Normal Probabilities

The position of the graphically represented keys can be found by moving your mouse on top of the graphic.

Probabilities on the Normal Distribution
The Problem: Given a normal distribution X with mean m and standard deviation s, what is the probability that X is between a and b?   P(a<X<b)
The Solution: Press The Menu Key: Row 1, Column 4 and arrow to the STAT picture. Press The E X E Key: Row 8, Column 5 and you'll be in the STAT mode. Press The F 5 Key: Fifth on the row of grey keys. (DIST for distribution), then The F 1 Key: First on the row of grey keys. (NORM for Normal). You'll see three choices. For this problem, press The F 2 Key: Second on the row of grey keys. ( Ncd for Normal, cumulative distribution). Enter the a value in the row that says Lower:. and The E X E Key: Row 8, Column 5. Enter the b value (in the row Upper:) and The E X E Key: Row 8, Column 5. The next value to type in is the standard deviation s. Press The E X E Key: Row 8, Column 5 and type in the mean m. Now press The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5 (yes, twice) to see the results.
(Note: the Save Res row is there if you want to Save Results.)

If you want to compute P(X < b), then make a very small.
If you want to compute P(X > a), then make b very large.
Examples: A normal distribution X has a mean of 100 and a standard deviation of 8.
  1. What is the probability that X is between 90 and 110?
  2. What is the probability that X is larger than 120?

Solutions: (Once in STAT mode)

  1. The F 5 Key: Fifth on the row of grey keys. (DIST) The F 1 Key: First on the row of grey keys. (NORM) The F 2 Key: Second on the row of grey keys. (Ncd) The Nine Key: Row 5, Column 3 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The One Key: Row 7, Column 1 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The Zero Key: Row 8, Column 1 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5
  2. The F 5 Key: Fifth on the row of grey keys. (DIST) The F 1 Key: First on the row of grey keys. (NORM) The F 2 Key: Second on the row of grey keys. (Ncd) The One Key: Row 7, Column 1 The Two Key: Row 7, Column 2 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The Zero Key: Row 8, Column 1 The Zero Key: Row 8, Column 1 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The Zero Key: Row 8, Column 1 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5

Answers: 

  1. p=0.78870045, z:Low = -1.25, z:Up = 1.25
  2. p=6.2097E-03 (that is, 0.0062097), z:Low = 2.5, z:Up = 112.5
Inverse Probabilities on the Normal Distribution
The Problem: Given a normal distribution X with mean m and standard deviation s, what x-value is larger than a percentage p of the data?  (p must be between 0 and 1, naturally.)
I.e., for what x is P( X < x) = p?
The Solution: Press The Menu Key: Row 1, Column 4 and arrow to the STAT picture. Press The E X E Key: Row 8, Column 5 and you'll be in the STAT mode. Press The F 5 Key: Fifth on the row of grey keys. (DIST for distribution) The F 1 Key: First on the row of grey keys. (NORM for Normal) You'll see three choices. For this problem, press The F 3 Key: Third on the row of grey keys. ( InvN for inverse normal). You will see the first row darkened in - Tail:Left. We want this for our problem. (But you can press The F 2 Key: Second on the row of grey keys. if your problem is P(X > x) = p or The F 3 Key: Third key in the topmost row of keys if your problem is P(-x < X < x) = p.) Press The down arrow: The lowest part of the big grey circular button. The next row says Area: type in the p value and The E X E Key: Row 8, Column 5. Next type in the standard deviation s and The E X E Key: Row 8, Column 5. Finally, enter the m value and press The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5 (yes, twice) for the results. 
Examples: A normal distribution has a mean of 20 and a standard deviation of 3.
  1. Find x such that P(X < x) = 70%
  2. Find x such that P(X > x) = 80%
  3. Find x such that P(-x < X < x) = 42%

Solutions: (Once in STAT mode)

  1. The F 5 Key: Fifth on the row of grey keys. (DIST) The F 1 Key: First on the row of grey keys. (NORM) The F 3 Key: Third on the row of grey keys. (InvN) Press The down arrow: The lowest part of the big grey circular button (Tail: Left) The Zero Key: Row 8, Column 1 The Seven Key: Row 5, Column 1 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Three Key: Row 7, Column 3 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5
  2. The F 5 Key: Fifth on the row of grey keys. (DIST) The F 1 Key: First on the row of grey keys. (NORM) The F 3 Key: Third on the row of grey keys. (InvN) Press The F 2 Key: Second on the row of grey keys. (Right) The down arrow: The lowest part of the big grey circular button The Zero Key: Row 8, Column 1 The Eight Key: Row 5, Column 2 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Three Key: Row 7, Column 3 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5
  3. The F 5 Key: Fifth on the row of grey keys. (DIST) The F 1 Key: First on the row of grey keys. (NORM) The F 3 Key: Third on the row of grey keys. (InvN) Press The F 2 Key: Second on the row of grey keys. (Right) The down arrow: The lowest part of the big grey circular button The Zero Key: Row 8, Column 1 The Four Key: Row 6, Column 1 The Two Key: Row 7, Column 2 The E X E Key: Row 8, Column 5 The Three Key: Row 7, Column 3 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5

Answers:

  1. x = 21.5732015
  2. x = 17.4751363
  3. x:Low = 18.3398458    x:Up = 21.6601552

Other Casio fx-9860g Pages:

Casio's online manual sample problem set home page  
Scatterplots and Histograms Basic Stats Hypothesis Tests Confidence Intervals