In today’s class we learnt to calculate chi-square value
manually and discussions were made.
Cross Tab or chi-square:
Every time we calculate chi-square we assume a hypothesis.
It is also called as the null hypothesis. If the chi-square value is high in
accordance with the standard value given in the table we reject the null
hypothesis else we accept it.
Let us understand this concept better by using an example.
Example:
Find the relation between a person’s background and the
marks scored.
Marks Scored out of 20
|
Commerce
|
Engineering
|
12.5
|
0
|
1
|
14
|
0
|
1
|
17
|
0
|
1
|
19.5
|
0
|
1
|
20
|
6
|
2
|
Total
|
6
|
6
|
Answer:
Step 1: Decide the null hypothesis
There is no relation between marks scored and the background
of the students.
Step 2: Calculate the expected frequency
Redraw the table to find the expected frequency
Marks Scored out of 20
|
Commerce
|
Engineering
|
12.5
|
1/12 x 6/12 x 12 = 0.5
|
1/12 x 6/12 x 12 = 0.5
|
14
|
1/12 x 6/12 x 12 = 0.5
|
1/12 x 6/12 x 12 = 0.5
|
17
|
1/12 x 6/12 x 12 = 0.5
|
1/12 x 6/12 x 12 = 0.5
|
19.5
|
1/12 x 6/12 x 12 = 0.5
|
1/12 x 6/12 x 12 = 0.5
|
20
|
8/12 x 6/12 x 12 = 4
|
8/12 x 6/12 x 12 = 4
|
Step 3: Calculate the chi-square values
The formula for chi-square is as follows:
Chi-square= ∑(observed-expected)2/expected
Marks Scored out of 20
|
Chi-Square Value Commerce
|
Chi-Square Value Engineering
|
12.5
|
0.5
|
0.5
|
14
|
0.5
|
0.5
|
17
|
0.5
|
0.5
|
19.5
|
0.5
|
0.5
|
20
|
1
|
1
|
Total of all the chi-square values = 6
Step 4: Find the degree of freedom
Degree of freedom is extremely important when rejecting or accepting
the null hypothesis.
For example, if you have to
take ten different courses to graduate, and only ten different courses are
offered, then you have nine degrees of freedom. Nine semesters you will be able
to choose which class to take; the tenth semester, there will only be one class
left to take - there is no choice, if you want to graduate.
Degrees of freedom are commonly discussed in relation to chi-square and other forms of hypothesis testing statistics. It is important to calculate the degree of freedom when determining the significance of a chi square statistic and the validity of the null hypothesis.
Degrees of freedom are commonly discussed in relation to chi-square and other forms of hypothesis testing statistics. It is important to calculate the degree of freedom when determining the significance of a chi square statistic and the validity of the null hypothesis.
Degree of freedom = (rows-1)(column-1)
Rows= 5
Column= 2
DOF = 4
Step 5: Compare the chi-square value with the standard value
For 4 degree of freedom look for 0.95 probability
For 4 degree of freedom look for 0.95 probability
The value according to the table is 0.711
The original value calculated is 6, which is very high. So, we
reject the null hypothesis.
This implies that there is a relationship between person and
background and marks.
T - test
:
If you have to take ten
different courses to graduate, and only ten different courses are offered, then
you have nine degrees of freedom. Nine semesters you will be able to choose
which class to take; the tenth semester, there will only be one class left to
take - there is no choice, if you want to graduate.
Degrees of freedom are commonly discussed in relation to chi-square and other forms of hypothesis testing statistics. It is important to calculate the degree(s) of freedom when determining the significance of a chi square statistic and the validity of the null hypothesis.
Degrees of freedom are commonly discussed in relation to chi-square and other forms of hypothesis testing statistics. It is important to calculate the degree(s) of freedom when determining the significance of a chi square statistic and the validity of the null hypothesis.
