07 testing of hypothesis t and f anova

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Testing of Hypothesis

Definition:

Testing of hypothesis is a procedure which enable us to decide whether to

accept or reject a particular statement or assumption about the population

parameter (s) on the basis of information obtained from sample data.

Types of Hypotheses:

1. Null Hypothesis

2. Alternative Hypothesis

3. Simple Hypothesis

4. Composite Hypothesis

Hypothesis: Hypothesis is astatement or assumption

about the population

parameter under theassumption that it is true.

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Types of Hypotheses

Null Hypothesis and Alternative Hypotheses

A hypothesis which is to be tested for possible rejection under the assumption

that is true is called, null hypothesis. On the other hand, if the null

hypothesis is rejected we consider another hypothesis which is called

alternative hypothesis. The null and alternative hypotheses are denoted

by H0 and H1 respectively. For example:

H0: There is no significant difference between the sale/production of

company A and B (1-2 = 0)

H1:There exist a significant difference between the sale/production of

company A and B (1-2 0)

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Mean Comparison: Testing of

HypothesisMean comparison of two different populations (i.e. two different companies

in terms of sale/production/saving/profit etc) can be done by using:

1. Two sample t-test (t-test for independent samples)

2. Paired t-test (t-test for dependent samples)

For example:

Comparison of mean production of two different companies. In this case both

the samples taken from company A and company B will be independent.

Comparison of daily mean production of company A in year-1 and year-2.

OR: Comparison of daily mean production of company A before and

after using new technology.

OR: Mean comparison before and after taking loan (credit).

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Mean Comparison: Testing of

HypothesisSome Basic Definitions:

1. Significance level

2. Test statistic

3. Critical region and critical values

4. One tailed and two-tailed tests

5. Type-I and Type-II errors

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Steps Involved in Testing of Hypothesis State/formulate the null and alternative hypotheses

Choose the level of significance, generally, 1%, 5% and 10% levels of

significance are used in literature

Choose the test statistic to be used i.e. Z-test, t-test, F-test etc.

Compute the value of test statistic from the sample data and available

information given under the null hypothesis, the value so obtain is called

calculatedvalue.

Define the critical value of the test statistic, called tabulated value; OR calculate

the P-value of the test statistic

Compare the calculated and tabulated values of the test statistic. Reject the null

hypothesis if calculated value of the test statistic is greater than the tabulated

value

Make the decision and conclude the results.

Main Steps in Testing of Hypotheses

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Steps Involved in Testing of Hypothesis

COMPARISON OF TWO MEANS

The t-test for independent samples

Populations variances are identical

Population variances are not identical

Paired t-test (dependent samples)

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The t-test for Independent Samples (populations have

identical variances)

In order to test the hypothesis that there is no significant difference between

the means of two populations, the following test statistic is applied:

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Independent Samples (populations variances are unequal )

In order to test the hypothesis that there is no significant difference betweenthe means of two populations, the following test statistic is applied:

2 2 2

1 1 2 2

2 2 2 2

1 1 2 2

1 2

2 2

1 2

[( / ) ( / )]

( / ) ( / )1 1

, and are the variances of

sample-1 and sample-2 respectively.

s n s n

s n s n

n n

where s s

Which under the null hypothesis has t-

distribution with degrees of freedom,where:

1 2

2 2

1 2

1 2

X Xt

s s

n n

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The t-test for dependent samples (Paired t-test)

0 1 2 0

1 1 2 1

The following hypothesis is considred

H : 0 or H : 0 VS

H : 0 or H : 0

To test the above hypothesis, the following t-test is used:

, which follow a t-distribution with (n-/

d

d

d

d

d

t s n

2

1) degrees of freedom

where, is the mean of " " values and

OR ; is the standard

deviation of all " " values and is computed as:

( )

1

B A A B d

d

dd d

n

d X X d X X s

d

d ds

n

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Company-A 12 13 14 13.5 10 11 12.5 13.8 15 11.6 15 16

Company-B 13 14 11 10 9 8 9.4 11.5 8 7 9 8.5

Example 1: Data showing the Monthly Profit in (0000) of Rs of two companies

Reject the null hypothesis of equal means and conclude the average profit

of both the companies differ significantly.

0 1 2

1 1 2

H : 0 (On the average, the monthly profit of both the companies is same)

VS

H : 0 (On the average, the monthly profit of both the companies is not same)

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Some Questions Regarding Example 1:

1. Write the hypothesis (both null and alternative) that there is no significant

difference between average profit of two companies.

2. Write about the significance of test and what does it indicate, decide on

the basis of P-value?

3. Which test is applied and why?

4. Interpret the result of Levenes F-test and what will be your hypothesis in

this case

5. Write 95% confidence interval for the difference between means (profit)

of the two companies.

