H2 Maths 6.3 - Normal Distribution: Finding Unknown Parameters

Step 1 - Set up the problem

Sometimes you know probabilities about a normal distribution but not its parameters.

Step 1 - Set up the problem

Here, X tilde N of mu, sigma squared. We are told P of X less than forty equals zero point one five eight seven, and P of X greater than sixty equals zero point zero six six eight.

Step 1 - Set up the problem

Our task is to find the mean mu and the standard deviation sigma.

Step 2 - Standardise the first condition

Convert the boundary to a Z-score using Z equals X minus mu divided by sigma.

Step 2 - Standardise the first condition

P of X less than forty is zero point one five eight seven, which is less than half, so the Z-score must be negative.

Step 2 - Standardise the first condition

From the standard normal, P of Z less than negative one equals zero point one five eight seven. So the first Z-score is negative one.

Step 3 - Standardise the second condition

For the second condition, P of X greater than sixty is zero point zero six six eight, so P of X less than sixty is zero point nine three three two.

Step 3 - Standardise the second condition

From the standard normal, P of Z less than one point five equals zero point nine three three two. So the second Z-score is one point five.

Step 3 - Standardise the second condition

That gives the second equation: mu plus one point five sigma equals sixty.

Step 4 - Solve simultaneously

We now have two linear equations in mu and sigma.

Step 4 - Solve simultaneously

Subtract equation one from equation two to eliminate mu.

Step 4 - Solve simultaneously

That gives two point five sigma equals twenty, so sigma equals eight, and substituting back gives mu equals forty eight.

Step 5 - State the distribution and verify

The mean is forty eight and the standard deviation is eight, so the variance is sixty four.

Step 5 - State the distribution and verify

Verify the first condition: forty minus forty eight over eight equals negative one, giving P of Z less than negative one equals zero point one five eight seven - correct.

Step 5 - State the distribution and verify

Verify the second: sixty minus forty eight over eight equals one point five, giving a right-tail probability of zero point zero six six eight - correct.