Topic 9.5 Carrying Out a Test for the Slope of a Regression Model: compute the t test statistic for the slope using the standard error, find the P-value with n minus 2 degrees of freedom, and state a conclusion in context.

What this topic is asking

The College Board (Topic 9.5) wants you to carry out and conclude a slope test: compute the t statistic $t = \dfrac{b}{SE_b}$, find the P-value with $n - 2$ degrees of freedom, compare to $\alpha$, and state a conclusion in context, completing the test set up in Topic 9.4.

The slope t statistic

The statistic is the sample slope divided by its standard error, the slope analogue of $\dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$. Both $b$ and $SE_b$ come from computer output: the slope row lists the coefficient $b$, its standard error $SE_b$, the t-statistic, and a P-value. You can compute $t = b/SE_b$ directly, or read it from the "t" column. The degrees of freedom are $n - 2$.

From t to the P-value and decision

Computer output usually prints the two-sided P-value. For a one-sided test, halve the reported P-value (when $b$ is in the alternative's direction). Match the tail to $H_a$. The conclusion states the decision, ties it to $P$ versus $\alpha$, and interprets in context ("there is convincing evidence of a positive linear relationship between ... and ..."). Never write "accept $H_0$"; the choices are reject or fail to reject.

Test and interval agree; association is not causation

Because the slope test and slope interval use the same $b$, $SE_b$, and $df = n - 2$, a two-sided test at level $\alpha$ and a $C = 1 - \alpha$ interval agree exactly: $H_0: \beta = 0$ is rejected precisely when the interval excludes $0$. So the zero-check on the interval doubles as a two-sided test. Two cautions complete the unit: a significant slope shows a linear association, which need not be the full story if the relationship is curved (always check the residual plot), and association is not causation, only a randomised experiment supports a causal claim. Reporting these limits, and the scope of inference, rounds out a complete answer.

Try this

Q1. Output gives $b = 1.5$, $SE_b = 0.5$, $n = 24$. Find the t statistic and degrees of freedom. [2 points]

• Cue. $t = \dfrac{1.5}{0.5} = 3.0$; $df = n - 2 = 22$.

Q2. Output reports a two-sided P-value of $0.04$, but your test is one-sided in the direction of $b$. What P-value do you use? [1 point]

• Cue. Halve it: $0.02$ (valid because $b$ is in the alternative's direction).