T-test and Linear regression are terms related to inferential statistics that is the statistical method that helps us in making generalizations and predictions about a population by taking a small but illustrative sample of that population. Three types of methodologies are typically used in inferential statistics- confidence intervals, hypothesis tests and regression analysis.

**T-test vs Linear Regression**

The difference between T-test and Linear Regression is that Linear Regression is applied to elucidate the correlation between one or two variables in a straight line. While T-test is one of the tools of hypothesis tests applied on the slope coefficients or regression coefficients derived from a simple linear regression.

While T-test is one of the tests used in hypothesis testing, Linear Regression is one of the types of regression analysis. Linear Regression is used to ascertain the extent of the linear relationship between the outcome variable (dependant variable) and one or more predictor variables (independent variables).

A T-test is one of the hypothesis tests conducted to find out that the difference between the averages of two groups is remarkable or not that is, whether those differences may have happened by chance or not.

## Comparison Table Between T-test and Linear Regression (in Tabular Form)

Parameter of Comparison | T-test | Linear Regression |
---|---|---|

Statistical Method | A T-test is one of the tools of hypothetical testing which in turn is a method of inferential statistics. | Linear Regression is one of the types of regression analysis which is also a method of inferential statistics. |

Usage | A T-test is used to compare the means of two different sets of observed data and to find to what extent such difference is ‘by chance’. | Linear Regression is used to find the relationship between one dependent or outcome variable and one or more independent or predictor variables. |

Types | T-tests are mainly of three types, namely Independent Sample t-test (comparison between the average of two sets of data), paired Sample T-test (comparing the averages of same sets of data as different intervals) and One Sample T-test (comparing the average of a single set of data with a known mean). | There are two types of Linear Regression, namely Simple Linear Regression (comprising one dependant and one independent variable) and Multiple Linear Regression (consisting of one dependant variable and two or more independent variables). |

Practical Applications | The T-test can be used for testing the returns from two different portfolios managed under two different strategies of investment. It was first used to check the consistent quality of stout in a brewing company. | Linear Regression is mainly used for observing customer behaviour, pricing, forecasting sales for a company, weather, GDP growth etc. |

The number of variables or sets that can be used. | Only two sets of data or groups can be used in a T-test. | While there is only one regressand, the number of regressors can be more than two. |

## What is T-test?

A T-test is one of the instruments used in hypothesis testing for comparing two different sets of data and their means or averages. Others are Analysis of Variance test, Z-test, Chi-Square Test and F-test.

A T-test is used to check the significant difference between two sets of data. It is used to determine how much of such difference is by chance.

It was used for the first time by William Sealy Gosset, a chemist who worked for a brewing company named Guinness to monitor the consistent quality of the stout.

Gradually, it was upgraded and now it refers to any hypothesis tests in which the data when analyzed is supposed to be equivalent to a t-distribution (a bell-shaped distribution curve with heavier tails) if the null hypothesis (the assumption that no relationship exists between the sets of data) proves to be right.

For standard interpretation and validation, it depends on certain assumptions about a sample population.

Such assumptions comprise of data, that is randomly sampled, data variables, which follow a normal distribution, a variance that is unknown and is considered to be homogenous and a scale of measurement which when applied to the data collected results in a continuous line.

There are three types of T-tests:

**Independent Samples T-test:**It is used to compare two different sets of observed data and their means.**Paired Sample T-test:**It is used to compare the average of a single set of observed data at different times.**One Sample T-test:**It makes a comparison between the mean of a single set of data and a known mean.

As an approach for testing hypothesis, T-test is quite conservative. It can be applied to only two sets of data and is considered suitable for only small sets of data.

## What is Linear Regression?

Linear Regression is a method of inferential statistics that tries to explain the correlation between a dependent variable(Y) and one or more independent variables(X) using a straight line. It mainly deals with three types of questions:

- Does a set of explanatory variables correctly predict the outcome variable?
- If it does, then which are the most prominent independent or explanatory variables that significantly affect the dependant or outcome variable?
- And lastly, to what extent a change in these independent or explanatory variables affect the outcome or dependant variable?

The relationship between the outcome variable and the explanatory variables is considered to be a positive one if an increase in the latter results in an increase in the former.

Similarly, a relationship between the dependant and the independent variable is said to be a negative one if the former decreases with an increase in the latter.

Linear Regression has three usages:

- For deciding the strength of independent variables i.e. to what extent they influence the independent variable.
- For forecasting the change in the dependant variable induced by the independent variables.
- For predicting future trends and values.

There are mainly two types of linear regressions: **Simple Linear Regression** which consists of one dependant variable and one independent variable and **Multiple Linear Regression** that comprises of dependant variable and two or more independent variables.

**Main Differences Between T-test and Linear Regression**

- Both the terms are associated with inferential statistics but fall under the purview of different methodologies. While T-test is one of the tests used in hypothesis testing, Linear Regression falls within the ambit of Regression analysis.
- A T-test can be conducted only when there are two sets of data and not more than that. While in Linear Regression, there can be more than two independent variables, though the dependent or outcome variable can only be one.
**The main difference between a Linear Regression and a T-test**is thata Linear Regression is used to explain the correlation between a regressand and one or more regressors and the extent to which the latter influences the former. While a T-test is used to compare two different sets of data and their averages and try to test whether any relation or significance exists between those sets of data or not.- Linear regression analysis can be done even with larger sets of data but a T-test is suitable for only smaller data sets.
- Finally, a Linear Regression can be used for observing customer behaviour, previous sales and predicting GDP growth, weather report and so on. While a T-test is used for testing the validity of certain assumptions related to two different sets of data which may be a population, investment portfolios and so on.

## Conclusion

Both T-test and Linear Regression fall within the broader framework of inferential statistics that are used to make assumptions on a particular population using a small sample. They play different roles and are essential tools for inferring the general characteristics of a population.

While linear regression helps in making certain predictions about a particular sample e.g. customer behaviour, T-test helps in testing the applicability of a hypothesis to a sample population.

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