37 Lumber Cutter The lengths of lumber a machine cuts are no

37. Lumber Cutter The lengths of lumber a machine cuts are norm distributed, with a mean of 96 inches and a standard deviation of 0.5 inch. (a) What is the probability that a randomly selected board cut by the machine has a length greater than 96.25 inches? (b) You randomly select 40 boards. What is the probability that their mean length is greater than 96.25 inches? (c) Compare the probabilities from parts (a) and (b).

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    96.25      
u = mean =    96      
          
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) / s =    0.5      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.5   ) =    0.308537539 [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 =    96.25      
u = mean =    96      
n = sample size =    40      
s = standard deviation =    0.5      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    3.16227766      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   3.16227766   ) =    0.000782701 [ANSWER]

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c)

Part a has a greater probability [ANSWER], because part a has a greater standard deviation than part b.