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

On this page, I will describe how to do the following functions:

Computing probabilities with normal distributions.
Inverse normal problems
A one-sample t-test
A one-sample z-test
A z-confidence interval
A t-confidence interval

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 Second Key: Row 1 Column 1 It's bright yellow. The Vars Key: Row 3 Column 4 (It should say DISTR above the key.) Press Two Key: Row 8 Column 3. The screen will now say "normalcdf(". Enter a, b, m, s in that order with a The Comma Key: Row 5 Column 2 in between each. Press Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5.

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:

  1. Second Key: Row 1 Column 1 It's bright yellow. Vars key: Row 3 Column 4 (DISTR) Two Key: Row 8 Column 3 The Nine Key: Row 6 Column 4 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The One Key: Row 8 Column 2 The One Key: Row 8 Column 2 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The One Key: Row 8 Column 2 The Zero Key: Row 9 Column 2 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The Eight Key: Row 6 Column 3 Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5. The answer should be .7887003221 or roughly 79%
  2. Second Key: Row 1 Column 1 It's bright yellow. Vars key: Row 3 Column 4 (DISTR) Two Key: Row 8 Column 3 The One Key: Row 8 Column 2 Two Key: Row 8 Column 3 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The One Key: Row 8 Column 2 The Zero Key: Row 9 Column 2 The Zero Key: Row 9 Column 2 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The One Key: Row 8 Column 2 The Zero Key: Row 9 Column 2 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 The Eight Key: Row 6 Column 3 Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5. The answer should be .0062096799 or roughly 0.62%
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 Second Key: Row 1 Column 1 It's bright yellow. Vars key: Row 3 Column 4 (It should say DISTR above the key.) Press Three Key: Row 8 Column 4. The screen will now say "invnorm(". Enter p, m, s in that order with a The Comma Key: Row 5 Column 2 in between each. Press Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5.

If you want to compute P(X > x) = p. Compute P(X < x) = 1 - p.
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%

Solutions:

  1. Second Key: Row 1 Column 1 It's bright yellow. Vars key: Row 3 Column 4 (DISTR) Three Key: Row 8 Column 4 The Decimal Key: Row 9 Column 3 The Seven Key: Row 6 Column 2 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 Three Key: Row 8 Column 3 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 Three Key: Row 8 Column 4 Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5. The answer should be 21.57320153
  2. Second Key: Row 1 Column 1 It's bright yellow. Vars key: Row 3 Column 4 (DISTR) Three Key: Row 8 Column 4 The One Key: Row 8 Column 2 The Minus Key: Row 7 Column 5 The Decimal Key: Row 9 Column 3 The Eight Key: Row 6 Column 3 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 Three Key: Row 8 Column 3 The Zero Key: Row 9 Column 2 The Comma Key: Row 5 Column 2 Three Key: Row 8 Column 4 Right Parenthesis Key: Row 5 Column 4 Enter Key: Row 9 Column 5. The answer should be 17.4751363.

 

T-tests and Z-tests
Note: a z-test and z-interval are used when the population standard deviation is known. If it not known, the t-test and t-interval are used. The sample standard deviation is computed in lieu of the population standard deviation. Technically, you should only use z-tests with data sets of more than 30 numbers. The examples shown here are thankfully smaller.
T-tests: one variable
The Problem: Given a list of numbers, is the mean of that list significantly different than m0?
The Solution: Enter your data in L1. That is: Press Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5. Enter each number and press Enter Key: Row 9 Column 5. When finished, press Stat Key: Row 2 Column 3. Use the arrow keys to highlight the word TESTS. Press Two Key: Row 8 Column 3 for t-tests, then Enter Key: Row 9 Column 5.
The first line will read: Inpt: Data Stats. Press Enter Key: Row 9 Column 5 The Down Arrow.
The next line will read: m0:         . Enter m0 - the mean you want to compare your list with.
The next line will read: List: L1. Press Enter Key: Row 9 Column 5.
The next line will read: Freq: 1.  Press Enter Key: Row 9 Column 5.
The next line will read: m0: =m0   <m0  >m0. Press Enter Key: Row 9 Column 5 to accept the two sided test or arrow to the test you want, then press Enter Key: Row 9 Column 5. This is the alternate hypothesis that you are selecting.
Press Enter Key: Row 9 Column 5 one more time to select Calculate. (Selecting Draw will give you a graph.)
Example: For the following list of numbers: 4  9  7  0  6:
  1. Run a t-test to check to test if the mean is different than 5.
  2. Run a t-test to test if the mean is larger than 5.

