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Completely Randomized Design (C.R.D.) /One-Way Classification

Consider the following results of k independent random samples each of size n, from k different populations :

 

Population      1 :                    x 1,            x12,     ... x 1n             :           Treatment 1

Population      2 :                    x21,      x22,      .. x2n                          :                   Treatment 2

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Population      k :                    xk1,       xk2,      ... xkn                        :                   Treatment k

 

Here xij refers to the j-th value of the i-th population and the corresponding random variables Xij which are all independent normally distributed with the respective means Jli and the common variance σ2.

 

Consider the model,

xij = µ + αi + eij, i = 1, 2, ... ,k,            j = 1, 2, ...... , n

 

Here, µ is referred to as the Grand mean,

 

αi  is referred to as treatment effects such that ∑ α = 0

 

and eij  random errors are identically distributed as N (0, σ2).

 

Hypothesis :

 

H0 : Population means are all equal

 

i.e.. µ1= µ2= ··· = µk

 

i.e. , samples were obtained from k populations with equal means.

 

Equivalently   αi = 0,  i = 1, 2, ... , i.e., there is no special effect due to any population.

H1 : αi ≠ 0 for at least one value of i.

 

Then we construct the following ANOVA TABLE :

Where                         SST = Total sum of squares

SS(Tr) = Treatment sum of squares

SSE = Error sum of squares,

SST = SS (Tr) + SSE,

Ti = Total of the values obtained for the i-th treatment

T.. = Grand total of all nk observations

MS (Tr) = SS (Tr) / (k-1) (Treatment mean square)

 

MSE  = SSE / k ( n _ 1)          (Error mean square)

 

F = Fcal = MS (Tr) / MSE        which follows F distribution.

 

Let L = level of significance, then

 

Ftab = Fα,α- 1, k (n-1)

 

Conclusion:

 

Reject H0 if     Fcal > Ftab

Accept H0 if    Fcal < Ftab

 

Example 1. A test was given to five students taken at random from the Xth class of three schools of a town. The individual scores are

Carry out the analysis of variance and state your conclusions.

 

Solution.         1.        H0 : α1 = α2 = α3 = 0 (No difference between the schools)

 

H1 : a ≠ 0 for at least one value of i.

 

2.         Let a= 0.01, Here k = 3, n = 5

 

Ftab = F0.01, 2. 12 = 6.93

 

3. Computations:

 

T1• = 77 + 81 + 71 + 76 + 80 = 385

T2• = 72 + 58 + 74 + 66 + 70 = 340

T3• = 76 + 85 + 82 + 80 + 77 = 400

T•• = T 1• + T2 • …. T3• = 1125

 

∑∑ xij2= 85041

 

SST = 85041 – 1 / 1.5  (1125)2 = 666

SS(Tr) = 1/ 5 [ (385?)2+ (340)2 + ( 400)2] – 1/1.5(1125)2

 

= 390

 

SSE = SST - SS(Tr) = 666 - 390 = 276

ANOVATABLE

4. Conclusion :

Since F cal > Ftab => H0 is rejected.

=> Students of different schools of class X are not same.

Example 2. The following are the number of typing mistakes made in five successive weeks by four typists working for a publishing company.

Test at the 0.05 level of significance whether the differences among the four sample means can be attributed to chance.

 

Solution. 1. H0 : µ1 = µ2 = µ3 = µ4

 

H1 : At least two of them are not equal.

 

2. Here k = 4, n = 5,   α = 0.05

F0.05,3.16 = 3-24

 

3. Computations :

 

T1• = 13 + 16 + 12 + 14 + 15 = 70

T 2• = 14 + 16 + 11 + 19 + 15 = 7 5

T 3• = 13 + 18 + 16 + 14 + 18 = 79

T 4• = 18 + 10 + 14 + 15 + 12 = 69

T••= T1• + T2• + T3• + T4• = 293

 

∑∑ xij2  =   4407

 

SST     =          4407 -1/ 20 (293)2      =          114.55+

SSE = SST - SS(Tr) = 101.6

 

ANOVATABLE

4. Conclusions :

Since Feat  <  Ttab, H0 is accepted and we conclude that the typing mistakes can be attributed to chance.

 

Note. 1. Since variance is independent of change of origin, then the origin can be shifted to an arbitrary data. Final variance ratio is independent of change of scale. Hence change of origin and scale, if necessary, can be employed which help to perform simpler arithmetic.

2. Each observations xij can be decomposed as

xij              =                                 +                  (i  -   )                   +                  ( xij -  )

Grand mean       Deviation due to                 Error

Treatment