2 The time that takes to complete a certain type of construc

2. 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?
Work:
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?
Work:
c. If the z-score for the completion time of a project is -2.6, find the completion time for that project.

Solution

a)

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

Here,

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% [ANSWER]

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b)

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      
          
Thus,          
          
z = (x - u) / s =    -1      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1   ) =    0.841344746 = 84.13% [answer]

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c)

As

x = u + z*sigma,

x = 35.5 + (-2.6)*1.5

x = 31.6 [answer]