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: The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (to get into MATH mode) The Seven Key: Row 7, Column 2 (For Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (for 8: UTPN, which stands for Upper Tail Probability Normal) The Enter Key: Row 10, Column 5.  UTPN will show up on the screen. Enter the mean m, followed by The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (for a comma). Enter the variance (s2, not the standard deviation s), followed by The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4. Enter the a value. Press The Right Arrow Key: Between the 1st and 2nd rows, Column 5 The Minus Key: Row 8, Column 5. Press The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (to get into MATH mode) The Seven Key: Row 7, Column 2 (For Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (for 8: UTPN, which stands for Upper Tail Probability Normal) The Enter Key: Row 10, Column 5. Then type in the mean, variance,  and b value separated by The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (commas). Finally, press The Enter Key: Row 10, Column 5. See the examples below for variations of this problem.

Note: If a and b seem backwards, it's because we're using the upper-tail probabilities.
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?
  3. What is the probability that X is less than 95?

Solutions: (s.d. = 8 means variance = 64)

  1. The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (MATH) The Seven Key: Row 7, Column 2 (Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (UTPN) The Enter Key: Row 10, Column 5 The One Key: Row 9, Column 2 The Zero Key: Row 10, Column 2 The Zero Key: Row 10, Column 2 The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Six Key: Row 8, Column 4 The Four Key: Row 8, Column 2 (8 squared) The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Nine Key: Row 7, Column 4 The Zero Key: Row 10, Column 2 The Right Arrow Key: Between the 1st and 2nd rows, Column 5 The Minus Key: Row 8, Column 5 The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (MATH) The Seven Key: Row 7, Column 2 (Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (UTPN) The Enter Key: Row 10, Column 5 The One Key: Row 9, Column 2 The Zero Key: Row 10, Column 2 The Zero Key: Row 10, Column 2 The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Six Key: Row 8, Column 4 The Four Key: Row 8, Column 2 (8 squared) The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The One Key: Row 9, Column 2 The One Key: Row 9, Column 2 The Zero Key: Row 10, Column 2 The Enter Key: Row 10, Column 5
  2. The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (MATH) The Seven Key: Row 7, Column 2 (Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (UTPN) The Enter Key: Row 10, Column 5 The One Key: Row 9, Column 2 The Zero Key: Row 10, Column 2 The Zero Key: Row 10, Column 2 The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Six Key: Row 8, Column 4 The Four Key: Row 8, Column 2 (8 squared) The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The One Key: Row 9, Column 2 The Two Key: Row 9, Column 3 The Zero Key: Row 10, Column 2 
  3. The One Key: Row 9, Column 2 The Minus Key: Row 8, Column 5 The Bent Left Key: Row 8, Column 1 The Symb Key: Row 4, Column 4 (MATH) The Seven Key: Row 7, Column 2 (Probability) The Enter Key: Row 10, Column 5 The Eight Key: Row 7, Column 3 (UTPN) The Enter Key: Row 10, Column 5 The One Key: Row 9, Column 2 The Zero Key: Row 10, Column 2 The Zero Key: Row 10, Column 2 The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Six Key: Row 8, Column 4 The Four Key: Row 8, Column 2 (8 squared) The Orange Arrow Key: Row 9, Column 1 The S P C (space) Key: Row 10, Column 4 (comma) The Nine Key: Row 7, Column 4 The Five Key: Row 8, Column 3  

Answers: 

  1. p=0.788700452666
  2. p=6.20966532578E-3, that is, 0.00620966532578.
  3. p=.265985529049

Other Hp50g Pages:

HP's online manual*  sample problem set home page
Basic Stats Confidence Intervals Hypothesis Tests

* Technically, this is the manual for the 49g+, but the commands are the same for both calculators.

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