Question

Suppose that the lifetimes of light bulbs are approximately normally​ distributed, with a mean of 57 hours and a standard deviation of 3.5 hours. With this​ information, answer the following questions.

​(a) What proportion of light bulbs will last more than 61 ​hours?

​(b) What proportion of light bulbs will last 52 hours or​ less?

​(c) What proportion of light bulbs will last between 57 and 62 ​hours?

​(d) What is the probability that a randomly selected light bulb lasts less than 45 ​hours?

Answer 1

Solution :

Given that ,

mean = = 57

standard deviation = = 3.5

(a)

P(x 61) = 1 - P(x   61)

= 1 - P[(x - ) / (61 - 57) / 3.5]

= 1 -  P(z 1.14)  

= 1 - 0.8729

= 0.1271

proportion = 0.1271

(b)

P(x 52)

= P[(x - ) / (52 - 57) /3.5 ]

= P(z -1.43)

= 0.0764

proportion = 0.0764

(c)

= P[(57 - 57 /3.5 ) (x - ) / (62 - 57 / 3.5) ]

= P(0 z 1.43)

= P(z 1.43) - P(z 0)

= 0.9236 - 0.5

= 0.4236

proportion = 0.4236

(d)

P(x 45)

= P[(x - ) / (45 - 57) / 3.5]

= P(z -3.43)

= 0.0003

probability = 0.0003