Statistics: What Is a t-Test?

T-Test

A t-test is commonly used to determine whether the mean of a population significantly differs from the mean of another population.

The t-score is the difference between two groups divided by the difference within the groups

The t-score is very similar to the z-score if you are familiar with that, except z-scores are used only for normal "symmetrical/bell-shaped" distributions while t-scores can be used for non-normal distributions.

The t-score formula is: $t =$$\frac{x̄-μ_0}{s/\sqrt{n}}$

Such that:

$x̄$ = sample mean

$μ_0$ = population mean or hypothesized value (value being compared to in null hypothesis statement)

$s$ = sample standard deviation

$n$ = sample size

How it may be written slightly differently to help tackle different kinds of t-tests such as one-sample and two-sample t-tests. But do not be afraid! These other formulas will be introduced in step-by-step solutions in their respective sections.

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• A high magnitude t-score tells you that the groups are different.
• A low magnitude t-score tells you that the groups are similar.

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A t-value of 0 or below some chosen critical value indicates that the sample results exactly equal the null hypothesis and there is no significant difference between the two group means and hence we would fail to reject the null hypothesis.

As the magnitude of the t-value increases and surpasses our chosen critical value, the difference between the sample data and the null hypothesis increases and hence we would reject the null hypothesis.

Note: We cannot "accept the null hypothesis" ever. Hypothesis tests can only determine whether or not be can reject or fail to reject the null hypothesis.

t-tests can be either:

• one-tailed or two-tailed
• one-sample or two-sample

You will need to be able to identify whether your t-test has 1 or 2 tails, and 1 or 2 samples based on the wording of your problem in order to create your hypothesis statements accurately.

Hit "Continue" for a table with hypothesis statements for each condition

one-sample:

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$H_0: \mu = m_0$

$H_A: \mu \neq m_0$ (two-tailed)

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$H_0: \mu \leq m_0$

$H_A: \mu > m_0$ (upper-tailed)

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$H_0: \mu \geq m_0$

$H_A: \mu < m_0$ (lower-tailed)

two-sample:

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$H_0: \mu_0 = \mu_1$

$H_A: \mu \neq \mu_1$ (two-tailed)

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$H_0: \mu_0 \leq \mu_1$

$H_A: \mu_0 > \mu_1$ (upper-tailed)

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$H_0: \mu_0 \geq \mu_1$

$H_A: \mu_0 < \mu_1$ (lower-tailed)

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Such that:

$m_0$ is the value you are comparing the mean to (aka comparison value)

and $\mu, \mu_0, \mu_1$ are your group means

(Video) How To Find T-Score With T-Table

by danirockyo

You will first need to know two pieces on information:

1. The degrees of freedom ($df$)

2. The confidence level (i.e., 90%, 95%, 99%, etc...)

Once you have these two values you just need to find their cross-section in a t-table! There's a handy t-table by San Jose State University that you can use to find a t-score.

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Note: After a sample size of 30 and higher, you can technically use the z-table.

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What Do T-Tests Actually Test For?

One-Sample T-Test

The one-sample t-test determines whether the sample mean is statistically different from a known/hypothesized population mean when the group size is $\leq$ 30.

Two-Sample T-Test

Similarly, the two-sample t-test is applied to compare whether the average difference between two groups when the group size is is $\leq$ 30 is significant or if it is due to random chance.

Welch's T-Test

Also known as an unequal variances t-test, this test tells us if there's a significant difference between the means of two independent groups, accounting for potentially unequal variances and sample sizes.

Example

Let's say we want to compare the average blood pressure of two different groups of people: one group who exercises regularly and another group who does not exercise regularly. We could use Welch's t-test to see if there's a significant difference in their average blood pressure levels. This would help us understand if exercise has an impact on blood pressure.

Another example could be: Imagine we have two groups of students: Group A and Group B. Group A consists of students who used a particular study technique (e.g., mnemonic devices) before a test, while Group B consists of students who did not use this technique. We want to test if using the study technique improves test scores, specifically if the average test score in Group A (technique users) is higher than in Group B (non-users). We can use a t-test for a random sample, by randomly selecting students for both Group A and Group B from a larger student population.

Looking Further

In the future, you will also learn about unpaired and paired t-tests. An independent samples t-test, also known as an unpaired t-test, compares the means of two independent groups to determine if the samples are significantly different.