Hypothesis Tests and Confidence Intervals

The position of the graphically represented keys can be found by moving your mouse on top of the graphic. The top row of function keys does not count as a row. Hence, row 1 starts with the yellow shift key

On this page, I will describe how to do the following functions:
A one-sample t-test
A one-sample z-test
A z-confidence interval
A t-confidence interval

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:  Get into STAT mode and type in your list. Press The F 3 Key: Third on the row of grey keys. (TEST) and different menu tags will appear. Press The F 2 Key: Second of the grey row of keys on top. for the t-test, then The F 1 Key: First of the grey row of keys on top. (1-S, for one sample).
The first line says Data:List. Press .
The second line says m ≠ m0. We want this so press . If you want the alternate hypothesis to be m < m0, press The F 2 Key: Second of the grey row of keys on top.. For m > m0, press The F 3 Key: Third on the row of grey keys..
The third line says m0:0. Type in your m0 value and press The E X E Key: Row 8, Column 5.
Press The E X E Key: Row 8, Column 5 a second time to see the results.
Example: For the following list of numbers: 7 6 5 7 4 3 4 4 6 5:
  1. Run a t-test to check to test if the mean is different than 4.
  2. Run a t-test to test if the mean is larger than 5.

Solution: 

  1. (Once in STAT mode) The Seven Key: Row 5, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The Seven Key: Row 5, Column 1 The E X E Key: Row 8, Column 5 The Four Key: Row 6, 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 Four Key: Row 6, Column 1 The E X E Key: Row 8, Column 5 The Four Key: Row 6, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The F 3 Key: Third on the row of grey keys. (TEST) The F 2 Key: Second of the grey row of keys on top. (t) The F 1 Key: First of the grey row of keys on top. (1-S) The Four Key: Row 6, Column 1 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5
  2. (Once in STAT mode) The Seven Key: Row 5, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The Seven Key: Row 5, Column 1 The E X E Key: Row 8, Column 5 The Four Key: Row 6, 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 Four Key: Row 6, Column 1 The E X E Key: Row 8, Column 5 The Four Key: Row 6, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The F 3 Key: Third on the row of grey keys. (TEST) The F 2 Key: Second of the grey row of keys on top. (t) The F 1 Key: First of the grey row of keys on top. (1-S) The F 3 Key: Third on the row of grey keys. (>) The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5

You should see the following for the first question:
m      ≠4
t        =2.53846154
p       =0.031790927
x       =5.1
xsn-1=1.37032032
n       = 10

Since p<0.05, we reject the null hypothesis and affirm that the mean is not 4.

For the second question, t=0.230769231 and p=0.41132727. 
     Since p>0.05, we cannot reject the null. That is, the mean may very well have been 5.
Z-tests: one variable
The Problem: Given a list of numbers, is the mean of that list significantly different than µ0?
The Solution: The Solution:  Get into STAT mode and type in your list. Press The F 3 Key: Third on the row of grey keys. (TEST) and different menu tags will appear. Press The F 1 Key: First of the grey row of keys on top. for the Z-test, then The F 1 Key: First of the grey row of keys on top. (1-S, for one sample).
The first line says Data:List. Press .
The second line says m ≠ m0. We want this so press . If you want the alternate hypothesis to be m < m0, press The F 2 Key: Second of the grey row of keys on top.. For m > m0, press The F 3 Key: Third on the row of grey keys..
The third line says m0:0. Type in your m0 value and press The E X E Key: Row 8, Column 5.
The fourth line says s:0. Type in your s value and press The E X E Key: Row 8, Column 5.
Press The E X E Key: Row 8, Column 5 a second time to see the results.
Examples: The following list of numbers: 1  9  6  5  3  8 come from a distribution with s = 2.75.
  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. The One Key: Row 7, Column 1 The E X E Key: Row 8, Column 5 The Nine Key: Row 5, Column 3 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, 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 Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The F 3 Key: Third on the row of grey keys. (TEST) The F 1 Key: First of the grey row of keys on top. (Z) The F 1 Key: First of the grey row of keys on top. (1-S) The Five Key: Row 6, Column 2 The Decimal Point Key: Row 8, Column 2 The Three Key: Row 7, Column 3 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The Decimal Point Key: Row 8, Column 2 The Seven Key: Row 5, Column 1 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5
  2. The One Key: Row 7, Column 1 The E X E Key: Row 8, Column 5 The Nine Key: Row 5, Column 3 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The Five Key: Row 6, 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 Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The F 3 Key: Third on the row of grey keys. (TEST) The F 1 Key: First of the grey row of keys on top. (Z) The F 1 Key: First of the grey row of keys on top. (1-S) The F 3 Key: Third on the row of grey keys. (>) The Five Key: Row 6, Column 2 The Decimal Point Key: Row 8, Column 2 The Three Key: Row 7, Column 3 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The Decimal Point Key: Row 8, Column 2 The Seven Key: Row 5, Column 1 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5

