Question

The time that takes to complete a certain type of construction projects has a mean of 35.5 months and a standard deviation of 1.5 months.

a. According to the Tchebysheff’s theorem, at least what percentage of these projects must have taken between 26.5 months and 44.5 months to complete?

b. If in addition we are told that the relative frequency curve for the completion times is a bell-shaped curve, then approximately what percentage of these projects would take more than 34 months to completed?

Answer 1

The time that takes to complete a certain type of construction projects has a mean of 35.5 months and a standard deviation of 1.5 months.

ANSWER :

a). According to the Tchebysheff’s theorem, at least what percentage of these projects must have taken between 26.5 months and 44.5 months to complete.

According to the theorem, at least 1 - 1/k^2 lie within k standard deviations from the mean.

k = (x - u) / s = (26.5 - 35.5)/1.5 = -6

Thus at least 1 - 1/k^2 = 1 - 1/6^2 = 0.972222222 = 97.22222%

b). If in addition we are told that the relative frequency curve for the completion times is a bell- shaped curve, then approximately what percentage of these projects would take more than 34 months to completed

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    34   

u = mean =    35.5      
          
s = standard deviation =    1.5      
            
          
z = (x - u) / s =    -1      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1   ) =    0.841344746 = 84.13%