# T Test in Statistics

#### t-value

Test statistic for the t-test family, it measures the difference between an observed statistic and its hypothesized population parameter in units of standard error. A t-test compares this observed t-value to a critical value on the t-distribution with (n-1) degrees of freedom to determine whether the difference between the estimated and hypothesized values of the population parameter is statistically significant.  Applications of t-values include:

comparing two sample means

comparing the means of matched pairs

determine the significance of a regression coefficient

comparing two regression coefficients

You can also use t-values in a 1-sample t-test. For example, you want to determine whether the length of a manufactured part meets its target value of 10cm. You take a sample of 50 parts, perform a two-sided 1-sample t-test on their mean length with the following hypotheses:

H0: m = 0 (the mean length of all parts meets the target value)

H1: m ≠ 0 (the mean length of all parts does not meet the target value)

The test produces a t-value of 2.5. On the t-distribution with (n-1 = 49) degrees of freedom, this t-value corresponds to a p-value of 0.0158. For most common alpha-levels, this result is statistically significant, so you reject the null hypothesis that the mean length meets the target, and conclude that the process needs improvement.

#### t-test

A hypothesis test of the mean of one or two normally distributed populations. Several types of t-tests exist for different situations, but they all use a test statistic that follows a t-distribution under the null hypothesis:

1-sample t-test

Purpose: Tests whether the mean of a single population is equal to a target value

Example:  Is the mean height of female college students greater than 5.5 feet?

2-sample t-test

Purpose: Tests whether the difference between the means of two independent populations is equal to a target value

Example:  Does the mean height of female college students significantly differ from the mean height of male college students?

Paired t-test

Purpose:  Tests whether the mean of the differences between dependent or paired observations is equal to a target value

Example:  If you measure the weight of male college students before and after each subject takes a weight-loss pill, is the mean weight loss significant enough to conclude that the pill works?

t-test in regression output

Purpose: Tests whether the values of coefficients in the regression equation differ significantly from zero.

Example:  Are high school SAT test scores significant predictors of college GPA?

An important property of the t-test is its robustness against assumptions of population normality – in other words, t-tests are often valid even when the samples come from non-normal populations. This property makes them one of the most useful procedures for making inferences about population means.