1. Inverse Normal

2. 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

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

4. 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)= Find x.
0.982 x
Step 2:
Use GC
GC select invn Area=.982
Answer: x=33.387

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

6. 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

7. 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= =0.95
Answer: k =

8. 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 = is bigger than the µ and is on the right of the mean – so answer ok

9. 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

10. 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

11. 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 = σ = 1 µ = 0
This gives z =

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