1 testing of hypothesis two sample test dr. t. t. kachwala

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Testing of Hypothesis Two Sample test

Dr. T. T. Kachwala

Objective of Hypothesis Testing for differences between Means:

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In many decision making situations, we need to determine whether the

parameters of the two populations are alike or different. For example:

i)Do female employees earn less than male employees for the same

work in a company?

ii)Do students of one division score more marks than other division?

In the above examples, we are concerned with the parameters of two

populations. We are not interested in the actual values of the population

means as we are in the relation between the values of the two population

means, i.e. how these population means differ.

The objective of two sample test is to assess whether or not there is a

significant difference between the two population means

Hypothesis Testing for difference between Means Procedure (Large Sample Size – z test):

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In testing of difference between two means, we must choose whether to use a one-tailed hypothesis test or a two-tailed hypothesis test.

If the test concerns whether two means are equal or are not equal, use a two tailed test.

However, if the test concerns whether one mean is significantly higher or significantly lower than the other, a one-tailed test is more appropriate.

H0 : 1 = 2

H1 : 1 2

H0 : 1 = 2

H1 : 1 2

orH0 : 1 = 2

H1 : 1 > 2

Hypothesis Testing for difference between Means Procedure (Large Sample Size – z test):

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Step (i) H0 : 1 = 2

H1 : 1 2 (assuming Two Tail Test)

Step (ii) Select α 0.05, select Normal distribution, select two tailed;

zcritical

Step (iii) 21

21

x - x

)XX(

statisticz

2

22

1

21

21 n

n

ss

x - xˆ

Hypothesis Testing for difference between Means (Large Sample Size – z test):

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x

z stat Accept H0

x

z stat Reject H0

Accept H1

If ‘z’ is in acceptance area, Accept H0 : 1 = 2 and we

conclude that there is no significant difference in the population mean.

If ‘z’ is in the rejection area, Reject H0 : 1 = 2

Accept H1 : 1 2 and we

conclude that the population means differ significantly.

Step (iv) & (v): Decision Rule & Conclusions

Hypothesis Testing for difference between Means Small Sample Size – t test: (assuming variations of two population are equal)

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When sample sizes are small, there are two changes in our procedure for testing the differences between means:

(i)Estimated standard error of the difference between two Sample mean

ii) Use of t distribution.

21

x - x

The following is the procedure for t test

Step (i) H0 : 1 = 2

H1 : 1 2 (assuming Two Tailed Test)

Step (ii) α = 0.05; t distribution, 2 tailed, = n1+n2 - 2t critical = (from table)

Note: Degree of freedom for two samples: = n1 – 1 + n2 – 1 = n1 + n2 - 2


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Step (iii) Calculation of tstatistic

t

21

21

xx

- statistic

-

XX

N

S ˆ

21 x - x

Where ‘S’ is the pooled estimate of standard deviation (weighted average), N is the combined sample size

2nn -

x x S

21

2

2

2

1

222

111

X - X

X - X

x

x

21

21

nnnn

N


8

x

t stat Accept H0

x

t stat Reject H0

Accept H1

Step (iv) & (v): Decision Rule & Conclusion

If ‘t’ is in acceptance areaAccept H0:

Conclusion: There is no significant difference in the population means.

If ‘t’ is in rejection areaReject H0:

Accept H1:

Conclusion: There is a significant difference in the population means

Paired difference Test (Dependent Samples) - Objective

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1. In the earlier examples, our samples were chosen independently of each

other; for example students from two different colleges, light bulbs of two

different manufacturers, height of sailors and soldiers, sample of chicken

with high protein diet and another sample with low protein diet etc.

2. Sometimes however it makes sense to take samples that are not

independent of each other. Often the use of such dependent (or paired)

samples enable us to perform a more precise analysis because they will

allow us to control for extraneous factors.

3. With dependent samples, we follow the same basic procedure that we

have followed when testing hypothesis about a single mean.

Paired difference Test (Dependent Samples)

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Step (ii) t distribution (critical value depends on Degree of Freedom , level

of significance and single tailed or two tailed test).

H0: = H0 {H

0 = d = Mean of difference value of the population}

H1: H0

Step (i)

Procedure

t critical = (from table)

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Step (iii)


nxs

ˆ

1-n x2

s

xˆ

t 0Hμ -Xd

Calculation of test statistics

{H0 = d }


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Step (iv)& (v) Decision Rule & Conclusion: There are two possibilities as indicated in diagram below:

x

t stat Accept H0

x

t stat Reject H0

Accept H1There is no significant difference between Sample Mean &

There is a significant difference between Sample Mean & 0Hμ

0Hμ

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Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Objective:

In many decision making situations, we need to determine whether the

proportions of the two populations are alike or different. For example:

i)Is the proportion defective of the first factory different from the

proportion defective of the second factory?

ii)Is the proportion of wheat consumers of one town different from the

proportion of wheat consumers of the second town?

In the above examples, we are concerned with the proportion of two

populations. We are not interested in the actual values of the proportion

as we are in the relation between the values of the two proportions, i.e.

how these proportions differ.

The objective of two sample test is to assess whether or not there is a

significant difference between the two population proportions

Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Procedure:

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In testing of difference between two proportions, we must choose whether to use a one-tailed hypothesis test or a two-tailed hypothesis test.

If the test concerns whether two proportions are equal or are not equal, use a two tailed test.

However, if the test concerns whether one Proportion is significantly higher or significantly lower than the other, a one-tailed test is more appropriate.

H0 : p1 = p2

H1 : p1 p2

H0 : p1

= p2

H1 : p1

p2

orH0 : p1 = p2

H1 : p1 > p2

Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Procedure:

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Summary Procedure

Step (i) H0 : p1

= p2

H1 : p1 p

2 (assuming Two Tailed Test)

Step (ii) α = 0.05, Normal Distribution, 2 tailed

zcritical=

and Calculate (iii) Stepstatistic

z21

p - p

21

p- p = Standard Error for the difference in proportion of success

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Hypothesis Testing for difference between Proportions (Large Sample Size – z test):

1

1

q p 21

21 nn p - p

σ

21

21

21

2211

n n

a a

n n

n n p

pp

p - 1 q

21

1

p - p

p - p z 2

statistic

is the best estimate of the overall proportion of success in the population (combined proportion of success or weighted proportion of success)p

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Hypothesis Testing for difference between Proportions (Large Sample Size – z test):

Step (iv)& (v) Decision Rule & Conclusion: There are two possibilities as indicated in diagram below:

If zstatistics lies in acceptance area

Accept H0 : p1

= p2

If zstatistics lies in rejection area

Reject H0 : p1

= p2

Accept H1 : p1

p2

Conclusion: There is a significant difference in the population proportions

Conclusion: There is no significant difference in the population proportions

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Thanks and Good Luck

Dr. T. T. Kachwala

1 testing of hypothesis two sample test dr. t. t. kachwala

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