6. Why the value of t-statistic for equal variances is considered?

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Company-A Company-B

Profit Production Sale Profit Production Sale

12 120 110 13 112 100

13 140 132 14 132 122

14 150 145 11 145 13213.5 140 123 10 120 100

10 103 90 9 100 70

11 115 100 8 90 80

12.5 123 122 9.4 95 70

13.8 140 135 11.5 115 100

15 160 145 8 90 70

11.6 120 115 7 75 72

15 162 150 9 95 90

16 165 145 8.5 90 88

Example 2: Monthly Profit, Production and Sales of Company-A and B during

one year (12 months data)

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SPSS out put

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Variable

Company-A Company-B

t-ratio P-value

Mean SE Mean SE

Profit 13.12 0.514 9.87 0.613 4.060** 0.001

Production 136.5 5.868 104.92 5.841 3.815** 0.001

Sale 126 5.607 91.17 5.967 4.254** 0.000

SE = Standard Error of Mean; ** indicates significant at 1% level of probability

Results Presentation

Table 1: Average comparison of Company-A and B for the year-XXXXX

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0

20

40

60

80

100

120

140

160

Profit Production Sale

Meanv

alue

Company A Company B

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Example 3: Monthly Profit, Production and Sales of Company-A before and after

adopting a new technology (12 months data)

Before New technology After new technology

Profit Production Sale Profit Production Sale

12 120 110 17 125 115

13 140 132 20 145 140

14 150 145 18 179 160

13.5 140 123 16 158 150

10 103 90 12 110 105

11 115 100 14 134 125

12.5 123 122 13 135 124

13.8 140 135 15 150 145

15 160 145 17 170 160

11.6 120 115 13 145 142

15 162 150 17 170 165

16 165 145 20 170 163

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Some Questions Regarding Example 3:

1. Write the null and alternative hypotheses for such a problem.

2. Which test is applied for comparison and why?

3. Write 95% confidence interval for the difference between means (profit,

production and sale).

4. Is there any impact on the profit, sale and production of adopting the new

technology, how, discuss it.

5. What does the p-value (sig.) indicate and how you will utilize this value in

results interpretation.

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Results Presentation

Variable

Before New Tech. After New Tech.

t-ratio P-value

Mean SE Mean SE

Profit 13.12 0.514 16 0.769 -5.436** 0.000

Production 136.50 5.868 149.25 6.064 -5.413** 0.000

Sale 126.00 5.607 141.17 5.771 -6.407** 0.000

SE = Standard Error of Mean; ** indicates significant at 1% level of probability

Average comparison of different items of the company before and after

adopting a new technology

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0

20

40

60

80

100

120

140

160

Profit Production Sale

Meanvalue

Company A Company B

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Steps Involved in Testing of HypothesisCOMPARISON OF MORE THAN TWO

MEANS

Analysis of Variance (ANOVA) technique

One-way ANOVA

Two-way ANOVA

Multi-way ANOVA

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ONE-WAY ANOVA (Analysis of Variance):

In case of One-way ANOVA, the data is classified according to one criteria, e.g.

profit, sales, production of more than two companies; different marketing policies

adopted by the same company etc. It (ANOVA) partition the total variation into

different components (between groups and within groups) of variation. i.e.

SS Total = Between SS + Within SS OR

SS Total = SS Treatments + SS ErrorSS Total = SSTr + SSE

ANOVA TABLE

SOV df SS MS F-ratio

Between groups (t-1) Bet. SS (Bet. SS)/(t-1) = MSB MSB/MSE

With in groups t(r-1) SS Error (SS Error)/t(r-1) = MSE

Total (tr-1) SS Total (SS Total)

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Company-A 40 38 35 42 44 37

Company-B 20 22 18 23 25 24

Company-C 45 46 50 48 54 56

Example 4: Profit (0000) of three different companies for every two months of

a particular year. Analyze the data and draw your conclusions.

Which test you will apply and why?

Is it possible to apply t-test, if Yes/No, why, explain.

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One-Way ANOVA Results: SPSS Out Put

Maximum profit = Company C

Minimum Profit = Company B

The P-value in the ANOVA

Table shows that the profits

of three companies are

significantly different.

Maximum

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Pair-wise comparison (application of two samples t-test)

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Pair-wise comparison -continued

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Example: Profit (0000) of three different companies for every two months of a

particular year

Company-A 12 10 9 15 8 12

Company-B 14 17 16 11 14 12

Company-C 18 19 17 14 12 19

Company-D 20 22 15 13 11 16

Compare the average profit of these companies at 5% level of significance and test the

hypothesis that, is there any significant difference among the profits of these companies?

Also apply LSD (least significant difference) test and separate the mean profits which

are significantly different from one another.

Compare means

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Significant (P < 0.05)

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TWO-WAY ANOVA (Analysis of Variance):

In two-way ANOVA, the data is classified according to two criteria (describing

one at rows and other at columns) e.g. sales of a particular commodity ofdifferent companies at various cities of the country; different marketing policies

adopted by a group of companies; etc. In this case

SS Total = Row SS + Column SS + Error SS

Or SS Total = SSR + SSC + SSE

Company

Marketing Policy

I II III IV

Company-A 40 38 35 42

Company-B 60 55 50 47

Company-C 45 46 43 48

Example: Sale of three different companies by adopting four different marketing policies.

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07 testing of hypothesis t and f anova

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