Question

The president of a company that manufactures car seats has been concerned about the number and cost of machine breakdowns. The problem is that the machines are old and becoming quite unreliable. However, the cost of replacing them is quite high, and the president is not certain that the cost can be made up in today’s slow economy. To help make a decision about replacement, he gathered data about last month’s costs for repairs in dollars (y) and the ages in months (x) of the plant’s 20 welding machines as recorded :

Ages X	Cost Y
110	655.34
113	753.36
114	785.04
134	886.28
93	685.24
141	952.32
115	649.48
115	677.96
115	866.9
142	1052.74
96	724.84
139	897.52
89	670.54
93	701.88
91	583.62
109	935.6
138	948.96
83	708.3
100	840.22
137	832.08

a) Find the estimated regression equation to predict the repair cost of a machine from its age.

b) Interpret the slope in the estimated regression equation.

c) Find and interpret the correlation between the repair cost of a machine and its age.

d) Find the coefficient of determination, and discuss what this statistic tells you.

Answer 1

Variables	Ages X	cost Y	(x-x̅)(y-y̅)	(x-x̅)2	(y- )
	110	655.34	452.48785	11.2225	18244.17504
	113	753.36	12.96785	0.1225	1372.776601
	114	785.04	-3.49115	0.4225	28.847641
	134	886.28	1979.69485	426.4225	9190.865161
	93	685.24	2140.22985	414.1225	11060.93924
	141	952.32	4476.78385	764.5225	26214.52428
	115	649.48	-232.53615	2.7225	19861.54676
	115	677.96	-185.54415	2.7225	12645.2274
	115	866.9	126.20685	2.7225	5850.567121
	142	1052.74	7515.72585	820.8225	68816.50424
	96	724.84	1137.65685	301.0225	4299.556041
	139	897.52	2747.34585	657.9225	11472.33788
	89	670.54	2918.85885	592.9225	14369.05664
	93	701.88	1801.60585	414.1225	7837.737961
	91	583.62	4621.77885	499.5225	42762.51768
	109	935.6	-631.57215	18.9225	21079.84572
	138	948.96	3908.23285	607.6225	25137.7854
	83	708.3	2492.06885	921.1225	6742.216321
	100	840.22	-664.95015	178.2225	2480.936481
	137	832.08	985.47185	559.3225	1736.305561
Total	2267	15808.22	35599.023	7196.55	311204.2692
Mean	113.35	790.411	1779.95115	359.8275	15560.21346

a) Regression equation is estimated by using the following equation

y = a + bx

where a and b are estimated by

  

Now according to the sum

Var(X) = 359.8275 VAR(Y) = 15560.21346

[ The detailed calculations are shown in the table ]

There fore a = 790.41 - (1779.95115 * 113.35 ) / 359.8275 = 229.7049

b = 1779.95115 / 359.8275 = 4.946679

Thus regression equation becomes

y = 229.7 + 4.95x

b) The slope is 4.95 which means for one month increase/decrease in age we will have an increment/decrement of 4.5 in the value of cost.

c ) correlation coefficient is found by the following formula

from the calculations done above we get the value of r to be

  

From this value it can be said that x and y are strongly correlated positively or in other words we can say there is a high positive corelation between the cost and the age of machine i.e. if the age of the machine increases then the cost of the machine increases and if the age decreases then the cost decreases.

d) Coefficient of determination is equal to r2 = (0.752234)2 = 0.5659

This means that 56.59% of the total variability of Y can be accounted for by its linear regression on X.R2 is a statistical measure of how close the data are to the fitted regression line. Therefore we can say that the data is 56.59% close to the fitted regression line.(y = 229.7 + 4.95x)