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Unit 9

9.5 Carrying Out a Test for the Slope of a Regression Model

2 min readapril 28, 2021

josh

Josh Argo


9.5: Carrying Out a Test for the Slope of a Regression Model

As we finished out on Section 9.4, we have set up the test for slope of a regression model, now we have to calculate our test statistics and make our conclusion. 🔥

T-score

The first thing we need to calculate is our critical value-our t-score. Our t-score can tell us how far away from the mean (null slope value) our sample is, so it gives us a scale to see how close our sample is to the expected slope value.
The formula to compute our t-score is similar to any other critical value. We must take our observed value, subtract the expected value and then divide the result by the standard deviation. In this case, we are going to base our t-score off of our degrees of freedom being n-2. When there is only one parameter we are testing, we will use n-1 as our degrees of freedom.

P-Value

The next, and most important statistic that we need to calculate is the p-value. Our p-value is the probability that our particular slope occurs if we assume the null slope (usually 0). 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F9-ZisBppKnSMXX.png?alt=media&token=9088a863-b4bc-47fe-88e0-b1f0098ea637

Image Taken From Stats Online

In the example above, our t score came out to be 2.0791 with 21 degrees of freedom. Therefore, since we are performing a two-tailed test, our p-value comes out to be 0.5 (typically your t score and p-value don’t come out to be exactly our significance level). Therefore, there is approximately a 5% chance of obtaining our given sample, assuming that the null slope of 0 is true. A 5% chance of something happening by random chance is pretty low, therefore, we would reject our null hypothesis in favor of the alternate due to the evidence from our sample.

Concluding Your Test

Once you have your t-score and p-value from your calculations, you are ready to conclude your test. As in with any other inference procedure you have performed, your conclusion is primarily based on your p-value that was obtained from your t-distribution and t-score.
If the p is lower than your significance level:
  • Since our p value is less than our significance level, we reject our Ho. We have significant evidence that the true slope of the regression line between ________ and ______ is (value in alternate hypothesis, usually not 0).
If the p is not lower than your significance level:
  • Since our p value is more than our significance level, we fail to reject our Ho. We do not have significant evidence that the true slope of the regression line between ________ and ______ is (value in alternate hypothesis, usually not 0).
For more practice on this concept and to see a problem worked through completely, move on to Section 9.6.
🎥  Watch: AP Stats Unit 9 - Inference for Slopes

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