Suppose that the lifetimes of light bulbs are approximately normally distributed

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 62 hours? (b) What proportion of light bulbs will last 52 hours or less? (c) What proportion of light bulbs will last between 58 and 62 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 62 hours is (b) The proportion of light bulbs that last 52 hours or less is (c) The proportion of light bulbs that lasts between 58 and 62 hours is (d) The probability that a randomly selected light bulbs lasts less than 45 hours is

Explanation / Answer

X~N(52,3.52)

Z=(x-52)/3.5

Z~N(0,1)

a.
P(X>62)

=P(Z>(62-52)/3.5)=P(Z>10/3.5)

=1-0.99786=0.00214

that is 21 in 10000

b. this will 0.5 intuitively. as 52 is the mean.

50% is the answer

c.P(58<X<62)

=P(X<62)-P(X<58)

=0.99786-0.95676=0.0411

4.11% is the answer

d.P(X<45)

=0.02275

Note : probabilities in parts b,c,d have also been counted after converting Xto Z.

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