Question

Suppose that X is a normal random variable with mean μ = 200 and standard deviation σ = 40. What is the probability that X will take a value greater than 228?

Answer 1

Answer:

          0.242  (approx)

Step-by-step explanation:

How you calculate this depends upon what tools you are using.

Converting to a standard normal distribution

It is possible that you have a table or a calculator that only gives probabilities for the standard normal distribution; that is, with mean μ = 0 and standard deviation σ = 1.  If that is the case, then...

Let Z = (X - μ) / σ = (X - 200) / 40.  Then Z has a standard normal distribution.  Now express the required probability in terms of Z:

  Pr(X > 228)

= Pr(X - 200 > 228 - 200)

= Pr( (X - 200)/40  >  (228 - 200)/40 )

= Pr(Z > 28/40)  =  Pr(Z > 7/10)  = Pr(Z > 0.7)

This we get from our table or calculator as 0.242 (approx).

Note:  If your tables only give probabilities for Pr(Z < ...), then there's one more step before using the table...

Pr(Z > 0.7)  =  1 - Pr(Z < 0.7)   =  Pr(Z < -0.7),  use either of these.

Sometimes your tables only give probabilities for Pr(0 < Z < ...).  In that case, use...

Pr(Z > 0.7) = 0.5 - Pr(0 < Z < 0.7)

Using a calculator or software that applies to any normal distribution

Some calculators and software do the conversion to a standard normally distributed Z for us.  Check the instructions carefully to see how to specify the mean and standard deviation.

For example, using R, the function pnorm(x, μ, σ) gives the probability Pr(X<x).  So...

Pr(X > 228) = 1 - Pr(X < 228) = 1 - pnorm(228, 220, 40) = 0.242 (approx).

Hope this helps you out.