Solution:

  1. Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The Four Key: Row 7 Column 2 Enter Key: Row 9 Column 5 The Seven Key: Row 6 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Zero Key: Row 9 Column 2 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key Two Key: Row 8 Column 3 Enter Key: Row 9 Column 5 The Down Arrow The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 (=m) The Down Arrow Enter Key: Row 9 Column 5
    You should get t=.130744091 and p=.9022895822. p is not high enough to claim that the mean is not equal to 5.
  2. Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The Four Key: Row 7 Column 2 Enter Key: Row 9 Column 5 The Seven Key: Row 6 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Zero Key: Row 9 Column 2 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key Two Key: Row 8 Column 3 Enter Key: Row 9 Column 5 The Down Arrow The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 The Right Arrow Key Enter Key: Row 9 Column 5 (< m) The Down Arrow Enter Key: Row 9 Column 5
    You should get t=.130744091 and p=.5488552089. p is not high enough to claim that the mean is not equal t
Z-tests: one variable
The Problem: Given a list of numbers, is the mean of that list significantly different than µ0?
The Solution: Enter your data in L1. That is: Press Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5. Enter each number and press Enter Key: Row 9 Column 5. When finished, press Stat Key: Row 2 Column 3.Use the arrow keys to highlight the word TESTS. Press Two Key: Row 8 Column 3 for t-tests, then Enter Key: Row 9 Column 5.
The first line will read: Inpt: Data Stats. Press Enter Key: Row 9 Column 5 The Down Arrow.
The next line will read: m0:         . Enter m0 - the mean you want to compare your list with.
The next line will read: s:       . Enter s - the population standard deviation.
The next line will read: List: L1. Press Enter Key: Row 9 Column 5
The next line will read: Freq: 1.  Press Enter Key: Row 9 Column 5.
The next line will read: m0: =m0   <m0  >m0. Press Enter Key: Row 9 Column 5 to accept the two sided test or arrow to the test you want, then press Enter Key: Row 9 Column 5. This is the alternate hypothesis you are selecting.
Press Enter Key: Row 9 Column 5 one more time to select Calculate. (Selecting Draw will give you a graph.)
Examples: The following list of numbers: 1  9  6  5  3  8 come from a distribution with s = 2.748.
  1. Run a z-test to test if the mean is 5.3.
  2. Run a z-test to test if the mean is larger than 5.

Solutions: 

  1. Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The One Key: Row 8 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 The Three Key: Row 8 Column 4 Enter Key: Row 9 Column 5 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key The One Key: Row 8 Column 2 Enter Key: Row 9 Column 5 The Down Arrow The Five Key: Row 7 Column 3 The Decimal Point Key: Row 9 Column 3 The Three Key: Row 8 Column 4 Enter Key: Row 9 Column 5 The Two Key: Row 8 Column 3 The Decimal Point Key: Row 9 Column 3 The Seven Key: Row 6 Column 2 The Four Key: Row 7 Column 2 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 (=m) The Down Arrow Enter Key: Row 9 Column 5
    You should get z=.0297123938 and p=.9762963205. p is large enough to conclude that the mean is not equal to 5.3. 
  2. Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The One Key: Row 8 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 The Three Key: Row 8 Column 4 Enter Key: Row 9 Column 5 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key The One Key: Row 8 Column 2 Enter Key: Row 9 Column 5 The Down Arrow The Five Key: Row 7 Column 3 The Decimal Point Key: Row 9 Column 3 The Three Key: Row 8 Column 4 Enter Key: Row 9 Column 5 The Two Key: Row 8 Column 3 The Decimal Point Key: Row 9 Column 3 The Seven Key: Row 6 Column 2 The Four Key: Row 7 Column 2 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 The Right Arrow Key (=m) Enter Key: Row 9 Column 5 The Down Arrow Enter Key: Row 9 Column 5
    You should get z=.0297123938 and p=.9762963205. p is large enough to conclude that the mean is not equal to 5.3. 

