Inverse Normal

Inverse Normal This is where you know the probability and have to find out either the value that X is ≥ < > ≤, σ or µ. Since it is working backwards to what we normally do we call it “inverse normal” Again we can use our GC: Stat Dist Norm INVN = inverse normal The x value given always has the shaded area to the left x

Example 1 X is a normally distributed random variable with µ = 25 and σ = 4. P(X<x)=0.982. Find x. Step 1: Draw a diagram 0.982 x

Step 2: 0.982 Use GC x GC select invn Area=.982 Answer: x=33.387 Example 1 X is a normally distributed random variable with µ = 25 and σ = 4. P(X<x)=0.982. Find x. 0.982 x Step 2: Use GC GC select invn Area=.982 Answer: x=33.387

Check: Does answer make sense? Example 1 X is a normally distributed random variable with µ = 25 and σ = 4. P(X<x)=0.982. Find k. 0.982 x Step 3: Check: Does answer make sense? x=33.387 is bigger than the µ and is on the right of the mean – so answer ok

0.05 Step 1: Draw a diagram k Example 2 X is a normally distributed random variable with µ = 2500 and σ = 8. P(X>k)=0.05 Find k. Step 1: Draw a diagram 0.05 k

0.05 Step 2: Use GC k GC select invn Area=1 - 0.05=0.95 Example 2 X is a normally distributed random variable with µ = 2500 and σ = 8. P(X>k)=0.05 Find k. 0.05 k Step 2: Use GC GC needs this area GC select invn Area=1 - 0.05=0.95 Answer: k = 2513.1

Check: Does answer make sense? Example 2 X is a normally distributed random variable with µ = 2500 and σ = 8. P(X>k)=0.05 Find k. 0.05 k Step 3: Check: Does answer make sense? k = 2513.1 is bigger than the µ and is on the right of the mean – so answer ok

Note: If you need to find σ or µ then you need to use: So first need to find z by using inverse for STANDARD normal

X is a normally distributed random variable with µ = 32 Find σ if Example 3 X is a normally distributed random variable with µ = 32 Find σ if P(X<40)=0.75 Step 1. Draw a diagram 0.75 µ=32 X

Step 2. Find z: GC: Stat Dist Norm INVN Area = 0.75 σ = 1 µ = 0 Standard normal normal is the same as µ=0 z µ=32 X Step 2. Find z: GC: Stat Dist Norm INVN Area = 0.75 σ = 1 µ = 0 This gives z =0.67448

Example 3 Step 3. Find σ: Sub X=40 µ=32 and z=0.67448 into: 0.67448 = 40 – 32 σ σ = 11.86