##
Overview
Section* *

This lesson presents two alternative methods for testing whether a linear association exists between the predictor *x* and the response *y* in a simple linear regression model:

\(H_{0}\): \(\beta_{1}\) = 0 versus \(H_{A}\): \(\beta_{1}\) ≠ 0.

One is the ** t-test for the slope** while the other is an

**analysis of variance (ANOVA)**.

*F*-testAs you know, one of the primary goals of this course is to be able to translate a research question into a statistical procedure. Here are two examples of research questions and the alternative statistical procedures that could be used to answer them:

- Is there a (linear) relationship between skin cancer mortality and latitude?
- What statistical procedure answers this research question? We could estimate the regression line and then use the
*t*-test to determine if the slope, \(\beta_{1}\), of the population regression line is 0. - Alternatively, we could perform an (analysis of variance)
*F*-test.

- What statistical procedure answers this research question? We could estimate the regression line and then use the
- Is there a (linear) relationship between height and grade point average?
- What statistical procedure answers this research question? We could estimate the regression line and then use the
*t*-test to see if the slope, \(\beta_{1}\), of the population regression line is 0. - Again, we could alternatively perform an (analysis of variance)
*F*-test.

- What statistical procedure answers this research question? We could estimate the regression line and then use the

We also learn a way to check for linearity — the "L" in the "LINE" conditions — using the **linear lack of fit test**. This test requires replicates, that is multiple observations of *y* for at least one (preferably more) values of *x*, and concerns the following hypotheses:

- \(H_{0}\): There is no lack of linear fit.
- \(H_{A}\): There is lack of linear fit.

## Objectives

- Calculate confidence intervals and conduct hypothesis tests for the population intercept \(\beta_{0}\) and population slope \(\beta_{1}\) using Minitab's regression analysis output.
- Draw research conclusions about the population intercept \(\beta_{0}\) and population slope \(\beta_{1}\) using the above confidence intervals and hypothesis tests.
- Know the six possible outcomes about the slope \(\beta_{1}\) whenever we test whether there is a linear relationship between a predictor
*x*and a response*y*. - Understand the "derivation" of the analysis of variance
*F*-test for testing \(H_{0}\): \(\beta_{1} = 0\). That is, understand how the total variation in a response*y*is broken down into two parts — a component that is due to the predictor*x*and a component that is just due to random error. And, understand how the expected mean squares tell us to use the ratio MSR/MSE to conduct the test. - Know how each element of the analysis of variance table is calculated.
- Know what scientific questions can be answered with the analysis of variance
*F*-test. - Conduct the analysis of variance
*F*-test to test \(H_{0}\): \(\beta_{1} = 0\) versus \(H_{A}\): \(\beta_{1} ≠ 0\). - Know the similarities and distinctions of the
*t*-test and*F*-test for testing \(H_{0}\):\(\beta_{1} = 0\). - Know the
*t*-test for testing that \(\beta_{1}\) = 0, the*F*-test for testing that \(\beta_{1}\) = 0, and the t-test for testing that \(\rho = 0\) yield similar results, but understand when it makes sense to report the results of each one. - Calculate all of the values in the lack of fit analysis of variance table.
- Conduct the
*F*-test for lack of fit. - Know that the (linear) lack of fit test only gives you evidence against linearity. If you reject the null, and conclude lack of linear fit, it doesn't tell you what (non-linear) regression function would work.
- Understand the "derivation" of the linear lack of fit test. That is, understand the decomposition of the error sum of squares, and how the expected mean squares tell us to use the ratio MSLF/MSPE to test for lack of linear fit.