# How do I find if intercept is significantly different from zero? (using R)

This is probably simple question, but after searching for a bit I didn't find (or alternatively understand) an answer. (I'm sure a question like this has been posted again and again, but it is quite hard to find the exact thing when your knowledge of the subject is very limited.)

I'm trying to figure how I can find if intercept is significantly different from 0 (or any number).

So I have some made-up numbers: I do remember that the Pr(>|t|) refers to the significance, but significance of what?

(In case the image doesn't show, this is the ugly version, couldn't figure how to format it here:)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.005         0.001   -2     0.08 .
mkt          1.2           0.03    50     <2e-16 ***

• For H0: intercept = 0, p =0.08 is in your output. – user158565 Nov 7 '18 at 15:17

Mathematically, you (and R) are testing:

The null hypothesis: $$H_{0}: \beta_{0} = 0$$

vs.

The alternative hypothesis: $$H_{0}: \beta_{0} \neq 0$$

When we say the word significant what we really mean is significantly different from zero.

1) Something that is said to be significant, is significantly different from zero.
2) Something that is said to be not significant, is not significantly different from zero.

In 1) there is enough evidence in the data to reject the null hypothesis. (You reject the statement that says $$H_{0}: \beta_{0} = 0$$).

In 2) there is not enough evidence in the data to reject the null hypothesis. Hence, you say that you fail to reject the null hypothesis.

NOTE: We never say that we "accept" the alternative hypothesis, your conclusion should be limited to reject or fail to reject the null hypothesis.

Finally, more details of your specific problem can be found in the section "2.2 Inferences Concerning $$\beta_{0}$$" of the book Applied Linear Statistical Models, 5th Ed, by Kutner, Nachtsheim, Neter and Li.

The column Pr(>|t|) refers to a test that the corresponding coefficient is zero. In your case it is greater than 0.05 and so there is insufficient evidence that it is non-zero. Having said all that your output does look rather unusual as a unit change in your other predictor leads to a massive change in the outcome. I think you should investigate the output of plot(model) to see if there is anything unusual to be seen there.

• Thanks for the answer. So basically this output says that there is 8% chance that the result could be 0 and if we use 0.05 significance level we can't say it is significantly different from 0? (Regarding the model, it is about monthly stock prices with mkt referring beta, but I did change the numbers to make it easier to read.) – Anssi Nov 7 '18 at 15:56
• The second half of your sentence starting "So basically" is OK but the first half is not. – mdewey Nov 7 '18 at 16:13