Two-Sample T-Test on TI-84 Calculator
Two-Sample T-Test on TI-84 Calculator, A two-sample t-test is a statistical method used to determine whether there is sufficient evidence that the means of two independent populations are different.
It is commonly used when comparing two separate groups, such as a treatment group and a control group. Unlike a paired t-test, the observations in a two-sample t-test do not need to be matched or measured on the same subjects.
This guide explains how to perform a two-sample t-test on a TI-84 calculator, using summary statistics, and how to interpret the resulting test statistic, degrees of freedom, and p-value.
When Should You Use a Two-Sample T-Test?
A two-sample t-test is appropriate when you want to compare the means of two independent groups.
For example, you might want to compare:
- Average test scores between two classes
- Average sales between two regions
- Mean blood pressure between two independent groups
- Average product performance between two formulations
- Fuel efficiency between treatment and control groups
The two groups should represent independent samples.
Hypotheses for a Two-Sample T-Test
Suppose the population means are:
- μ₁ = mean of population 1
- μ₂ = mean of population 2
For a two-sided test, the hypotheses are:
H₀: μ₁ = μ₂
H₁: μ₁ ≠ μ₂
The null hypothesis states that the two population means are equal, while the alternative hypothesis states that they are different.
Depending on your research question, you can also perform a one-sided test:
Left-tailed: H₁: μ₁ < μ₂
Right-tailed: H₁: μ₁ > μ₂
Example: Two-Sample T-Test on a TI-84
Suppose researchers want to determine whether a new fuel treatment changes the average MPG of a particular type of car.
They randomly select two independent groups:
- Control group: 12 cars without the treatment
- Treatment group: 12 cars with the treatment
The summary statistics are:
| Group | Sample Size | Mean MPG | Sample SD |
|---|---|---|---|
| Control | 12 | 21.00 | 2.73 |
| Treatment | 12 | 22.75 | 3.25 |
We want to determine whether there is a statistically significant difference in average MPG between the two groups.
Because we are comparing two independent groups, a two-sample t-test is appropriate.
Step 1: Open 2-SampTTest
On your TI-84 calculator:
- Press STAT.
- Use the arrow key to move to TESTS.
- Select 2-SampTTest.
- Press ENTER.
The calculator will display the input screen for the two-sample t-test.
Step 2: Enter the Summary Statistics
Because we already know the means, standard deviations, and sample sizes, we can use the Stats option rather than entering the individual observations.
Set:
Inpt: Stats
Then enter the following information.
First Group
x₁ = 21
This is the sample mean for the control group.
Sx₁ = 2.73
This is the sample standard deviation for the control group.
n₁ = 12
This is the number of observations in the control group.
Second Group
x₂ = 22.75
This is the sample mean for the treatment group.
Sx₂ = 3.25
This is the sample standard deviation for the treatment group.
n₂ = 12
This is the number of observations in the treatment group.
Step 3: Select the Alternative Hypothesis
Next, select the appropriate alternative hypothesis.
Because our research question asks whether the average MPG is different, rather than specifically higher or lower, we use a two-tailed test.
Select:
μ₁ ≠ μ₂
This corresponds to:
H₁: μ₁ ≠ μ₂
The other options are:
- μ₁ < μ₂ → left-tailed test
- μ₁ > μ₂ → right-tailed test
For this example, select ≠ μ₂.
Step 4: Choose the Variance Option
The TI-84 will also ask:
Pooled: Yes or No
For this example, select:
No
This performs the test without assuming that the two populations have equal variances. This is commonly referred to as Welch’s two-sample t-test.
Unless your statistical procedure specifically requires the equal-variance assumption, using the unpooled option is often a reasonable choice.
After entering all the information, highlight:
Calculate
and press ENTER.
Step 5: Read the TI-84 Results
For these summary statistics, the calculator produces approximately:
t = -1.4283
p = 0.1677
df = 21.36
The calculator will also display the sample statistics you entered.
What Does the T-Statistic Mean?
The t-statistic measures how far the observed difference between the sample means is from zero relative to the estimated standard error.
In this example:
t ≈ -1.428
The negative sign occurs because the first group’s mean is smaller than the second group’s mean:
21.00 − 22.75 = -1.75
The sign indicates the direction of the difference. For a two-tailed test, the magnitude of the statistic is particularly important when assessing evidence against the null hypothesis.
What Does the P-Value Mean?
The p-value is approximately:
p = 0.1677
Using a significance level of:
α = 0.05
we compare:
0.1677 > 0.05
Therefore, the result is not statistically significant at the 5% level.
We fail to reject the null hypothesis.
This means the sample does not provide sufficient statistical evidence to conclude that the population mean MPG differs between the control and treatment groups.
