One-Sample T-Test on TI-84 Calculator

One-Sample T-Test on TI-84 Calculator, A one-sample t-test is a statistical method used to determine whether the mean of a population is significantly different from a specified value. It is commonly used when the population standard deviation is unknown and the sample standard deviation is used instead.

In this guide, we’ll show you how to perform a one-sample t-test on a TI-84 calculator, using both summary statistics and raw data.

What Is a One-Sample T-Test?

A one-sample t-test evaluates the null hypothesis: H0​:μ=μ0​

against one of the following alternative hypotheses: Ha​:μ=μ0​ Ha​:μ>μ0​

or Ha​:μ<μ0​

where:

• μ = true population mean
• μ0​ = hypothesized population mean
• xˉ = sample mean
• s = sample standard deviation
• n = sample size

The test statistic is calculated as: t=s/n​xˉ−μ0​​

The resulting p-value is used to determine whether there is sufficient statistical evidence against the null hypothesis.

When Should You Use a One-Sample T-Test?

A one-sample t-test is appropriate when:

• You have one quantitative variable.
• You want to compare its population mean with a specific value.
• The population standard deviation is unknown.
• Observations are independent.
• The population is approximately normally distributed, particularly when the sample size is small.

For larger samples, the t-test is generally more robust to moderate departures from normality, although extreme outliers can still affect the results.

Example: Testing Car Miles Per Gallon

Suppose researchers want to determine whether a particular type of car has an average fuel economy of 20 miles per gallon (mpg).

A random sample of 74 cars produces the following summary statistics:

• Hypothesized mean: μ0​=20
• Sample mean: xˉ=21.29
• Sample standard deviation: s=5.78
• Sample size: n=74

We want to determine whether the true average fuel economy differs from 20 mpg.

Step 1: State the Hypotheses

Because we want to determine whether the population mean is different from 20 mpg, this is a two-tailed test.

Null hypothesis: H0​:μ=20

Alternative hypothesis: Ha​:μ=20

We will use a significance level of: α=0.05

Step 2: Open the T-Test on the TI-84

On your TI-84 calculator:

1. Press STAT.
2. Use the arrow keys to move to TESTS.
3. Select T-Test.
4. Press ENTER.

The calculator will display options for entering your data.

Step 3: Enter Summary Statistics

Because we already know the sample mean, standard deviation, and sample size, select Stats.

Enter:

• μ0: 20
• x̄: 21.29
• Sx: 5.78
• n: 74

Under the alternative hypothesis, select:

μ ≠ μ0

This specifies a two-tailed test.

Finally, select Calculate and press ENTER.

Step 4: Calculate the Test Statistic

The t-statistic can also be calculated manually: t=5.78/74​21.29−20​

This gives approximately: t=1.920

The degrees of freedom are: df=n−1=74−1=73

The calculator then determines the corresponding p-value.

For this example: p≈0.0588

Step 5: Interpret the TI-84 Results

The calculator output will include information such as:

• Alternative hypothesis: μ=20
• t: approximately 1.9199
• p: approximately 0.0588
• x̄: 21.29
• Sx: 5.78
• n: 74

The most important value for making the statistical decision is the p-value.

Step 6: Make the Statistical Decision

Compare the p-value with the significance level: p=0.0588 α=0.05

Because: 0.0588>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 mean fuel economy differs from 20 mpg.

It is important to note that failing to reject the null hypothesis does not prove that the population mean equals 20 mpg. It means that the available evidence is not strong enough to establish a statistically significant difference at the chosen significance level.

How to Perform a One-Sample T-Test Using Raw Data

If you have the individual observations rather than summary statistics, you can enter the data directly into the TI-84.

Step 1: Enter the Data

1. Press STAT.
2. Select 1:Edit.
3. Enter the observations into L1.

For example:

L1182221...

Step 2: Open T-Test

Press:

STAT → TESTS → T-Test

Instead of Stats, select:

Data

Then enter:

• μ0: your hypothesized mean
• List: L1
• Freq: 1

Select the appropriate alternative hypothesis.

For a two-tailed test, choose:

μ ≠ μ0

Then select Calculate.

One-Sample T-Test vs. Z-Test

The choice between a t-test and z-test depends largely on whether the population standard deviation is known.

When the population standard deviation is unknown, the sample standard deviation is used and the t-distribution accounts for the additional uncertainty.

Understanding the T-Test Output

The TI-84 provides several useful statistics.

t-statistic

The t-statistic measures how far the sample mean is from the hypothesized mean relative to the estimated standard error.

A t-statistic close to zero indicates that the sample mean is relatively close to the hypothesized value.

p-value

The p-value represents the probability of obtaining a test statistic at least as extreme as the observed result, assuming the null hypothesis is true.

A small p-value provides stronger evidence against the null hypothesis.

A common decision rule is:

• p < 0.05: Reject H0​
• p ≥ 0.05: Fail to reject H0​

The significance level should be selected before conducting the test.

One-Tailed Tests

The TI-84 also allows you to perform one-tailed tests.

Testing whether the mean is greater than a value

For: Ha​:μ>μ0​

select:

μ > μ0

Testing whether the mean is less than a value

For: Ha​:μ<μ0​

select:

μ < μ0

Testing whether the mean is different from a value

For: Ha​:μ=μ0​

select:

μ ≠ μ0

Always select the alternative hypothesis based on the research question rather than choosing the direction after looking at the results.

Relationship Between a One-Sample T-Test and Confidence Intervals

A one-sample t-test and a confidence interval provide closely related information.

For example, a 95% confidence interval for the population mean can be used to assess a two-sided hypothesis test at the 5% significance level.

If the hypothesized mean falls outside the 95% confidence interval, the corresponding two-sided test will reject the null hypothesis at α=0.05.

If the hypothesized mean falls inside the interval, the test will generally fail to reject the null hypothesis.

Common Mistakes When Performing a One-Sample T-Test

Using the wrong test

Do not automatically use a t-test simply because the sample size is small. The choice depends on the research design and whether the population standard deviation is known.

Confusing standard deviation with standard error

The TI-84 expects the sample standard deviation Sx when using summary statistics for a t-test. Do not enter the standard error unless the calculator specifically asks for it.

Choosing the wrong alternative hypothesis

A test of: μ=μ0​

is different from: μ>μ0​

or: μ<μ0​

Choose the alternative hypothesis before interpreting the result.

Interpreting a large p-value incorrectly

A p-value greater than 0.05 does not prove that the null hypothesis is true. It simply means there is insufficient evidence to reject it at the 5% significance level.

One-Sample T-Test Formula

The one-sample t-test statistic can be summarized as: t=s/n​xˉ−μ0​​

with: df=n−1

where:

• xˉ = sample mean
• μ0​ = hypothesized population mean
• s = sample standard deviation
• n = sample size
• df = degrees of freedom

Conclusion

A one-sample t-test on a TI-84 calculator provides a convenient way to determine whether a sample mean differs significantly from a specified population value.

In our car MPG example, the test produced a t-statistic of approximately 1.92 and a p-value of approximately 0.0588. Because the p-value is greater than 0.05, we fail to reject the null hypothesis.

Therefore, at the 5% significance level, there is not enough statistical evidence to conclude that the true mean fuel economy differs from 20 mpg.

The TI-84 makes the calculation straightforward, but the most important step is correctly defining the hypotheses and interpreting the p-value in the context of the research question.

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