How To Do A Paired T Test On Ti 84

Introduction

A paired t‑test (also called a dependent‑samples t‑test) is a statistical procedure used to determine whether the mean difference between two related measurements is significantly different from zero. Typical situations include pre‑test/post‑test designs, matched‑pair studies, or any scenario where each observation in one sample can be naturally linked to a corresponding observation in the other sample.

On the TI‑84 graphing calculator, the paired t‑test is not a separate menu item; instead, you first compute the differences between the paired observations and then run a one‑sample t‑test on that difference list. This approach leverages the calculator’s built‑in T‑Test function (found under STAT → TESTS → 2:T‑Test).

The following article walks you through the entire process—from data entry to interpretation—providing a thorough, step‑by‑step guide, a concrete example, the underlying theory, common mistakes to avoid, and a set of frequently asked questions. By the end, you will be able to confidently perform a paired t‑test on your TI‑84 and understand what the results mean for your research or classroom project.

Detailed Explanation

What the Paired t‑Test Evaluates

The null hypothesis (H₀) for a paired t‑test states that the true mean of the difference scores (μ_d) is zero:

[ H_0: \mu_d = 0 ]

The alternative hypothesis (H₁) can be two‑sided (μ_d ≠ 0), left‑sided (μ_d < 0), or right‑sided (μ_d > 0), depending on the research question.

The test statistic is calculated as

[ t = \frac{\bar{d}}{s_d/\sqrt{n}} ]

where

• (\bar{d}) = sample mean of the differences,
• (s_d) = sample standard deviation of the differences,
• (n) = number of paired observations.

Under H₀ and the assumption that the differences are approximately normally distributed, this statistic follows a t‑distribution with (df = n-1) degrees of freedom. The calculator returns the t‑value, the associated p‑value, and often the confidence interval for μ_d.

Why Compute Differences First?

The TI‑84’s T‑Test function is designed for a single sample (or a single list of data). By forming the difference list (L_3 = L_1 - L_2) (or (L_2 - L_1) depending on the direction of interest), we convert the paired problem into a one‑sample problem: we now ask whether the mean of this single list differs from zero Surprisingly effective..

Assumptions to Check

1. Independence of pairs – each pair is unrelated to any other pair.
2. Normality of the difference scores – especially important when (n < 30). With larger samples the test is dependable to mild deviations.
3. Scale of measurement – the data should be at least interval‑level (e.g., test scores, reaction times).

If the normality assumption is questionable, you can inspect a histogram or a normal probability plot of the difference list (the TI‑84 can produce these via STAT PLOT).

Step‑by‑Step Procedure

Below is a complete workflow you can follow on any TI‑84 Plus (or TI‑84 Plus CE) calculator Most people skip this — try not to..

1. Enter the Paired Data

1. Press STAT → 1:Edit….
2. Clear any existing data in the lists you plan to use (highlight the list name, press CLEAR, then ENTER).
3. Enter the first measurement of each pair into L1 (press ENTER after each value).
4. Enter the second measurement of each pair into L2 in the same order.

Tip: Keep the pairs aligned row‑by‑row; the first value in L1 must correspond to the first value in L2, and so on Simple, but easy to overlook. That's the whole idea..

2. Compute the Difference List

1. Move the cursor to the header of an empty list, say L3.
2. Type the expression L1 - L2 (or L2 - L1 if you prefer the opposite sign) and press ENTER.
3. The calculator will automatically fill L3 with the element‑wise differences.

3. Run the One‑Sample t‑Test on L3

1. Press STAT, then right‑arrow to TESTS.

2. Scroll down to 2:T-Test and press ENTER.

3. Choose the Data input method (highlight Data and press ENTER).

4. Set the following fields:

   • Inpt: Data (already selected)
   • μ₀: 0 (the hypothesized mean difference under H₀)
   • List: L3 (the difference list)
   • Freq: 1
   • μ: choose ≠ μ₀ for a two‑tailed test, < μ₀ for left‑tailed, or > μ₀ for right‑tailed, according to your hypothesis.

5. Highlight Calculate and press ENTER Most people skip this — try not to..

4. Interpret the Output

The calculator returns:

• t – the test statistic.
• p – the p‑value (probability of observing a t as extreme as, or more extreme than, the one calculated, assuming H₀ is true).
• (\bar{x̄x – the sample mean of the differences ((\bar{d})).
• Sx – the sample standard deviation of the differences ((s_d)).
• n – the number of pairs.

If the p‑

value is less than your chosen significance level (α, typically 0.That said, 05), you reject the null hypothesis and conclude that the mean difference is statistically different from zero. If the p‑value is greater than α, you fail to reject the null hypothesis; there is insufficient evidence to claim a systematic difference between the paired measurements Simple, but easy to overlook..

Most guides skip this. Don't.

Worked Example

Suppose a researcher measures the reaction time (in milliseconds) of 10 subjects before and after a caffeine dose.

Following the steps above:

1. Define L3 = L1 - L2 (positive values indicate slowing; negative values indicate speeding up).
2. 2. Enter Before in L1 and After in L2.
      Run T-Test on L3 with μ₀ = 0 and μ: ≠ μ₀.

The calculator returns:

• t = 8.94
• p = 1.2 × 10⁻⁶
• x̄ = 20
• **Sx = 7.

Because p < 0.05, we reject H₀. The data provide strong evidence that caffeine reduces reaction time (the mean difference of 20 ms is significantly different from zero).

Reporting the Results

When writing up a paired t‑test, include:

• The test name (paired-samples t‑test).
• The test statistic, degrees of freedom (df = n – 1), and p‑value.
• The sample mean difference and its standard deviation (or standard error).
• A 95% confidence interval for the mean difference (the TI‑84 provides this if you select Draw instead of Calculate, or you can compute it manually: (\bar{d} \pm t_{\alpha/2, n-1} \cdot \frac{s_d}{\sqrt{n}})).

Example write‑up:

A paired-samples t‑test indicated that reaction times after caffeine (M = 280, SD = 15.8) were significantly faster than before caffeine (M = 300, SD = 15.8), t(9) = 8.94, p < .001, 95% CI [14.3, 25.7] ms.

Common Pitfalls

Conclusion

The paired t‑test on the TI‑84 is a streamlined, reliable method for determining whether the mean of a set of difference scores deviates from zero. On the flip side, remember to verify the independence and normality assumptions—especially with small samples—and to state your hypotheses clearly before choosing a one- or two-tailed alternative. Consider this: by entering the raw paired measurements, creating a difference list in a single step, and running the built-in one-sample T-Test, you obtain the test statistic, p‑value, and descriptive statistics needed for both inference and reporting. With these practices in place, the calculator becomes a powerful ally for analyzing repeated-measures and matched-pairs designs in any research or classroom setting.