Paired T-Test on TI-84 Calculator

Paired T-Test on TI-84 Calculator, A paired samples t-test, also called a paired t-test or dependent samples t-test, is used when two measurements are collected from the same subjects or when observations in two groups are naturally matched.

Unlike an independent two-sample t-test, a paired t-test focuses on the difference within each pair. This makes it useful for before-and-after studies, repeated measurements, matched subjects, and experiments where the same individual is measured under two conditions.

This guide explains how to perform a paired t-test on a TI-84 calculator, including data entry, calculation of paired differences, hypothesis testing, and interpretation of the result.

When Should You Use a Paired T-Test on TI-84 Calculator?

A paired t-test is appropriate when the observations are dependent.

Common examples include:

• Measuring blood pressure before and after treatment
• Comparing test scores before and after training
• Measuring product performance before and after a formulation change
• Comparing fuel efficiency with and without a treatment
• Comparing the same subjects under two experimental conditions

The key idea is that every observation in one condition has a corresponding observation in the other condition.

For example, if 11 cars are tested before and after a fuel treatment, each car provides one pair of observations.

Paired T-Test Hypotheses

Suppose we define the difference for each pair as:

d = Before − After

The hypotheses for a two-sided paired t-test are:

H₀: μd = 0

H₁: μd ≠ 0

Here, μd represents the population mean of the paired differences.

If the mean difference is zero, there is no systematic difference between the two conditions.

If the mean difference is not zero, there is evidence of a difference between the conditions.

Example: Paired T-Test on a TI-84 Calculator

Suppose researchers want to determine whether a fuel treatment changes the fuel efficiency of cars.

They measure the MPG of 11 cars before and after the treatment.

Because each car is measured twice, the observations are paired.

We will use the following example data:

The exact numbers in your own dataset can be entered in the same way.

Step 1: Enter the Paired Data

First, enter the two sets of measurements into the TI-84 calculator.

1. Press STAT.
2. Select 1:Edit.
3. Enter the first measurement for each pair in L1.
4. Enter the corresponding second measurement in L2.

It is important that the rows remain correctly matched.

For example, the first value in L1 must correspond to the first value in L2, the second value in L1 to the second value in L2, and so on.

Step 2: Create a Difference List

The easiest way to perform a paired t-test on a TI-84 is to calculate the difference between every pair and then perform a one-sample t-test on those differences.

Suppose we define:

Difference = Before − After

To create the difference list:

1. Highlight the name L3 at the top of the third column.
2. Press 2nd → 1 to insert L1.
3. Press the subtraction key −.
4. Press 2nd → 2 to insert L2.
5. Press ENTER.

The calculator will automatically calculate the difference for every pair.

The expression should look like:

L1-L2

Your L3 column now contains the paired differences.

Why Do We Create a Difference List?

This is the central idea behind a paired t-test.

Instead of comparing two separate groups, we transform the problem into a one-sample test of the mean difference.

In other words:

Paired t-test → Calculate differences → Test whether mean difference = 0

This approach accounts for the fact that the two observations come from the same subject or matched pair.

Step 3: Open the T-Test Menu

Now perform a one-sample t-test using the difference values in L3.

1. Press STAT.
2. Use the right arrow to select TESTS.
3. Select 2:T-Test.
4. Press ENTER.

The TI-84 will display several options.

Step 4: Enter the T-Test Information

Set the options as follows.

Inpt

Select:

Data

because the paired differences are stored in L3.

μ0

Enter:

0

The null hypothesis assumes that the average paired difference is zero.

List

Enter:

L3

To enter L3, press:

2nd → 3

Freq

Leave the frequency as:

1

Alternative Hypothesis

For a two-sided test, select:

≠μ0

This corresponds to:

H₁: μd ≠ 0

If your research question is specifically directional, you can instead select:

• < μ0 for a left-tailed test
• > μ0 for a right-tailed test

Finally, highlight Calculate and press ENTER.

Step 5: Understand the TI-84 Output

The calculator will display several results, including:

• t = test statistic
• p = p-value
• x̄ = mean of the paired differences
• Sx = standard deviation of the differences
• n = number of pairs

For example, your output might look similar to:

μ ≠ 0t = -1.8751p = 0.0903x̄ = -1.5455Sx = 2.7336n = 11

The exact results depend on the data entered into the calculator.

How to Interpret the P-Value

The most important value for the hypothesis test is the p-value.

Suppose the calculator gives:

p = 0.0903

Using a significance level of:

α = 0.05

we compare:

0.0903 > 0.05

Therefore, we fail to reject the null hypothesis.

