So, let X be the number of hours of life span of a bulb.
X ~ N(1200, 200^2)
P(X < x) = P(Z < z) = 0.05
Find the z value that corresponds to this probability, then convert it into hours using:
A manufacturer of electric light bulbs finds that his bulbs have an average life span of 1200 hours and a S.D of 200 hours.Assume that the distribution of lifetimes is normal.
What should be the guaranteed life of the bulbs if the manufacturer is prepared to replace 5% of the bulbs sold?
How should I go about doing this question?
X is normally distributed random variable measuring the life span of light bulbs.
You have to determine the value A (number of hours such) that the life span of 95% of light bulbs will last longer than A hours. That means that only 5% of light bulbs will have life span shorter than A and thus such A should be the guaranteed life of light bulbs.
Now use the table values for standard normal distribution and you'll see that so you form the equation
That means that you are asked to use a z table different from mine.
So, from where I left:
Is your graph like this: http://p1cture.me/images/51394697395543890592.png
Or like this: http://p1cture.me/images/88238005700832427228.png
?
Ok.
You need to find the z value for this area: http://p1cture.me/images/17640253637132137970.png
But you don't have this in your table. The equivalent of this area is this area: http://p1cture.me/images/50823189974773945850.png
However, this isn't in your table either. But you have this area: http://p1cture.me/images/30470114981255486093.png
The probability of the last area is 0.5 - 0.05 = 0.45, okay?
What is the value of z for this area? in your table, you should read z = 1.65.
But when you look at the original distribution (first graph), you know that z must be negative as it's value is on the left of the mean, so, the z value is -1.65.
From there, you can then work back the value of X, the maximum lifetime of the bulb for it to fall in the 5% worst quality.
Is it clear now?
is the value of the distribution function of the random variable X at point A.
a uppercase Greek letter phi, commonly used to denote the distribution function of the standard normal random variable.
If X is normal random variable with parameters called expectation and variance, then distribution function of that random variable is usually denoted with and it is defined as
Standard normal random variable is normal random variable with parameters 0 and 1, and to distinct it from other normal variables its distribution function, although defined in the same manner, is usually denoted with :