Question

15. United Widget Manufacturing has a problem with defective widgets. Employees make hundreds of thousands of them each day, and many are defective. United has instituted training for the workers, and you would like to predict the number of defects per week based on the number of days of training an employee has received. You obtain the following data: Employee number 1015 2023 1153 40291117 0012 Days of training 4. 5 6 4 3 2 Defects per week (100s) 19 17 7 14 22 23 (a) Use the method of least squares to determine the regression equation. (b) Interpret the meaning of the slope using the words of the problem.

Answer 1

15)

a)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	24.00	102.00	10.00	174.0	-37.00
mean	4.00	17.00	SSxx	SSyy	SSxy

Sample size,   n =   6      
here, x̅ = Σx / n=   4.000          
ȳ = Σy/n =   17.000          
SSxx =    Σ(x-x̅)² =    10.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   -37.0      
              
estimated slope , ß1 = SSxy/SSxx =   -37/10=   -3.7000      
intercept,ß0 = y̅-ß1* x̄ =   17- (-3.7 )*4=   31.8000      
              
Regression line is, Ŷ=   31.800   + (   -3.700   )*x

b) for every unit increase in days of training, predicted defect per week get decrease by 370
c)

Predicted Y at X=   6   is          
Ŷ=   31.80000   +   -3.70000   *6=   9.600

d)

R² =    (SSxy)²/(SSx.SSy) =    0.787
Approximately    78.68%   of variation in observations of variable Y, is explained by variable x