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.

1. 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:

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.



2. 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.)

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:

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.

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.



3. 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:

4. 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.

5. 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.

• 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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