### What is a 'T-Test'

A t-test is an analysis of two populations means through the use of statistical examination; a t-test with two samples is commonly used with small sample sizes, testing the difference between the samples when the variances of two normal distributions are not known.

### Explaining 'T-Test'

A form of hypothesis testing, the t-test is just one of many tests used for this purpose. Statisticians must use tests other than the t-test to examine more variables, as well as for test with larger sample sizes. For a large sample size, statisticians use a z-test. Other testing options include the chi-square test and the f-test.

### Statistical Analysis of the T-Test

The formula used to calculate the test is a ratio: The top portion of the ratio is the easiest portion to calculate and understand, as it is simply the difference between the means or averages of the two samples. The lower half of the ratio is a measurement of the dispersion, or variability, of the scores. The bottom part of this ratio is known as the standard error of the difference. To compute this part of the ratio, the variance for each sample is determined and is then divided by the number of individuals the compose the sample, or group. These two values are then added together, and a square root is taken of the result.

### Example

For example, consider that an analyst wants to study the amount that Pennsylvanians and Californians spend, per month, on clothing. It would not be practical to record the spending habits of every individual (or family) in both states, thus a sample of spending habits is taken from a selected group of individuals from each state. The group may be of any small to moderate size — for this example, assume that the sample group is 200 individuals.

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#### What is a t-test?

A t-test is an analysis of two populations means through the use of statistical examination.

#### What are these samples composed of?

These samples are composed of individuals from different states.

#### What are the parts of this ratio?

The top portion of the ratio is the easiest portion to calculate and understand, as it is simply the difference between the means or averages of two samples.

#### How do you determine variance in each sample?

To determine variance in each sample, you need to find how much each individual spent on clothing per month. This can be done by dividing their total spending by 12 (months) and then multiplying that number by 100 (to get it into percentages). Then, add up all their monthly totals and divide that number by how many people were sampled (in this case 2). This will give you your variance for each group.

#### How does the formula for a t-test work?

The formula used to calculate the test is a ratio.