The lifetime of a warning lamp on a piece of equipment is no

The lifetime of a warning lamp on a piece of equipment is normally distributed with mean 8.20 years and standard deviation 1.40 years.

Solution

a) mean= 8.2 , s.d = 1.4

x= life time of the lamp

prob( lamp will last more than 10 years ) = p( x >10) = p( (x-mean)/ s.d > (10-8.2) / 1.4 ) = p(z>1.285714286)
= 1 - p( z <= 1.285714286 ) = 0.0992714

b) p( x<= 8)= p( z<= (8-8.2) / 1.4 ) =p(z <= -0.142857)= 0.4432015
so, it is the 44.32015 % percentile!

c) z-score for the 60% confidence =0.2533471 ................

(x- mean) / s.d = z i.e, x = mean + z* s.d = 8.2 + (0.2533471*1.4) ..... i.e, x = 8.554686.......

d) p( 9 < x < 11) = p( (9-8.2) / 1.4 < z < (11-8.2) / 1.4 ) = p( 0.571428571 < z < 2 ) = p( z<2) - p( z < 0.571428571)
= 0.9772499 - 0.7161454 = 0.2611045