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It is also called as Distribution Free Test. This means population may not be normal. Hence, the null hypothesis should be taken as-
H0 = Observations are in good agreement with hypothetical distribution.
Suppose Oi be observed frequency and ei be expected frequency where, i = 1, 2, …… n. the statistic will be written as-
X2=
The above equation follows chi-square distribution. It will be made with (n – 1) degree of freedom. When this calculated value becomes greater than tabulated value, null hypothesis will not be accepted (at level of significance ‘a’).
8A: Chi-Square Test of Independence
Consider r x c table with following data to name it as contingency table-
| c1 | c2 | c3 | Total | |
| r1 | O11 | O12 | O13 | rt1 |
| r2 | O21 | O22 | O23 | rt2 |
| r3 | O31 | O32 | O33 | rt3 |
| Total | ct1 | ct2 | ct3 | Grand Total |
Where,
ri and ci = Attributes
Oij = Observed Frequencies
Hence, expected frequency can be calculated by using the formula,
eij=
eij= , etc.
The above equation will follow the chi-square distribution with (c – 1)(r – 1) degree of freedom. As attributes are independent, the null hypothesis will be-
H0 😛ij = P12 ….. = Pic, i = 1, 2, 3….r
Where,
Pijis the probability of obtaining observation that belongs to i-th row and j-th column.
Thus,
If X2cal> X2tab, then H0 will be rejected and
IF Xcal< X2tab, then H0 will be accepted.
8B. A 2 x 2 Table (Simplified Form)
| Attribute B | Attribute A | Total | |
| a | b | R1 = a + b | |
| c | d | R2 = c + d | |
| Total | c1 = a + c | c2 = b + d | Grand Total = N
= R1 + R2 = c1 + c2 |
Here, statistics is calculated as-
X2 =
The degree of freedom will be (2 – 1) (2 – 1) = 1
8C. Yate’s Correction
Suppose the degree of freedom is one for four cell frequencies. If the total rows and columns are fixed, the Yate’s Correction has suggested the following-
- ad >bc, reduce a and d by the value 0.5 and increase the other two (b and c) by 0.5
- ad <bc, increase a and d by the value 0.5 and reduce the other two (a and d) by 0.5
X2 (Corrected) =
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
Links of Next Statistics Topics:-
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory
