7.7 Inference about the slope
Is the slope we found distinguishable from zero?
\(\hat{\beta}_1\) is an estimator with a standard error, so everything in Unit 4 applies without modification.
One assumption is quietly added at this point, and it is the last of the six.
CLRM6 — normality. The errors are normally distributed: \(u \sim N(0, \sigma^2)\), independently of \(x\).
This is what makes the \(t\) statistic below follow a \(t\) distribution exactly, in any sample however small. CLRM1 to CLRM5 gave us an unbiased estimator and a correct standard error; CLRM6 gives us the distribution needed to turn them into a \(p\)-value.
Unlike the others, this assumption is one to hold lightly.
Why normality matters less than it appears to.
Nothing established so far depends on it. Unbiasedness came from CLRM1 to CLRM4. The variance formula came from CLRM5. The estimator, the standard error and the interpretation are all untouched by the shape of the error distribution.
Normality earns its keep only in small samples, where it makes the \(t\) distribution exact rather than approximate. In larger samples the Central Limit Theorem of Section 1.6 does the work instead: \(\hat{\beta}_1\) is a weighted sum of the errors, and sums tend towards normality whatever they are summing. The \(t\) statistic is then approximately correct regardless of how the errors are distributed.
So the practical position is this. With a few dozen observations and visibly skewed errors, treat small \(p\)-values with some caution. With a few hundred, the assumption is close to irrelevant.
That is a far weaker requirement than CLRM4, which cannot be relaxed by collecting more data, and which no amount of sample size will repair. It is worth keeping the two firmly apart: normality is a convenience that large samples make unnecessary, while zero conditional mean is the assumption everything rests on.
The hypothesis of interest is almost always that \(x\) has no effect at all:
\[H_0 : \beta_1 = 0 \qquad H_A : \beta_1 \neq 0\]
and the test statistic has the form it always has — estimate minus hypothesised value, over standard error:
\[t = \frac{\hat{\beta}_1 - 0}{\mathrm{se}(\hat{\beta}_1)}\]
compared against \(t\) with \(n - 2\) degrees of freedom.
#> Estimate Std. Error t value
#> (Intercept) 1.562 0.28692 5.444
#> attendance 0.189 0.01074 17.589
#> Pr(>|t|)
#> (Intercept) 0.000000072863221377091288988811794338079641519811957550700754
#> attendance 0.000000000000000000000000000000000000000000000000000000002468\(\hat{\beta}_1 = 0.189\) with a standard error of 0.0107, giving \(t = 17.6\) on 678 degrees of freedom and a \(p\)-value indistinguishable from zero. We reject the claim that attendance is unrelated to GPA.
The confidence interval is the more informative report:
#> 2.5 % 97.5 %
#> (Intercept) 0.9987 2.125
#> attendance 0.1679 0.210Each additional class is associated with between 0.168 and 0.210 of a GPA point, with 95% confidence. The interval excludes zero, agreeing with the test as it must.
“Statistically significant” is a claim about distinguishability from zero, not about magnitude. With 680 observations, very small slopes clear the bar easily.
Always ask what the slope means in the units of the problem. Here 0.19 GPA points per class is large. In the regression of final examination score on attendance in the same dataset, the slope is 0.117 marks per class and equally “significant” — and 0.117 marks is nothing at all.