Types of t-Test:
1) One sample t-test
2) Independent sample t-test
3) Paired samples t-test
One Sample t-test:
A one sample t-test means that you have ONE GROUP (e.g., your class of 8th grade students) who you are comparing to A KNOWN MEAN SCORE (say the national mean on a normed test).
Independent Sample t-test:
A two sample t-test means that you have TWO GROUPS (e.g., your class of 8th grade students compared to your LAST YEAR'S group of students).
Paired Sample t-test:
A two sample t-test means that you have TWO GROUPS that you are comparing against one another, but the members of each group are related in some way to a specific member of the other group (e.g., study partners, siblings, married couples, etc.).
The following example was
taken in the class:
MCDonalds
Quality index was created based on various parameters. Each
Parameter has got points like dress, mosquitoes.
Report
Quality Index
Location of Franchise
|
Mean
|
N
|
Std. Deviation
|
Delhi
|
321.998514
|
16
|
.0111568
|
Mumbai
|
322.014263
|
16
|
.0106913
|
Pune
|
321.998283
|
16
|
.0104812
|
Bangalore
|
321.995435
|
16
|
.0069883
|
Jaipur
|
322.004249
|
16
|
.0092022
|
NOIDA
|
322.002452
|
16
|
.0086440
|
Calcutta
|
322.006181
|
16
|
.0093303
|
Chandigarh
|
321.996699
|
16
|
.0077085
|
Total
|
322.002009
|
128
|
.0108224
|
Benchmark Quality index – 322
One-Sample
Statistics
Location of Franchise
|
N
|
Mean
|
Std. Deviation
|
Std. Error Mean
|
|
Delhi
|
Quality Index
|
16
|
321.998514
|
.0111568
|
.0027892
|
Mumbai
|
Quality Index
|
16
|
322.014263
|
.0106913
|
.0026728
|
Pune
|
Quality Index
|
16
|
321.998283
|
.0104812
|
.0026203
|
Bangalore
|
Quality Index
|
16
|
321.995435
|
.0069883
|
.0017471
|
Jaipur
|
Quality Index
|
16
|
322.004249
|
.0092022
|
.0023005
|
NOIDA
|
Quality Index
|
16
|
322.002452
|
.0086440
|
.0021610
|
Calcutta
|
Quality Index
|
16
|
322.006181
|
.0093303
|
.0023326
|
Chandigarh
|
Quality Index
|
16
|
321.996699
|
.0077085
|
.0019271
|
One-Sample
Test
Location of Franchise
|
Test Value = 322
|
||||||||||
t
|
Df
|
Sig. (2-tailed)
|
Mean Difference
|
95% Confidence Interval of the Difference
|
|||||||
Lower
|
Upper
|
||||||||||
Delhi
|
Quality Index
|
-.533
|
15
|
.602
|
-.0014858
|
-.007431
|
.004459
|
||||
Mumbai
|
Quality Index
|
5.336
|
15
|
.000
|
.0142629
|
.008566
|
.019960
|
||||
Pune
|
Quality Index
|
-.655
|
15
|
.522
|
-.0017174
|
-.007302
|
.003868
|
||||
Bangalore
|
Quality Index
|
-2.613
|
15
|
.020
|
-.0045649
|
-.008289
|
-.000841
|
||||
Jaipur
|
Quality Index
|
1.847
|
15
|
.085
|
.0042486
|
-.000655
|
.009152
|
||||
NOIDA
|
Quality Index
|
1.134
|
15
|
.274
|
.0024516
|
-.002154
|
.007058
|
||||
Calcutta
|
Quality Index
|
2.650
|
15
|
.018
|
.0061813
|
.001210
|
.011153
|
||||
Chandigarh
|
Quality Index
|
-1.713
|
15
|
.107
|
-.0033014
|
-.007409
|
.000806
|
||||
Min. values we need in order to predict the remaining
values.
<.05 then reject.
Hence, we reject Mumbai, Bangalore and Calcutta.
Blog written by: Priya Jain (2013210)
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