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? ​(a) The proportion of light bulbs that last more than 60 hours is nothing. ​(Round to four decimal places as​ needed.)

ANSWER for all PARTS please.

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 < 3 / 3.5 )

= 1 - P ( z < 0.86 )

Using z table

= 1 - 0.8023

= 0.1977

Probability = 0.1977

b )P ( x < 50 )

P ( x -  / ) < ( 50 - 57 / 3.5 )

P ( z < -7/ 3.5 )

P ( z < - 2 )

Using z table

= 0.0228

Probability = 0.0228

c ) P (57 < x < 61 )

P( 57- 57 / 3.5 )< ( x -  / ) < ( 61 - 57 / 3.5 )

P ( 0 / 3. 5 < z < 4 / 3.5 )

P ( 0 < z < 1.14 )

P ( z < 1.14 ) - P ( z < 0 )

Using z table

= 0.8729 - 0.5000

= 0.3729

Probability = 0.3729

d ) P ( x < 45)

P ( x -  / ) < (45- 57 / 3.5 )

P ( z < -12/3.5 )

P ( z < - 3.43 )

Using z table

= 0.0003

Probability = 0.0003