Statistical Hypothesis Testing and Pic

Pages: 5 (763 words) Published: June 19, 2013
Analysis of Variance (ANOVA)

One Way Classification

Random samples of size n are selected from each of k populations. It will be assumed that the k populations are independent and normally distributed with means [pic][pic] and common variance [pic]. We wish to derive appropriate methods for testing the hypothesis:

[pic] [pic]
[pic] at least two of the means are not equal.
Table 1
K random samples
| |Population | | | |1 |2 |……… |i |……… |k | | | |[pic] |[pic] |……… |[pic] |……… |[pic] | | | |[pic] |[pic] |……… |[pic] |……… |[pic] | | | |. |. | |. | |. | | | |. |. | |. | |. | | | |. |. | |. | |. | | | |. |. | |. | |. | | | |[pic] |[pic] |……… |[pic] |……… |[pic] | | |Total |[pic] |[pic] |……… |[pic] |……… |[pic] |[pic] | |Mean |[pic] |[pic] |……… |[pic] |……… |[pic] |[pic] |

One way Sum of Squares Identity

[pic]

Total Sum of Square = SST = [pic]
Sum of Squares for column mean = SSC = [pic]
Error Sum of Square = SSE = [pic]
According to one way sum of squares identity
SST = SSC + SSE

Steps Of Working:

1. Set the null hypothesis [pic]
e.g. [pic] [pic]

2.Set the alternative hypothesis [pic]
e.g [pic] at least two of the means are not equal.

3.Level of significance [pic] [pic](also decide the case, either belong to the one tailed or two tailed) 4.Check the table
a.FOR EQUAL SAMPLE SIZE
(As it is the case of comparison of progress there for we use the table of F distribution.) [pic] b.FOR UNEQUAL SAMPLE SIZE
The degree freedom and its calculation is different when the sample size are not equal . [pic]
Note:(The formula for degree of freedom will remain same irrespective to the sample size) 5.Computations: (FOR EQUAL SAMPLE SIZE)

Sum of Squares Computational Formulae

SST = [pic]
SSC = [pic]
SSE = SST – SSC

Table 2-a
ANOVA (ONE-WAY CLASSISFICATION)

|Source of |Sum of |Degrees of |Mean |Computed | |Variation |Squares |Freedom |Square |f | |Column means |SSC |k-1 |[pic] |[pic] | |Error |SSE |K(n-1) |[pic] | | |Total |SST |nk - 1 | | |

Computations: (FOR UNEQUAL SAMPLE SIZE)

Sum of Squares Computational Formulae

SST = [pic]
SSC = [pic]
SSE = SST – SSC

Table 2-b
ANOVA (ONE-WAY CLASSISFICATION)...