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 58 and 61 hours? 

(d) What is the probability that a randomly selected light bulb lasts less than 46 hours? 

(a) The proportion of light bulbs that last more than 60 hours is (Round to four decimal places as needed.) 

(b) The proportion of light bulbs that last 50 hours or less is (Round to four decimal places as needed.) 

(c) The proportion of light bulbs that lasts between 58 and 61 hours is (Round to four decimal places as needed.) 

(d) The probability that a randomly selected light bulbs lasts less than 46 hours is (Round to four decimal places as needed.)

Answer 1

Here mean lifetime = = 57 hours

standard deviation = = 3.5 hours

(a) If x is the lifetime of a random light bulb. Then,

P(x > 60 hours) = 1 - NORMDIST(x <60 hours; 57 hours; 3.5 hours)

z = (60 - 57)/3.5 = 0.857

P(x > 60 hours) = 1 - NORMSDIST(0.857) = 1 - 0.8043 = 0.1957

(b) P(x < 50 hours) = NORMDIST(x < 50 hours ; 57 hours; 3.5 ours)

z = (50 - 57)/3.5 = -2

P(x < 50 hours) = 0.02275

(c) P(58 hours < x < 61 hours) = NORMDIST(x < 61 hours; 57 hrs; 3.5 hours) - NORMDIST(x < 58 hours; 57 hrs; 3.5 hrs)

z₁ = (61 - 57)/3.5 = 1.143

z₂ = (58 - 57)/3.5 = 0.286

P(58 hours < x < 61 hours) = NORMSDIST(1.143) - NORMSDIST(0.286) = 0.8735 - 0.6125= 0.2610

(d) P(x < 46 hours) = NORMDIST(x < 46 hours; 57 hrs; 3.5 hrs)

z = (46 - 57)/3.5 = -3.143

P(x < 46 hours) = NORMSDIST(-3.143) = 0.00084