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 60 ​hours? ​(b) What proportion of light bulbs will last 50 hours or​ less? ​(c) What proportion of light bulbs will last between 57 and 61 ​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 60) = 1 - P(x   60)

= 1 - P[(x - ) / (60 - 57) / 3.5]

= 1 -  P(z 0.86)  

= 1 - 0.8051

= 0.1949

proportion = 0.1949

(b)

P(x 50)

= P[(x - ) / (50- 57) / 3.5]

= P(z -2)

= 0.0228

proportion = 0.0228

(c)

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

= P(0 z 1.14)

= P(z 1.14) - P(z 0)

= 0.8729 - 0.5

= 0.3729

proportion = 0.3729

(d)

P(x 45)

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

= P(z -3.43)

= 0.0003

probability = 0.0003