27 with28 degrees of freedom is less than 0.05. Thus, the hypothesis isrejected. That is, the two production lines do appear to produce

packets of different weights.


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F TEST

F - test is based on F -distribution and is used tocompare the variance of the two-independent samples.

This test is also used in the context of analysis of variance (ANO VA) for judging the significance of morethan two sample means at one and the same time. It isalso used for judging the significance of multiplecorrelation coefficients.


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IM PORT A NC E

The application of hypothesis testing in quality management is being promoted. Both parametric test (t-test and z-test) andnonparametric test (sign test and Wilcoxon rank-sum test) areappropriate for use in a manufacturing environment.

Data collection establishes the foundation for appraising quality of a product or service. But without correct data processing, it becomeschallenging to make an objective conclusion.

U sed widely in production processes to determine a better process.E.G pharmaceutical industry.

U sed in Psychology.

U sed in HR processes like determining the impact of new trainingmethod versus the old method.


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L IM IT A TIONS(i) The tests should not be used in a mechanical fashion. It should be

kept in view that testing is not decision-making itself; the tests are onlyuseful aids for decision-making. Hence proper interpretation of statistical evidence is important to intelligent decisions.

(ii) Test do not explain the reasons as to why does the difference exist,say between the means of the two samples. They simply indicatewhether the difference is due to fluctuations of sampling or because of other reasons but the tests do not tell us as to which is/are the other reason(s) causing the difference.

(iii) Results of significance tests are based on probabilities and as suchcannot be expressed with full certainty. When a test shows that adifference is statistically significant, then it simply suggests that the

difference is probably not due to chance.(iv) Statistical inferences based on the significance tests cannot be saidto be entirely correct evidences concerning the truth of the hypotheses.This is specially so in case of small samples where the probability of drawing erring inferences happens to be generally higher. For greater reliability, the size of samples be sufficiently enlarged.


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CONCL U SION

In social science, where direct knowledge of population parameter(s) is rare, hypothesis testing isthe often used strategy for deciding whether a sampledata offer such support for a hypothesis thatgeneralisation can be made.

Thus hypothesis testing enables us to make probability statements about population parameter(s).

Decision-makers often face situations wherein they

are interested in testing hypotheses on the basis of available information and then take decisions on the basis of such testing.

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