**Introduction**

In mathematics, statistics is a field that deals with the collection, interpretation, and tabulation of numerical data. It uses different quantitative methods to produce a set of experimental data. Statistics makes work easy and simple by providing a clear picture of work done by an individual regularly. The collection and analysis of data by forming statistics is called statistical analysis. The types of statistical analysis are descriptive analysis, inferential analysis, predictive analysis, prescriptive analysis, exploratory data analysis, and causal analysis. Out of these types, T-Test from inferential statistical analysis and Linear Regression from predictive statistical analysis are different methods providing distinguished variables and equations.

**T-test vs Linear Regression**

A t-test is a statistical test used to compare the means of two groups. It is used in hypothesis testing to determine if a process affects the population of interest or if the two groups are different from each other. T-test is a parametric test and is also known as a single sample t-test. The method is important for post-testing analysis in statistics to validate data findings between the two groups.

Linear Regression is a method that is used to identify and understand the relationships between mean of variables.

**Difference Between T-Test and Linear Regression in Tabular Form**

Parameters of Comparison | T-Test | Linear Regression |

Definition | T-test is a statistical method used to determine if there is a significant difference between the means of the two given groups. | Linear regression is an analysis used to identify the relationship between the mean value of one variable and the corresponding values of other variables. |

Assumption | The popular standard deviation is unknown in the t-test. | Popular standard deviation is known in the linear regression. |

Comparison | A T-test is appropriate for comparing numerical variables between two independent groups. | Linear regression is appropriate in comparing and considering the means of many variables at the same time. |

Types | T-tests are of three types which include- independent sample t-test, paired sample t-test, and one sample t-test. | Linear regressions are of two types which include- simple linear regression and multiple linear regression. |

Sample Size | A sample size of at least 40 is required in the t-test. | Linear regression requires a minimum sample size of 30. |

**What is a T-Test?**

A t-test is an inferential statistic used to determine whether there is a significant difference between the means of two groups and their relations. It is generally used in hypothesis testing. If you want to compare more than two groups, then the ANOVA test or post-hoc test is used. T-tests are parametric tests as they can be used when the sample satisfies the conditions of normality, equal variance, and independence. In simple words, it assumes whether the data is independent, normally distributed, and has a homogeneity of variance.

A t-test studies a set of data collected from two same or different groups to determine the probability of a difference in the outcome than what is usually obtained. The use of the distribution patterns and the variants influencing the gathered samples determines the accuracy of the test. The test is conducted and a t-value is obtained. For example- to find the mean of the length of petals of a flower belonging to two different species, a t-test can be done. The final t-test interpretation is obtained in two ways:-

- By using null hypothesis- A null hypothesis indicates that the difference between the means is zero and both the means are shown as equal.
- By using alternate hypothesis- An alternate hypothesis signifies that the difference between the given means is different rather than zero. It rejects the null hypothesis by stating that the data set is accurate and not by chance.

Some of the assumptions of the t-test are:-

- The data are continuous.
- The scale of measurement applied to the data collected follows a continuous or ordinal scale, such as the scores of an IQ test.
- The observations in one sample are independent of the observations in the other sample.
- The sample data have been randomly sampled from a population.
- A reasonably large sample size is used in the data.
- Both samples are approximately normally distributed.
- There is homogeneity of variance in the data.

The three common types of t-tests are:-

**One sample t-test**- One sample t-test is a statistical hypothesis test used to determine if an unknown population mean is different from a specific value. It is also referred to as a single-sample test. The null hypothesis is denoted by the symbol H_{0 }and the alternative hypothesis is denoted by the symbol H_{1. }It is expressed in these forms in the one sample t-test:

H_{0}: *μ*= *μ*_{0 }[the population mean is equal to the proposed population mean]

H_{1}: *μ* ≠ *μ*_{0} [the population mean is not equal to the proposed population mean]

Here, *μ* is the true population mean and u_{0 }is the proposed mean.

The formula to calculate the value in one sample test is:-

t = *x* - *μ*_{0} / s*x* = *s**√n*, where *μ*_{0} is the test value, *x*is the sample mean, n is the sample size, s is the sample standard deviation and s*x* is the estimated standard error of the mean.

**Independent two-sample t-test**- Independent two-sample t-test is a statistical hypothesis test in which the samples from two independent groups are compared to determine whether the means of the associated populations are significantly different or not. It is also referred to as an unpaired two-sample t-test. The two-sample t-test is different based on variation within the two different groups. When the variances of populations are not equal:-

In the above formula, the standard error is the square root term.

When the two populations’ variances or standard deviations are equal, then the formula is:-

Here sp is the pooled standard deviation.

