In a murder trial in Los Angeles, a shoe expert stated that the range of heights

Question

In a murder trial in Los Angeles, a shoe expert stated that the range of heights of men with a size 12 shoe is 70 inches to 78 inches. Suppose the heights of all men wearing size 12 shoes are normally distributed with a mean of 73.7 inches and a standard deviation of 2.6 inches. What is the probability that a randomly selected man who wears a size 12 shoe. (do not round the z values. round answersto 4 decimal places)

a) Has a height outside the range 70 inches to 78 inches?

b) Is 76 inches or taller?

C) Is shorter than 69.5 inches?

Explanation / Answer

Solution:- mean = 73.7, standard deviation = 2.6

a)P( 70>x>78) =  1- P( [70 - ] / < [x - ] / < [78 - ] / )

= 1- P( [70 - 73.7] / 2.6 < z < [78 - 73.7] / 2.6 ) ; z = [x - ] / is the standard normal variable

= 1 - P( -1.4231 < z < 1.6538)

= 1 -  0.8727

= 0.1273  

b) P( X >= 76) = P(z >= (76-73.7)/2.6)

= P(z >=  0.8846)

=  10.8106

= 0.1894

c)  P( X < 69.5) =  P( z < (69.5-73.7)/2.6)

=   P( z < -1.6154)

= 0.0526

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