Question 2 of 6: Paired T-Test for Two SO₂ Measurement Times

Question 2: Paired T-Test for Two SO₂ Measurement Times (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Which Test, and the Conclusion

Given.

Find. The appropriate t-procedure, its test statistic, and whether Time 1 is significantly different from Time 2 at $\alpha=0.05$.

Approach. Determine whether the two columns are independent or paired, then apply the matching t-procedure to the 10 differences and compare the statistic to the supplied t-table.

1. Identify the design. Time 1 and Time 2 were both measured at the same 10 locations in the building – each pair $(T1_i,T2_i)$ shares a common location, so the two columns are naturally correlated rather than independent. The paired t-test is therefore the statistically appropriate procedure: it works directly on the 10 differences $d_i=T1_i-T2_i$, so location-to-location variability common to both readings cancels out of the comparison, leaving only the genuine systematic difference between the two times. An independent two-sample test would ignore that correlation and misstate the standard error of the difference.
2. Hypotheses. $H_0:\mu_d=0$ (Time 1 is not significantly different from Time 2) vs. $H_a:\mu_d\ne0$ (two-sided, matching the "not significantly different" wording, a statement of equality).
3. Test statistic. $\bar d=-0.64$, $s_d=4.479$, $SE_{\bar d}=s_d/\sqrt{n}=4.479/\sqrt{10}=1.416$. $$t=\frac{\bar d}{SE_{\bar d}}=\frac{-0.64}{1.416}$$ $$t=\boxed{-0.452}\qquad(df=n-1=9)$$
4. Critical value and decision. For a two-tailed test at $\alpha=0.05$, each tail carries 0.025; from the supplied t-table at $df=9$, $t_{0.025,9}=2.262$. Since $|t|=0.452<2.262$, we fail to reject $H_0$. (At the scenario sentence's 10% level, $t_{0.05,9}=1.833$ – still $|t|=0.452<1.833$, the same decision.)

Conclusion. At both the instructed 5% level and the scenario's stated 10% level, there is no statistically significant difference between the Time 1 and Time 2 SO₂ readings – the control measures show no detectable shift in pollutant concentration between the two measurement times, which is exactly what the exam scenario set out to demonstrate.

(b) Main Assumption and Graphical Check

The paired t-test's main assumption is that the population of paired differences $d_i=T1_i-T2_i$ is (at least approximately) normally distributed. With only $n=10$ pairs, the Central Limit Theorem cannot be relied on to normalize a non-normal difference distribution on its own, so approximate normality of the $d_i$ themselves is required for the t reference distribution used in part (a) to be valid.

A simple graphical check is a normal probability (quantile-quantile) plot of the 10 sorted differences against their theoretical normal quantiles: if the points fall close to a straight line, normality is a reasonable working assumption; systematic curvature or a lone far-outlying point would instead signal a problem.

(c) Alternative Tests

If the normality of the differences were in doubt (e.g. the Q–Q plot in part (b) had shown clear curvature or a severe outlier), the standard distribution-free alternative to the paired t-test is the Wilcoxon signed-rank test, which ranks the absolute differences $|d_i|$, restores the sign of each rank, and tests whether the signed-rank sum is consistent with a symmetric distribution centred on zero – it uses more information than a simple sign count and is the usual first choice when normality fails but the differences are still reasonably symmetric. A cruder but assumption-free alternative is the sign test, which only counts how many differences are positive versus negative (ignoring magnitude) and compares that count to a binomial(n, 0.5) reference – it is less powerful than the Wilcoxon signed-rank test but requires no distributional assumption at all beyond independence of the pairs. A third option, if the underlying process is believed genuinely independent rather than paired, would be an independent two-sample Welch's t-test (unequal-variance t-test), though that would be the wrong model here given the paired design identified in part (a).