In a particular city the age of the roofs of homes are appro

In a particular city, the age of the roofs of homes are approximately uni-

formly distributed between 2 and 10 years. (Note this implies the mean is µ = 6 years and

the standard deviation is = 2:309 years). If a random sample of 64 homes is selected, what

is the probability that the sample mean roof age will be between 5.4 and 6.2 years?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    5.4      
x2 = upper bound =    6.2      
u = mean =    6      
n = sample size =    64      
s = standard deviation =    2.309      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -2.078822001      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    0.692940667      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.018816857      
P(z < z2) =    0.755826606      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.737009749   [ANSWER]