When the two populations’ variances or standard deviations are equal:-

**Paired sample t-test**- Paired sample t-test is the hypothesis testing conducted when two groups belong to the same group or population. The formula to find the t-value is:-

T = *m**s/√n*, where t is the t-statistic, m is the mean of the group, s is the standard deviation of the group and the n is sample size.

**Equal variance t-test**- Equal variance t-test is the hypothesis test conducted when the sample size in the population is the same or the variance of the two sets of data is similar. It is also referred to as a pooled t-test.

t-value = mean1 – mean2 / (n1-1) x var1^{2} + (n2-1) x var2^{2} *1**n1**+**1**n2*, where mean 1 and mean 2 is the average value of each set, var1 and var2 is the variance of each set of samples and n1 and n2 are the number of the records in each set.

**What is Linear Regression?**

Linear regression is a predictive analysis used to predict the value of a variable based on the value of another variable. The value that is to be predicted is known as the dependent variable. The value of the variable that is used to predict the value of the other variable is known as the independent variable.

The equation of the regression is given by the formula: y = c + b*x, where y is the estimated dependent variable score, c is constant, b is the regression coefficient and x is the score on the independent variable. The dependent variable of linear regression is also referred to as outcome variable, criterion variable, endogenous variable, or regress variable. The independent variable is also referred to as the exogenous variable, predictor variable, or regressor variable.

Linear regression is mainly used in many fields like economics, finance, and social sciences. It is used to model a wide variety of economic relationships in economics. It is used to tell how a GDP can affect sales for a company in finance and is used to conduct preliminary data analysis and predict future trends in social science.

Some of the common assumptions while using linear regression are:-

- The relationship between the dependent and independent variables should be linear.
- There should be no correlation between the residual (error) terms.
- Observations are independent of each other.
- The chosen sample is representative of the population.
- The error term of a model is normally distributed.
- The number of training data or observations should be always greater than the number of test or prediction data.
- The residuals from the model should be homogenous or equal spaces.

Linear regression is generally classified into two types:

**Simple Linear Regression**- Simple linear regression is used to estimate the relationship between two quantitative variables. It also can be used to find the value of the dependent variable at a certain value of the independent variable. For example- the amount of soil erosion at a certain level of rainfall. As simple linear regression is a parametric test, it has some assumptions:-

- Homogeneity of variance
- Independence of observations
- Normality

The formula of the simple linear regression is:-

Y = *β*_{0} + *β*_{1}X + *Ïµ*, where y is the predicted value of the dependent variable, B_{0} is the intercept, B_{1} is the regression coefficient, x is the independent variable and e is the error of the estimate.

**Multiple Linear Regression**- Multiple linear regression is used to predict the outcome of a response variable. It aims at modeling the linear relationship between the explanatory variables and response variables.

The formula of the multiple linear regression is:-

Y_{i} = *β*_{0} + *β*_{1}x_{i1} + *β*_{0} + *β*_{2}x_{i2} + … + *β*_{0} + *β*_{p}x_{ip} + *Ïµ*, where y_{i} is the dependent variable, x_{i} is the explanatory variable, *β*_{0} is the y-intercept and *β*_{p} is the slope coefficient for each explanatory variable and *Ïµ* is the model’s error term.

Other types include logistic regression, ordinal regression, and multinomial logistic regression.

**Main Differences Between T-test and Linear Regression (In Points)**

- T-test is a statistical used to determine if a particular variable is significant statistically in the model. While Linear regression is an analysis used to identify and predict future outcomes.
- The term t-test was first coined by William Sealy Gosset in 1908. On the other hand, the term linear regression was coined by Francis Galton in 1886.
- T-test is used to determine the means of the two groups. Whereas, linear regression is used to determine and predict the relationship between the variables.
- T-test is used to test the returns from two different portfolios managed under two different strategies of investment. While, linear regression is used to observe the price, customer behavior, weather, GDP growth, etc.
- T-test focuses on the t-distribution in a group or sample. On the other hand, linear regression focuses on the conditional probability distribution of the values.
- The degree of freedom of the t-test is n-1. Whereas, the linear regression model has two degrees of freedom.
- T-tests can be used by a doctor to know if some new drug leads to a significant reduction in blood pressure compared to the currently used drug. While, linear regression can be used in business, finance, and stock markets.
- T-test can be used to find if the mean of the length of the petals of a flower belonging to different species is the same or not. Whereas, linear regression can be used to describe the age and height of an individual.

**Conclusion**

In short, t-tests involve the use of categorical predictors, linear regression involves the use of continuous regression. T-tests are used in linear regression to determine whether a particular variable is statistically significant in the given model. Both are distinguished from each other based on uses, analysis, formula, etc.

**References**

- vitalflux.com
- ajodo.org