Using the TI-83 to Perform a Statistical Test

THE TWO-SAMPLE t-TEST {13-13}

Here, the difference within a single population (group, in this investigation)
is being measured. The null hypothesis is defined as **µ1=µ2**.
µ1 represents the mean of the data in list 1. µ2 represents
the mean of the data in list 2. Therefore, this mathematical expression
indicates that there is no difference between the means of the data in
list 1 & list 2, which would be expected of two samples from the same
population. The statistical test is used when you have two samples (or
populations) when neither sample (or population) standard deviation is
known. In this example, sample 1 refers to list 1 and sample 2 refers to
list 2.

- Press [
**STAT**] to enter statistics mode. Press [] twice to display the TESTS menu. Your screen should look like this:

Press [**4**] to select "4:2-SampTTest..."
You will see **2-SampTTest** at the top of the window. Position the
cursor over the word **Data **and press [**ENTER**]
to select the Data window. You will see a window with several parameters.
Using the blue arrow keys, position the cursor over each parameter and
make the appopriate changes. Your screen should look like this:

- Shown below are the results of the 2-sample t-test. (The right hand screen displays the parameters if you scroll down using the [] key.)

List1:L1 |
This is the name of the first list of data being tested. |

List2:L2 |
This is the name of the second list of data being tested. |

Freq1:1 |
This is the frequency of the occurrence of a particular value of L1.
Each value in L1 is being listed separately. |

Freq2:1 |
This is the frequency of the occurrence of a particular value of L2.
Each value in L2 is being listed separately. |

µ1:µ2 |
This is the alternative hypothesis against which the null hypothesis is tested. |

Pooled:Yes |
This pools the data in computing the variance. |

After you have made all changes in the Data window, position the cursor
over **Calculate **and press [**ENTER**].
The calculator will then process your request.

and

Here is a brief explanation, with mathematical forumulae, of each of
the parameters. Substitute **L1 **for sample 1 & **L2 **for sample
2.

**µ1µ2
**(This is the alternative hypothesis against which the null hypothesis
is tested.)

**t** represents the test statistic used in a t-test.

**p** represents the p-value. It indicates probability.

**df** indicates the degrees of freedom. It is the number of free
variables in a system. For a 2-sample t-test, the formula is:

**df **= **n1 **+ **n2 **- 2

**x1 **is the sample mean for sample 1. The formula for the arithmetic
mean (average) **x1 **is:

x1(n1) is the average of captures (of plain circles on a plain background) for each team.

**x2 **is the sample mean for sample 2. The formula for the arithmetic
mean (average) **x2 **is:

x2(n2) is the average of captures (of print circles on a plain background) for each team.

The standard deviation is a standard measurement to assess distance. It is a measure of the spread or dispersion of data values. Mathematically, it is the root mean squared deviation. (The square of the standard deviation is variance.)

**Sx1 **is the sample standard deviation of x for sample 1. The formula
for the standard deviation **Sx1 **is:

**Sx2 **is the sample standard deviation of x for sample 2. The formula
for the standard deviation **Sx2 **is:

**Sxp **is the pooled standard deviation for both lists, **L1 **&
**L2**.

- Press [
**STAT**] to return to the statistics mode. Press [] twice to display the TESTS menu. Press [**4**] to select "4:2-SampTTest..." You will see**2-SampTTest**at the top of the window. Position the cursor over the word**Stats**and press [**ENTER**] to select the Statistics window. You will see a window with several parameters from the Data window. Your screens should look like this:

**n1 **is the number of data points for list 1. It corresponds to
the number of elements in **L1**.

**n2 **is the number of data points for list 2. It corresponds to
the number of elements in **L2**.

and

- After you have made all changes in the Statistics window, position
the cursor over
**Draw**and press [**ENTER**]. The calculator will then process your request. Your screen should look like this:

The **t**-statistic can be any number and is the number of population
standard deviations away from the population mean. In other words, **t**
measures how close is the sample difference to the expected difference
of zero. The closer **t **is to zero, the more likely two samples are
from the same population. The **p**-value is the probablity, which ranges
from zero to one, and indicates how likely a sample difference (with the
**t**-statistic) is from zero. The closer **p **is to zero, the more
unlikely the two samples are from the same population. The null hypothesis
is then rejected, and it is concluded that the two samples are from two
different populations. In this example, the **p**-value is not close
enough to zero to reject the null hypothesis. However, if the data from
Groups 6, 9 & 11 are not considered, the **t**-statistic is -1.348617178
and the **p**-value is 0.1988726469. Due to the low **p**-value,
the null hypothesis can be rejected and one can state that the two samples
(**L1** & **L2**) are from two different populations, which intuitively
makes sense given the conditions under which each sample was collected.

- Repeat this entire process (steps 1-4) for list 4 & list 5. Replace
list 4 each place you see list 1 and list 5 each place you see list 2.
If all the data are considered, the
**t**-statistic is 1.94333593 and the**p**-value is 0.0661836666. Here, because all of the**L4**values are greater than or equal to their corresponding**L5**values, it is not necessary to discount some of the data. (If the data from Groups 6, 9 & 11 are not considered, the**t**-statistic is 1.364552406 and the**p**-value is 0.193926306.) Again, due to the low**p**-value, the null hypothesis can be rejected and one can state the two samples (**L4**&**L5**) are from two different populations, which intuitively makes sense given the conditions under which each sample was collected.

Go to other sections in "Using the TI-83 to Perform a Statistical Test":

- Opening Page
- The Basics
- Entering Data Values and Computing Averages
- The t-Test
- The One-Sample t-Test
- Acknowledgements & Where to Get More Information

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