 

Confidence intervals
Using the t-distribution
The Problem: Find an interval for which you can be p% confident that it contains the population mean.
The Solution: Enter your data in L1. That is: Press Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5. Enter each number and press Enter Key: Row 9 Column 5. When finished, press Stat Key: Row 2 Column 3.Use the arrow keys to highlight the word TESTS. Press Eight Key: Row 6 Column 3 for t-interval, then Enter Key: Row 9 Column 5.
The first line will read: Inpt: Data Stats. Press Enter Key: Row 9 Column 5 The Down Arrow.
The next line will read: List: L1  . Press Enter Key: Row 9 Column 5.
The third line will read: Freq: 1.  Press Enter Key: Row 9 Column 5.
The next line will read: C:     . Enter p as a decimal. (So a 95% confidence interval would be entered as 0.95. Press Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5. (Yes, twice.) 
You should now see the confidence interval, along with the mean x, the standard deviation sx, and the number of data values n.
Example: For the following list of numbers: 4 7 9 0 6, construct a 95% confidence interval.

Solution: Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The Four Key: Row 7 Column 2 Enter Key: Row 9 Column 5 The Seven Key: Row 6 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Zero Key: Row 9 Column 2 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 The Down Arrow Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 The Decimal Point Key: Row 9 Column 3 The Nine Key: Row 6 Column 4 The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5
You should get (.95286, 9.4471)
Using the z-distribution
The Problem: Find an interval for which you can be p% confident that it contains the population mean. The standard deviation is known to be s.
The Solution: Enter your data in L1. That is: Press Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5. Enter each number and press Enter Key: Row 9 Column 5. When finished, press Stat Key: Row 2 Column 3.Use the arrow keys to highlight the word TESTS. Press Seven Key: Row 6 Column 2 for z-interval, then Enter Key: Row 9 Column 5 The Down Arrow.
The first line will read: Inpt: Data Stats. Press Enter Key: Row 9 Column 5.
The second line will read: s:    Enter the standard deviation. Press Enter Key: Row 9 Column 5.
The next line will read: List: L1  . Press Enter Key: Row 9 Column 5.
The next line will read: Freq: 1.  Press Enter Key: Row 9 Column 5.
The next line will read: C:     . Enter p as a decimal. (So a 95% confidence interval would be entered as 0.95. Press Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5. (Yes, twice.) 
You should now see the confidence interval, along with the mean x, the standard deviation sx, and the number of data values n.
Example: The following list of numbers: 1  9  6  5  3  8 come from a distribution with s = 2.748. Construct a 99% confidence interval.

Solution: Stat Key: Row 2 Column 3 Enter Key: Row 9 Column 5 The One Key: Row 8 Column 2 Enter Key: Row 9 Column 5 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 The Six Key: Row 7 Column 4 Enter Key: Row 9 Column 5 The Five Key: Row 7 Column 3 Enter Key: Row 9 Column 5 The Three Key: Row 8 Column 4 Enter Key: Row 9 Column 5 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Stat Key: Row 2 Column 3 The Right Arrow Key The Right Arrow Key The Seven Key: Row 6 Column 2 Enter Key: Row 9 Column 5 The Down Arrow The Two Key: Row 8 Column 3 The Decimal Point Key: Row 9 Column 3 The Seven Key: Row 6 Column 2 The Four Key: Row 7 Column 2 The Eight Key: Row 6 Column 3 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5 The Decimal Point Key: Row 9 Column 3 The Nine Key: Row 6 Column 4 The Nine Key: Row 6 Column 4 Enter Key: Row 9 Column 5 Enter Key: Row 9 Column 5
You should get (2.4436, 8.2231)

Other TI-84 Pages:

Basic Stats Statistical Graphs Normal Probabilities Hypothesis tests Confidence Intervals
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