In a manufacturing process the useful life of a cutting tool

In a manufacturing process, the useful life of a cutting tool is linearly related to the speed at which the tool is operated. The given data were derived from life tests for a particular brand of cutting tools currently used in the production process. Calculate the equation of the estimated regression line. What percentange of observed variation in the useful life of the tool can be attributed to the linear relationship with the cutting speed? What is the probability that, on average, a cutting speed of 55 m/min yields a useful life above 4.0 hrs.

Solution

A)

Using technology, we get              
              
slope =    -0.111          
intercept =    9.356666667          
              
Thus, the regression line is              
              
y^ =    -0.111x   + 9.356666667 [ANSWER]

***************************
B)

Also, getting the correlation,              
              
r =    -0.950776834          
              
Thus, the coefficient of determination is              
              
r^2 =    0.903976587          

Thus, about 90.398% of the variation in useful life can be eplained by the cutting speed. [ANSWER, 90.398%]

************************

c)

Thus, if x =    55  
      
Then      
      
y^ =    3.251666667  

The standard error of the estimate is              
              
sy =    0.549568595          
              
We first get the z score for the critical value. As z = (x - y^) / sy, then as          
          
x = critical value =    4      
y^ = mean =    3.251666667      
          
sy = standard deviation =    0.549568595      
          
Thus,          
          
z =    1.36167412      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.36167412   ) =    0.086650376 [ANSWER]