Topic 9.2 Confidence Intervals for the Slope of a Regression Model: check the regression conditions and construct a t-interval for the population slope using the sample slope, its standard error, and n minus 2 degrees of freedom.

What this topic is asking

The College Board (Topic 9.2) wants you to construct a confidence interval for the population slope $\beta$: check the regression conditions, read the sample slope $b$ and its standard error $SE_b$ (usually from computer output), and use the $t$-distribution with $n - 2$ degrees of freedom.

The interval and the standard error

The structure is the familiar "estimate $\pm$ critical value $\times$ standard error." The estimate is the sample slope $b$; the standard error $SE_b$ measures how much $b$ varies from sample to sample (it shrinks with more data, a wider spread of $x$-values, and a tighter fit). On the exam, $b$ and $SE_b$ are typically given in computer output, where the slope row lists the coefficient, its standard error, a t-statistic, and a P-value. You read $b$ and $SE_b$ from that row. The degrees of freedom are $n - 2$, because estimating both the intercept and the slope uses up two.

The regression conditions (LINER)

You check these mainly from residual plots and a description of the data: a residual plot with no pattern supports linearity, roughly constant vertical spread supports equal variance, and a histogram or normal plot of residuals supports normality. State each condition and the evidence for it; "the residual plot shows no pattern and constant spread" addresses two conditions at once. These conditions earn the $t$-model for the slope, just as the normal/large-sample conditions earned it for a mean.

Interpreting the interval

Interpret a slope interval in the units of the relationship: "We are $C\%$ confident that the true slope, the mean change in $y$ for each one-unit increase in $x$, is between [low] and [high]." The decisive feature is whether the interval contains $0$. A slope of $0$ means no linear relationship (changing $x$ does not change the predicted $y$). So if the interval excludes $0$, there is evidence of a real linear association; if it contains $0$, no relationship is plausible. This zero-check is the slope analogue of the difference-interval zero-check and previews the test of $H_0: \beta = 0$. As with all intervals, higher confidence widens it and a larger sample (or a wider spread of $x$) narrows it.

Try this

Q1. A regression with $n = 30$ gives $b = 2.0$, $SE_b = 0.6$. Find the degrees of freedom and a rough $95\%$ margin of error (use $t^{*} \approx 2.05$). [2 points]

• Cue. $df = n - 2 = 28$; margin of error $\approx 2.05 \times 0.6 = 1.23$, interval about $(0.77, 3.23)$.

Q2. What does it mean if a slope interval contains $0$? [1 point]

• Cue. A slope of $0$ (no linear relationship) is plausible, so there is not convincing evidence of a linear association between $x$ and $y$.