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

51

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

46

​hours?

Answer 1

Solution :

Given that ,

(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 51)

= P[(x - ) / (51 - 57) / 3.5]

= P(z -1.71)

= 0.0436

proportion = 0.0436

(c)

P( 57< x < 61) = 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 < 46) = P[(x - ) / < (46 - 57) / 3.5]

= P(z < -3.14)

= 0.0008

Probability = 0.0008