One-Sample T-Test on a TI-84 Calculator
One-Sample T-Test on a TI-84 Calculator, A one-sample t-test is used to determine whether the mean of a population is significantly different from a specific hypothesized value.
It is especially useful when the population standard deviation is unknown and the analysis is based on sample data.
In this guide, we’ll walk through how to perform a one-sample t-test on a TI-84 calculator, using a practical example and explaining how to interpret the calculator output.
What Is a One-Sample T-Test?
A one-sample t-test compares a sample mean with a hypothesized population mean.
The hypotheses are typically written as:
- Null hypothesis: H0:μ=μ0
- Alternative hypothesis: Ha:μ=μ0
Depending on the research question, the alternative hypothesis can also be one-sided:
- Left-tailed: Ha:μ<μ0
- Right-tailed: Ha:μ>μ0
- Two-tailed: Ha:μ=μ0
The t-statistic is calculated as: t=s/nxˉ−μ0
where:
- xˉ = sample mean
- μ0 = hypothesized population mean
- s = sample standard deviation
- n = sample size
Example: One-Sample T-Test on a TI-84
Suppose researchers want to determine whether a particular type of car gets an average of 20 miles per gallon (mpg).
They randomly select 74 cars and obtain:
- Sample mean = 21.29 mpg
- Sample standard deviation = 5.78 mpg
- Sample size = 74
- Hypothesized mean = 20 mpg
We want to test whether the true population mean differs from 20 mpg.
The hypotheses are: H0:μ=20 Ha:μ=20
Because we are testing for a difference in either direction, this is a two-tailed one-sample t-test.
Step 1: Open the T-Test Function
On your TI-84 calculator:
- Press STAT.
- Use the arrow keys to move to TESTS.
- Select 2:T-Test.
- Press ENTER.
The calculator will display the T-Test input screen.
Step 2: Enter the Summary Statistics
Because we already have the sample mean, standard deviation, and sample size, select Stats rather than entering individual observations.
Enter the following values:
| Calculator Field | Value |
|---|---|
| Inpt | Stats |
| μ₀ | 20 |
| x̄ | 21.29 |
| Sx | 5.78 |
| n | 74 |
| Alternative | ≠ μ₀ |
For the alternative hypothesis, select:
≠ μ₀
This tells the calculator that the test is two-tailed.
Once everything has been entered, highlight Calculate and press ENTER.
Step 3: Read the TI-84 Results
The calculator will display the test results, including the alternative hypothesis, t-statistic, p-value, sample mean, standard deviation, and sample size.
For this example, the results are approximately:
- Alternative hypothesis: μ ≠ 20
- t = 1.9199
- p = 0.0588
- x̄ = 21.29
- Sx = 5.78
- n = 74
The t-statistic tells us how many estimated standard errors the sample mean is away from the hypothesized mean.
The p-value is particularly important when making the statistical decision.
Step 4: Interpret the P-Value
Assume a significance level of: α=0.05
The calculator gives: p≈0.0588
Compare the p-value with the significance level: 0.0588>0.05
Because the p-value is greater than 0.05, we fail to reject the null hypothesis.
This means the sample does not provide sufficient statistical evidence at the 5% significance level to conclude that the true average fuel economy differs from 20 mpg.
Importantly, failing to reject the null hypothesis does not prove that the population mean is exactly 20 mpg. It means that the available sample provides insufficient evidence of a difference.
Understanding the TI-84 Output
Here is what each value means:
μ ≠ 20
This represents the alternative hypothesis. We are testing whether the population mean is different from 20 mpg.
t = 1.9199
This is the t-test statistic. It measures the difference between the observed sample mean and the hypothesized mean relative to the estimated standard error.
p = 0.0588
This is the p-value. It represents the probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis is true.
x̄ = 21.29
This is the sample mean.
Sx = 5.78
This is the sample standard deviation.
n = 74
This is the number of observations in the sample.
What If You Want a One-Tailed Test?
The TI-84 also allows you to perform one-tailed tests.
For example, if your research question is whether the cars get more than 20 mpg, the hypotheses would be: H0:μ=20 Ha:μ>20
Select > μ₀ on the T-Test screen.
If instead you want to determine whether the average is less than 20 mpg, use: Ha:μ<20
and select < μ₀.
The direction of the alternative hypothesis should be determined before examining the test results.
One-Sample T-Test vs. Z-Test
A common question is whether a t-test or z-test should be used.
A one-sample t-test is generally appropriate when:
- The population standard deviation is unknown.
- You estimate variability using the sample standard deviation.
- The observations are independent.
- The sample comes from an appropriate random or representative process.
A z-test for a population mean is generally used when the population standard deviation σ is known.
TI-84 One-Sample T-Test Quick Reference
For summary statistics, the basic workflow is:
STAT → TESTS → 2:T-Test → Stats
Then enter:
μ₀ → x̄ → Sx → n → Alternative → Calculate
For the example:
20 → 21.29 → 5.78 → 74 → ≠μ₀ → Calculate
The calculator returns approximately: t=1.9199,p=0.0588
Since p>0.05, we fail to reject H0.
Final Conclusion
The TI-84 calculator makes a one-sample t-test quick to perform when you have either raw observations or summary statistics. In the car-fuel example, the sample mean was 21.29 mpg, compared with a hypothesized mean of 20 mpg.
Although the sample mean is higher than 20 mpg, the resulting p-value of approximately 0.0588 is slightly above the 0.05 significance level. Therefore, there is not sufficient statistical evidence at the 5% level to conclude that the true mean mpg differs from 20 mpg.
When reporting a one-sample t-test, it is good practice to report the sample mean, t-statistic, degrees of freedom, p-value, significance level, and conclusion in the context of the research question.
