Hypothesis

literally the word hypothesis means something which is assumed to be true . A statistical hypothesis is a tentative statement logically drawn relating to population parameter.

According to professor morris hamburg ” A hypothesis in statistics is simply a quantitative statement about a population . it is an assumption which my or may not be true about a population parameter .

Standard Error

standard deviation of the sampling distribution is known as standard error. it is so called because it measures the sampling variability due to chance or random forces . it is also known as standard error of estimate . with the help of standard error of Mean, a decision maker can easily understand the scattering of sample mean and evaluate the precision of a sample .

Uses of standard error

• it is used as an instrument in testing a given hypothesis .
• it provides n idea about the unreliability of sample
• the reciprocal of standard error is I/SE . it is a measure of reliability or precision of the sample
• with the help of standard error we can determine the limits with in which the parameter values are expected to lie .

Problems related with one way ANOVA

Below are given yield (in kg) per acre for 5 plots of 4 varieties of yields. State analysis of variance

Plot numbers	1	2	3	4
1	42	48	68	80
2	50	66	52	94
3	62	68	76	78
4	34	78	64	82
5	52	70	70	66
Total	240	330	330	400

Solution

1. H₀ = H₁ (that is the means are equal )

H⁰≠ H₁

• T= sum of all observation in rows and columns

T= 1300 (240+330+330+400)

• T²/ N = 1690000/20= 84500

N= total number of observations

• SST = sum of squares of all observation – T²/N

= 88736- 84500 = 4236

• SSC = (∑x1 )²/n1 + (∑x2)²/n2 +etc….- T²/N

Were ∑x1, ∑x2…etc are the column total

= 240²/5 + 330²/5 + 330²/5 +400²/5 – 84500

= 57600/5 + 108900/5 + 108900/5 + 160000/5 – 84500

= 11520 + 21780 + 21780 + 32000 – 84500  

= 87080 – 84500

= 2580

• SST – SSC

= 4236 – 2580 = 1656

• K =  4 (number of columns )

MSC = SSC / K-1

= 2580 / 4-1

= 2580 / 3

= 860

• MSE = SSE / N-K

= 1656 / 20-4

= 1656 / 16

= 103.5

• F Ratio = MSC / MSE

= 860 / 103. 5

= 8.309

Calculate value is = 8. 31

Level of significance is 5% that is 0.05

Degree of freedom is  (k-1 , N-k )

Therefore

Table value of F at 5% level of significance for degree of freedom (3, 16) is 3.24

Since the CV 8.3 is > Table value 3.24  

So

Null hypothesis (H₀) is rejected that is all means are not equal.

ANOVA TABLE

Sources of variations	Sum of squares	Degree of freedom	Mean square
Between samples	2580 (SSC)	3 (k-1)	860 (MSC)
Within samples	1656 (SSE )	16 (N-k )	103.5 (MSE )
Total	4236 (SST)	19 (N- 1)

Two way analysis of variance – procedure

1. a) Assume means of all columns are equal

b) Assume means of all rows are equal

• compute T = sum of all the values
• SST = sum of squares of all observation-T²/N
• SSC = ∑(x1)²/n1 + ∑(x2)²/n2 etc – T²/N

(Note: where ∑x1 , ∑x2 etc are column totals )

• SSR = ∑(x1)²/n1 + ∑(x2)²/n2 etc – T²/N

(Note: where ∑x1, ∑x2 etc are row totals)

• SSE = SST- SSC-SSR
• MSC = SSC / c-1 , MSR = SSR/r-1 , MSE = SSE / (c-1)(r-1)

(Note : Where c = number of columns , r = number of rows )

• F_C = MSC/MSE
• F_R  = MSR /MSE

(Note F= large variance / small variance)

Degree of freedom for F_c = [(c-1), (c-1) ×(r-1)]

Degree of freedom for F_r = [ (r-1 ), (c-1) × (r-1)]

ANOVA TABLE

Sources of variations	Sum of squares	Degree of freedom	Means square	F ratio
Between column	SSC	c-1	MSC	FC
Between rows	SSR	r-1	MSR	FR
Residual	SSE	(c-1( (r-1)	MSE
Total	SST	N – 1

One way analysis of variance – procedure

1. Assume that mean of the samples are equal. that is effects of all factors are equal
2. Compute mean square between samples, that is MSC and mean square within the sample MSE .

For computing MSC and MSE the following calculations are made

• T = sum of all observations in rows and columns
• SST = sum of squares of all observations – T²/N
• SSC = (∑x1 )²/n1 + (∑x2)²/n2 +etc – T²/N

Were ∑x1, ∑x2…etc are the column total

• SSE= SST –  SSC
• MSC = SSC / k-1 (were k= number of column )
• MSE = SSE / N-k

3) Calculate F ratio = MSC / MSE

4) Obtain the table values of F (k-1, N-k degree of freedom)

5) If the calculated value of F is less than the table value accepts the null hypothesis. That is sample means are equal.

ANOVA TABLE

Sources of variance	Some of squares	Degree of freedom	Mean square
Between sample	SSC	k-1	MSC
Within sample	SSE	N-k	MSE
Total	SST	N-k

Two way classification of data (variance )

In two way classification 2 type of data are observed. The samples taken from the columns and samples taken from the rows are denoted by “ c and r respectively .the different type of variance in two way classification are follows

• Variance between samples due to column variable
• Variance between samples due to raw variables
• Variance within the samples
• Total variance for all observations

one way classification of data ( variance )

In one way classification only one variable is use. Here “k “is taken as number of samples . the following are the types of variances in one way classification .

• Variance between samples
• Variance within the samples
• Total variance for all observations together.

Features of analysis of variance

• It makes statistical analysis of variance of 2 or more samples
• It test whether the difference in the mean of different sample is due to chance or due to any significance causes.
• It uses the statistical test called F test by finding the appropriate variance ratio.

Assumptions of analysis of variance

• Populations from which samples have been drawn are normally distributed.
• Population from which samples are drawn have same variance
• The observations , in the samples are randomly selected from the population .
• The observation are not correlated random variable
• Any observation is the some of the effect of the factors influence it.

Analysis of variance

Analysis of variance may be defined as a technique which analyses the variance of two or more samples for determining the significance of difference in their arithmetic mean, and for determining whether different samples under the study are drawn from same population or not, with the help of statistical techniques called F test