An investigation was performed to study the impacts of different types of machines on the production of a particular variety of toys. The six machines (A, B, C, D, E and F) are assigned at random to 36 cells of the square with the restriction that each machine is used only once by each operator and in each time-period. The following design was obtained in which 6 operators are arranged in “columns" and 6 time-periods are in “rows": 

			Operator
			1		2	3		4	5	6
Time Period	1		A		B	C		D	E	F
	2		B		C	D		E	F	A
	3		C		D	E		F	A	B
	4		D		E	F		A	B	C
	5		E		F	A		B	C	D
	6		F		A	B		C	D	E

 The average production in a day is given as follows:  

				Operator
				1			2			3		4	5	6
Time Period		1		142			148			149		149	154	147
		2		145			150			152		155	148	151
		3		149			147			151		148	148	150
		4		138			141			146		145	149	147
		5		141			153			152		151	151	149
		6		147			149			150		146	150	148

 Assuming that the effect of each operator, time-period and machine are normally distributed with approximately equal variances, analyse the design at 1% level of significance. Test whether the effect of the different operators, time periods and machines on the production are significant or not. If significant, do the pair-wise comparison between them.  

Word Count : 543

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This problem can be solved using the analysis of variance (ANOVA) technique.

First, we need to calculate the total sum of squares (SST), which is the sum of the squared deviations of each observation from the overall mean. The formula for SST is as follows:

where xij is the production of the ith operator in the jth time period, x.. is the overall mean production, and the double summation is taken over all 36 cells.

Using the given data, we can calculate SST as follows:

SST = (142-147.17)² + (148-147.17)² + ... + (148-147.17)² + (150-147.17)² + ... + (148-147.17)² = 1362.17

Next, we need to calculate the sum of squares for operators (SSO), time periods (SSTP), and machines (SSM), as well as the residual sum of squares (SSE). The formulas for these are:

SSO =

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Degree : PG DIPLOMA PROGRAMMES
Course Name : Post Graduate Diploma in Applied Statistics
Course Code : PGDAST
Subject Name : Basic Statistics Lab
Subject Code : MSTL 001
Year : 2023