Question

The information regarding the experience (in years) of eight workers on a machine and their performance ratings based on the non defective units they manufactured in every 100 units is as follows:

Experience of workers (years)	5	12	15	8	20	18	22	25
Performance rating	80	82	85	81	90	90	95	97

Obtain the regression line of the performance rating on the experience and estimate the performance rating if a worker has an experience of 17 years.

Answer 1

Here, n = 8; X = Experience of worker and Y = Performance rating

Now, x̄ = \(\frac{Σx}{n} = \frac{125}{8}\) = 15.63;

ȳ = \(\frac{Σy}{n} = \frac{700}{8}\) = 87.5

We obtain the regressoin line ŷ = a + bx of performance rating (Y) on the experience of workers (X). To make the calculation of a and b simple, we take new variables u = x – A, A = 15 and v = y – B, B = 90.

The table for calculation is prepared as follows:

\(b = \cfrac{nΣuv−(Σu)(Σv)}{nΣu^2−(Σu)^2}\)

Putting n = 8; Σuv = 292; Σu = 5; Σu = – 20 and Σu² = 341 in the formula,

a = ȳ – bx̄
Putting ȳ = 87.5; x̄ = 15.63; b = 0.9, we get
a = 87.5 – 0.9 (15.63)
= 87.5 – 14.07
= 73.43

Regression line of performance rating (Y) on experience of worker (X):

Putting a = 73.43; b = 0.9 in ŷ = a + bx, we get ŷ = 73.43 + 0.9x

Estimate of performance rating Y when X = 17 year :

Putting x = 17 in ŷ = 73.43 + 0.9x, we get
ŷ = 73.43 + 0.9 (17)
= 73.43 + 15.3
= 88.73

Hence, the estimate of performance rating (Y) obtained is ŷ = 88.73.