# What Is a T-Test? - Procedure, Interpretation & Examples

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• 0:00 Overview Of T-Tests
• 2:30 Formula For An…
• 2:50 Evaluating T
• 4:25 Types Of T-Tests
• 5:15 Assumptions Of T-Tests
• 5:55 Lesson Summary
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Lesson Transcript
Instructor: Orin Davis
T-tests are used to compare two means to assess whether they are from the same population. T-tests presume that both groups are normally distributed and have relatively equal variances. The t-statistic is distributed on a curve that is based on the number of degrees of freedom (df). There are three kinds of t-tests: independent-samples, paired-samples, and one-sample.

## Overview Of T-Test

It's Battle of the Sexes, Round 6172. DING! We want to compare the guys and the gals on one last question before declaring a winner. So we tested 100 guys, and they scored an average of 80% with a standard deviation of 3%. The 100 gals, however, scored an average of 81% with a standard deviation of 4%. Can the gals declare victory and call it a day?

Not so fast!

Note that, although the gals have a higher score than the guys, they have a wider range of scores, too. Assuming a normal distribution, note that nearly all of the guys have scores higher than 74 (2 standard deviations below their mean -- recall that, in a normal distribution, more than 95% of the scores are within two standard deviations of the mean), while some of the gals have scores below 74 (as 2 standard deviations below the gals' mean is 73!).

Since, barring any outliers, the gals have both lower and higher scores than the guys, we are going to have to see if the gals have done better in light of both their averages and their standard deviations. To do this, we use a t-test.

Essentially, a t-test is used to compare two samples to determine if they came from the same population. Whenever we draw a sample from the population, we can reasonably expect that the sample mean will deviate from the population mean a little bit. So, if we were to take a sample of guys, and a sample of gals, we would not expect them to have exactly the same mean and standard deviation.

The question is, are their means so different that we are willing to call it unlikely that they are indeed from the same population? After all, despite the so-called Martian and Venusian origins and multiple jests to the contrary, both guys and gals are human. Might the two groups just be slight variations on the human population, or are guys and gals really different on this measure?

Thus, our null hypothesis states that the guys and gals are from the same population, and thus their means are equal. The alternative hypothesis states that the guys and gals are not from the same population, and thus their means are not equal. Since we will reject the null hypothesis if a group did either better or worse, we make a two-tailed test and split our minimum probability for rejecting the null hypothesis (alpha) into two rejection regions of .025 instead of one rejection region of .05.

To determine the probability that the results are true given the null hypothesis, let's compute the t-statistic:

## Formula for an Independent Samples T-test

This is the formula for an independent samples t-test. Where the X's ('X-bar-one' and 'X-bar-two') are the means of the two independent samples (hence this is called an independent-samples t-test), the s represents the standard deviation for each group, and the N's each represent the respective sample sizes of each group.

In the case of the guys vs. gals battle example, we have this:

## Evaluating t

We can evaluate the t-statistic on the t-distribution, which varies in shape slightly depending upon the number of degrees of freedom (df). The t-distribution varies in shape slightly depending on the df. The formula for degrees of freedom in an independent samples t-test is:

df = N1+N2-2

We subtract 2 because each of the two means we computed costs us one degree of freedom.

We use the area under the curve of the t-distribution to determine the probability of obtaining a value for t that is higher/lower than the one calculated (in this case, we want the probability of obtaining t >= 2, on a t-distribution based on 198 (100+100-2) degrees of freedom). To do this, we will need a table of t-distributions.

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