# Hypothesis test for the slope

e conduct hypothesis test for the slope **to see if there is evidence for a relationship between X and Y**. As we do not know the true regression line, we can only estimate it, and therefore it makes sense to test.

**Key points on hypothesis testing of the regression slope**

- Hypothesis test on the regression slope typically
**tests the relationship between X and Y** **The H**β̂_{0}is typically set to_{1}**=0**claiming no relationship, but any other value can be tested as well

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**The purpose of hypothesis testing for the slope**

A hypothesis test for the slope is based on the fundamentals of hypothesis testing.

The main purpose of regression analysis is to **explore the relationship between the explanatory variable (X) and the dependent variable (Y)**. We pretend to be as confident as possible about this relationship which is expressed by the slope. Therefore, a hypothesis test for the slope (β̂_{1})=0 is the most common, but we can test for any value. If the slope is 0, it means that our sample statistics indicate no relationship.

**How to set up the hypothesis test**

We recall our estimated regression model:

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When conducting a hypothesis test for the slope, **the** **null hypothesis claims that there is no linear relationship** between X and Y:

The formula of the t-statistics says: β̂_{1} minus the hypothesized value, divided by the SE. As the hypothesized value typically will be 0, we can write as expressed below. In case, we test for another value, this value is applied instead of the 0.

**Worked example**

Back to the home-made example where we explore the relationship between persons’ height and their size of gloves:

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As explained in Standard error of the slope and Confidence intervals for the slope, we first calculate the sample standard deviation which then is plugged into the SE formula, and with the SE, we can calculate the t-statistic:

The critical t-value for df = 13 at an alpha level of 0.05 is 2.160 (t-table or statistical software), and our t-statistic = 10.402 which is “far” beyond this critical value. We therefore reject the null hypothesis concluding that the provided data indicates that there is a strong relation between X and Y.

The output of Excel for this example:

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As the table shows the **p-value** for our data is 1.15 × 10^{-7}= 0.000000115. This is an extremely low p-value giving us **very high evidence against the H _{0} hypothesis** and thus against the claim that there is no relationship.

We have very strong evidence that the true slope is not zero and that therefore is a relationship between X and Y. This extremely low p-value expresses that it would be extremely unlikely to see the increase of our line is due to chance alone. **There definitely seems to be a relationship between X and Y**.

**Hypothesis test for the slope in MS Excel**

Hypothesis test for the slope in MS Excel can be run through **Data >> Data Analysis >> Regression**, where the p-value is given. With the p-value we can conclude on the likelihood of getting the slope you get from your sample data due to chance alone.

**If the p-value is “beyond” your alpha level, you would reject the H _{0}**and conclude that there is evidence for a relationship between X and Y.

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**Learning resources, hypothesis test for the slope**

- Econometrics with R (bookdown text format): Testing Two-Sided Hypotheses Concerning the Slope Coefficient:
- JBstatistics (video 6:56): Inference on the slope (the formulas)
- JBstatistics (video 7:00): Inference on the slope (an example)
- Khan Academy (video 7:15): Running through an example: Calculating t statistics for slope of regression line
- Khan Academy (video 6:24): Using a P-value to make conclusions in a test about a slope

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Confidence intervals

##### Simple linear regression, fundamentals

##### Two-sample inference

##### ANOVA & the F-distribution

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