Working with the Normal distribution

The cumulative probability distribution function for a N(0,1) variable, the one that is tabulated, is often called Φ (the Greek letter capital Phi).

If X is a random variable distributed N(0,1):
P(X<z) = Φ(z)
P(X>z) = 1-Φ(z) = Φ(-z)

The sums you need to do to use the tables are not hard, but to keep track of them consider using Φ in your working.

For example, if z is negative write

Φ(-z) = 1-Φ(z)

If you see something asking for a range of values you can write:
P(a < z < b) = Φ(b)-Φ(a)

The special case follows:
P(-a < z < a) = Φ(a)-(1-Φ(a)) = 2Φ(a)-1

More hints

e.g. Let W=weight of parts ~ N(40,5²)

Remember that the variance goes in the N() expression but the standard deviation goes in the expression to standardise to Z. Be very careful that you get the right one.

Next, write the probability you are looking for,

P(15 < W < 35)

Then transform into the standard normal, Z

=P((15-μ)/σ < Z < (35-μ)/σ)

=φ((35-μ)/σ) - φ((15-μ)/σ)

Another way to do the question is to draw a diagram with probabilities, Z and the actual variable all on the same diagram. Write on to the diagram all the facts that you know and it should make it clear what you have to do.

For example, if you had a weight variable W with mean 32, and you had to find the weight which 5% of objects exceeded, the diagram might look like this. normal