Topic 9.4 Setting Up a Test for the Slope of a Regression Model: state the null and alternative hypotheses about the population slope, identify the significance level, and verify the regression conditions for a t-test.

What this topic is asking

The College Board (Topic 9.4) wants you to set up a significance test for a regression slope: write the hypotheses about the population slope $\beta$ (usually testing $\beta = 0$), identify the significance level, and check the regression conditions for the $t$-test.

Hypotheses about the slope

Hypotheses concern the population slope $\beta$, not the sample slope $b$; $b$ is the evidence. The null value is almost always $0$, because "$\beta = 0$" is precisely "no linear relationship," the claim we test against. Choose the alternative from the wording: a general "is there a relationship?" is two-sided ($\ne$); "higher $x$ goes with higher $y$" is one-sided positive ($>$); "$x$ is negatively associated with $y$" is one-sided negative ($<$). Define $\beta$ in words first ("let $\beta$ be the true slope of the regression of ... on ...").

Choosing the significance level

Set $\alpha$ before seeing the data, as the false-alarm risk you will accept for declaring a relationship that is not there. This is the line the P-value (Topic 9.5) will be compared against.

Checking the conditions (LINER)

These are the same conditions as the slope interval, checked from residual plots and the data description. A residual plot with no pattern supports linearity; constant vertical spread supports equal variance; a roughly symmetric residual distribution supports normality. State each condition with its evidence. These conditions earn the $t$-model with $n - 2$ degrees of freedom that the test in Topic 9.5 uses.

Try this

Q1. Write the hypotheses to test for any linear relationship between $x$ and $y$. [1 point]

• Cue. $H_0: \beta = 0$ versus $H_a: \beta \ne 0$ (two-sided, about the population slope).

Q2. Name the five regression conditions (LINER). [2 points]

• Cue. Linear, Independent, Normal residuals, Equal variance, Random.