AP Statistics : How to test of significance for the slope of a least-squares regression line

Example Questions

Example Question #1 : Ap Statistics

A statistician conducts a regression analysis and obtains a p-value of 0.1. It is more likely than not that there is a relationship between the variables in the study.

Possible Answers:

True

False

Correct answer:

True

Explanation:

A p-value of 0.1 is generally not sufficient to reject the null hypothesis, but this is only because we want a high degree of confidence before finding a relationship between variables.  Here, there is most likely a relationship between the variables even though the statistician could not reject the null hypothesis.

Example Question #1 : Mean And Linear Regression

For a data set, the least-squares regression line has a  confidence interval for the slope of .

Based on this confidence interval, what can you do with a hypothesis test at  significance level where and ?

Possible Answers:

Reject the null hypothesis because this confidence interval does not include .

Fail to reject the null hypothesis because this confidence interval does not include .

Not enough information to be able to decide.

Reject the null hypothesis because this confidence interval does not include .

Fail to reject the null hypothesis because this confidence interval does not include .

Correct answer:

Reject the null hypothesis because this confidence interval does not include .

Explanation:

Notice that the interval does not include . This means that the P-value for the hypothesis test would be under 5%, which would lead us to reject our null hypothesis.

Any confidence interval can be used to create a hypothesis test by inverting it, and it is fairly simple, but the concept is tested into graduate-level statistics theory.

Example Question #1 : Inference

Which of the following is an incorrect condition requirement for regression inference?

Possible Answers:

A linear relationship between  and

A trend/pattern of some sort in the residual plot

The standard deviation of the response must be constan

Response must vary normally about the regression line for any given value of

Ordered pairs must be independent of each other

Correct answer:

A trend/pattern of some sort in the residual plot

Explanation:

All of the following choices are correct conditions except for the choice concerning a trend/pattern of some sort in the residual plot. For regression inference to be accurate, we need to look at the residual plot of the data of interest and make sure there is random scatter. Random scatter indicates that the ordered pairs are indeed independent of each other. Any sort of pattern present in the residual plot would not satisfy that requirement, and therefore would not enable us to successfully use regression inference.