# Hypothesis testing on linear regressions

The goal of a simple linear regression is to approximate the population as a whole, not just the sample given. There are two common ways of evaluating how well simple linear regression models are likely to resemble the overall population, and whether there is a relationship between two variables.

With all of the tests below, we calculate a t-statistic. From the t-statistic, we take the t-statistic and reference a t-distribution table. A t-distribution table is organized by degrees of freedom (rows) and levels of certainty (columns). If the absolute value of the t-statistic is greater than the number in the t-distribution table, we reject our null hypothesis and accept our hypothesis.

Description | |
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We will reject the null hypothesis. In other words, we will conclude: "There is sufficient evidence at this | |

We will accept the null hypothesis. In other words, we will conclude: "There is not enough evidence at this |

## t-test for the population correlation coefficient

The correlation coefficient for the sample is

Hypothesis | Statement | Description |
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There is no relationship between the two variables. | ||

There is a relationship between the two variables. |

## t-test to determine linear association

This t-test is a.k.a the "slope" test. It tests whether a relationship exists, making it similar to the previous test. We fundamentally have either a hypothesis

The hypothesis

Hypothesis | Statement | Description |
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There is no relationship between the two variables. | ||

There is a relationship between the two variables. |

Calculating the likelihood that the population slope is some value

In full notation,

Generally, when we test if a relationship exists, we will use

## F-test to determine if a line or a curve is the best fit

The F-test is also called the "analysis of variance" (ANOVA) test.

Hypothesis | F-test |
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