For question 1, you should get z=0.02969079 and p=0.97631355.
For question 2, you should get z=0.29690785 and p=0.383268496.
In both cases, we cannot reject the null.

 

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: Get into STAT mode and enter the data. Press The F 4 Key: Fourth on the row of grey keys. (INTR, for interval). Press The F 2 Key: Second of the grey row of keys on top. to choose the t-interval, then The F 1 Key: First of the grey row of keys on top. (1-s, for one-sample).
The first line reads Data: List. Simply press .
The next reads C-level. Enter p as a decimal and press The E X E Key: Row 8, Column 5
The next line reads List  :List1. We want this so press The E X E Key: Row 8, Column 5 and the result screen should show.
Example: For the following list of numbers: 8 4 3 8 1 2 2 0 0 6, construct a 95% confidence interval.

Solution: (once in STAT mode)  The Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The Four Key: Row 6, 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 Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The E X E Key: Row 8, Column 5  The Two Key: Row 7, Column 2 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The E X E Key: Row 8, Column 5 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The F 4 Key: Fourth on the row of grey keys. (INTR) The F 2 Key: Second of the grey row of keys on top. The F 1 Key: First of the grey row of keys on top. (1-s) The Zero Key: Row 8, Column 1 The Decimal Point Key: Row 8, Column 2 The Nine Key: Row 5, Column 3 The Five Key: Row 6, Column 2 The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5

You should see the following:

Left  =1.23546245
Right=5.56453755
x       =3.4
xsn-1=3.02581486
n       =10
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: Get into STAT mode and enter the data. Press The F 4 Key: Fourth on the row of grey keys. (INTR, for interval). Press The F 1 Key: First of the grey row of keys on top. to choose the Z-interval, then The F 1 Key: First of the grey row of keys on top. (1-s, for one-sample).
The first line reads Data: List. Simply press .
The next reads C-level. Enter p as a decimal and press The E X E Key: Row 8, Column 5
The next line reads s: Enter s and press The E X E Key: Row 8, Column 5 The E X E Key: Row 8, Column 5.
Example: For the following list of numbers: 8 4 3 8 1 2 2 0 0 6, construct a 95% confidence interval. Take s to be 3.

Solution: (once in STAT mode)  The Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The Four Key: Row 6, 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 Eight Key: Row 5, Column 2 The E X E Key: Row 8, Column 5 The One Key: Row 7, Column 1 The E X E Key: Row 8, Column 5  The Two Key: Row 7, Column 2 The E X E Key: Row 8, Column 5 The Two Key: Row 7, Column 2 The E X E Key: Row 8, Column 5 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Zero Key: Row 8, Column 1 The E X E Key: Row 8, Column 5 The Six Key: Row 6, Column 3 The E X E Key: Row 8, Column 5 The F 4 Key: Fourth on the row of grey keys. (INTR) The F 1 Key: First of the grey row of keys on top. (Z) The F 1 Key: First of the grey row of keys on top. (1-s) The Zero Key: Row 8, Column 1 The Decimal Point Key: Row 8, Column 2 The Nine Key: Row 5, Column 3 The Five Key: Row 6, 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 E X E Key: Row 8, Column 5

You should see the following:

Left  =1.5406149
Right=5.2593851
x       =3.4
xsn-1=3.02581486
n       =10
 

Other Casio fx-9860g Pages:

Casio's online manual sample problem set home page
Scatterplots and Histograms Basic Stats Normal Probabilities