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8/2/2019 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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