Question

The lifetimes of a certain type of light bulbs follow a normal distribution. If 4% of the bulbs have lives exceeding 462 hours, and 40% have lives exceeding 372 hours, what are the mean and standard deviation of the lifetimes of this particular type of light bulbs? Round your answer to the nearest integer.

Mean = hours

Standard deviation = hours

Answer 1

Given X : Life time of Bulb

X follows normal distribution

We have given 4% of the bulbs have lives exceeding 462 hours,

P ( X > 462 ) = 0.04

so we get

P ( X < 462 ) = 1 - 0.04

P ( X < 462 ) = 0.96

Now we find the Z score for Area = 0.96

We Use excel command

Select empty cell from excel and do not forget to type "=" sign

=NORMSINV(0.96)

You get Z =1.750686071

So we get by using the Formula of Z score

we get Equation 1 as

     Equation 1

Now for 40% have lives exceeding 372 hours

We get

P( X > 372 ) = 0.40

So we get

P ( X < 372 ) = 1 - 0.40 = 0.60

P ( X < 372 )   = 0.60

Now we find the Z score for area = 0.60

=NORMSINV(0.60)

So we get the Z score

Z = 0.253347103

We apply the formula of Z score

So we get the equation 2 as

   equation 2

Subtract equation 1 and 2 we get

So we get the standard deviation as

Round it to nearest integer so we get

Now we plug value of standad deviation is equation 1 to calculate the mean

we plug      "

round the mean to nearest integer

So we get the final answer as :-