There is not enough statistical evidence at the 5% significance level to conclude that the population mean difference is different from zero.

In the fuel-treatment example, this means the sample does not provide sufficient evidence that the treatment changes the average MPG.

Important Statistical Wording

It is better to say:

“We fail to reject the null hypothesis.”

rather than:

“We accept the null hypothesis.”

A non-significant result does not prove that the two conditions are identical. It means that the available evidence is insufficient to establish a statistically significant difference.

Understanding the T-Statistic

The paired t-test statistic is based on the mean difference, its variability, and the number of pairs.

The general formula is:

t = (d̄ − μd) / (sd / √n)

Where:

• d̄ = mean paired difference
• μd = hypothesized population mean difference
• sd = standard deviation of the paired differences
• n = number of pairs

For the usual paired t-test:

μd = 0

so the formula becomes:

t = d̄ / (sd / √n)

A larger absolute value of t provides stronger evidence against the null hypothesis, all else being equal.

How to Perform a One-Tailed Paired T-Test

Sometimes the research question is directional.

For example:

Does the fuel treatment increase average MPG?

If you define:

d = After − Before

then the hypotheses could be:

H₀: μd = 0

H₁: μd > 0

On the TI-84, select:

> μ0

instead of ≠ μ0.

Be careful about how you define the difference. If you reverse the subtraction order, the direction of the alternative hypothesis also changes.

Paired T-Test vs Two-Sample T-Test

These tests are often confused.

Paired T-Test

Use a paired t-test when observations are related or matched.

Examples:

• Before vs after
• Same person under two conditions
• Same machine before and after modification
• Matched subjects

Two-Sample T-Test

Use an independent two-sample t-test when observations come from two independent groups.

Examples:

• Treatment group vs unrelated control group
• Men vs women when individuals are independently sampled
• Machine A vs a separately sampled Machine B

The choice of test depends on the study design, not simply on the fact that there are two columns of numbers.

Checking the Paired T-Test Assumptions

Before interpreting a paired t-test, consider the following assumptions.

1. The Data Are Paired

Each observation in one condition should have a meaningful counterpart in the other condition.

2. The Pairs Are Independent

Different pairs should generally be independent of one another.

For example, measurements from Car 1 should not influence measurements from Car 2.

3. The Differences Are Approximately Normally Distributed

The t-test is applied to the differences, not separately to the two original variables.

This assumption becomes particularly important when the number of pairs is small.

You can examine the distribution of the values in L3 using graphical methods such as a histogram or boxplot.

4. The Difference Variable Does Not Contain Extreme Outliers

A few extreme differences can have a substantial impact on the mean and consequently on the paired t-test.

Reporting the Results

A clear statistical conclusion should include the test statistic, degrees of freedom, p-value, and conclusion.

For example:

A paired-samples t-test was conducted to compare MPG before and after the fuel treatment. The test produced a t-statistic of -1.875 with 10 degrees of freedom and a p-value of 0.0903. Because p > 0.05, we fail to reject the null hypothesis and conclude that there is insufficient evidence of a statistically significant difference in mean MPG.

For a paired t-test:

df = n − 1

Therefore, with 11 pairs:

df = 11 − 1 = 10

Frequently Asked Questions

Can I perform a paired t-test directly on a TI-84?

The practical approach on a TI-84 is to calculate the paired differences in a separate list and then use the T-Test function on that difference list.

Why do I need to calculate the differences?

A paired t-test is fundamentally a one-sample t-test applied to the differences within each pair.

What should μ0 be for a paired t-test?

For testing whether there is any difference between the two conditions, use:

μ0 = 0

What does a significant paired t-test mean?

A statistically significant result provides evidence that the population mean difference is different from the hypothesized value, usually zero.

What does p > 0.05 mean?

It means the result is not statistically significant at the 5% significance level. You would generally fail to reject the null hypothesis.

Conclusion

A paired t-test on a TI-84 calculator can be performed efficiently by converting the two related measurements into a single list of paired differences. Once the differences are calculated, the TI-84’s T-Test function can determine whether the average difference is statistically different from zero.

The most important steps are:

1. Enter the two paired measurements.
2. Create a difference list.
3. Run STAT → TESTS → T-Test.
4. Use μ0 = 0.
5. Select the appropriate alternative hypothesis.
6. Examine the t-statistic and p-value.
7. Draw your conclusion based on the chosen significance level.

This method is useful for before-and-after experiments, repeated measurements, matched studies, and many other situations where observations naturally occur in pairs.