Importantly, this does not prove that the two population means are exactly equal. It means that the observed difference is not statistically convincing enough under the selected significance level.
Understanding the Degrees of Freedom
The TI-84 gives approximately:
df = 21.36
When the variances are not assumed to be equal, the degrees of freedom can be non-integer.
This is normal for Welch’s two-sample t-test.
You should not be concerned if the TI-84 displays a decimal value for the degrees of freedom.
Interpreting the Difference Between the Means
The observed sample means are:
Control = 21.00 MPG
Treatment = 22.75 MPG
The observed difference is:
21.00 − 22.75 = −1.75 MPG
Or, expressed in the opposite direction:
22.75 − 21.00 = 1.75 MPG
So, the treatment group had an average MPG that was 1.75 MPG higher than the control group in this sample.
However, the hypothesis test indicates that this observed difference is not statistically significant at α = 0.05.
How to Perform the Test Using Raw Data
If you have the individual observations rather than summary statistics, you can also perform the test directly from two lists.
For example:
- Enter the control observations into L1.
- Enter the treatment observations into L2.
Then:
- Press STAT.
- Move to TESTS.
- Select 2-SampTTest.
- Choose Data under
Inpt. - Set List1 = L1.
- Set List2 = L2.
- Set Freq1 = 1.
- Set Freq2 = 1.
- Select the appropriate alternative hypothesis.
- Choose the appropriate pooled setting.
- Select Calculate.
The TI-84 will calculate the sample statistics and perform the test from the raw observations.
Two-Sample T-Test vs Paired T-Test
It is important not to confuse these two tests.
Two-Sample T-Test
Use it when the two groups are independent.
Example:
12 different cars receive treatment and another 12 different cars form the control group.
Paired T-Test
Use it when observations are naturally matched.
Example:
The same 12 cars are tested before and after receiving the treatment.
The experimental design determines which test is appropriate.
What If the P-Value Is Less Than 0.05?
Suppose your TI-84 produces:
p < 0.05
Then you would reject the null hypothesis at the 5% significance level.
You would conclude that there is statistically significant evidence that the two population means differ.
For the current example, however:
p ≈ 0.1677
so we fail to reject H₀.
Common Mistakes When Using 2-SampTTest
Using a Paired Test for Independent Groups
If the observations come from separate, unrelated groups, do not use a paired t-test.
Using the Wrong Alternative Hypothesis
If your question asks whether the means are different, select:
μ₁ ≠ μ₂
If you specifically want to determine whether one mean is greater than the other, use the corresponding one-tailed alternative.
Automatically Choosing Pooled Variances
The pooled option assumes equal population variances. If that assumption is not justified, use the unpooled option.
Interpreting a Non-Significant Result as Proof of Equality
A p-value greater than 0.05 does not prove that the two means are equal. It indicates that there is insufficient evidence to establish a difference at the selected significance level.
How to Report a Two-Sample T-Test
A concise statistical report could look like this:
An independent two-sample t-test was performed to compare the average MPG of the control and fuel-treatment groups. The control group had a mean MPG of 21.00, while the treatment group had a mean MPG of 22.75. The test produced t = -1.43, df = 21.36, and p = 0.168. Since 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 between the two groups.
Frequently Asked Questions
What does 2-SampTTest do on a TI-84?
The 2-SampTTest function compares the means of two independent populations using sample data or summary statistics.
Should I select Stats or Data?
Select Stats when you already know the sample means, standard deviations, and sample sizes.
Select Data when you have the individual observations stored in calculator lists.
Should I use pooled or unpooled?
Use Pooled: No when you do not want to assume that the two population variances are equal. Use the pooled option only when the equal-variance assumption is appropriate for your analysis.
What does a p-value of 0.1677 mean?
At α = 0.05, it means the result is not statistically significant. You would fail to reject the null hypothesis.
Can the degrees of freedom be a decimal?
Yes. With the unpooled version of the two-sample t-test, the degrees of freedom can be fractional.
Conclusion
The two-sample t-test on a TI-84 calculator provides a convenient way to compare the means of two independent groups. When summary statistics are available, you can enter the two sample means, standard deviations, and sample sizes directly into 2-SampTTest.
For the fuel-treatment example, the observed difference between the sample means is 1.75 MPG, but the p-value of approximately 0.1677 is greater than 0.05. Therefore, there is insufficient evidence to conclude that the treatment and control populations have different mean MPG values.
The key workflow is simple:
STAT → TESTS → 2-SampTTest → Stats/Data → Enter values → Select hypothesis → Choose pooled/unpooled → Calculate
Understanding both the calculator procedure and the statistical assumptions behind the test will help you use the TI-84 correctly rather than simply relying on the calculator’s output.
