34.5 Example: Removal efficiency

In wastewater treatment facilities, air from biofiltration is passed through a membrane and dissolved in water, and is transformed into harmless byproducts. The removal efficiency $y$ (in %) may depend on the inlet temperature (in ${}^{\circ }$C; $x$).

The RQ is

In treating biofiltation wastewater, is the removal efficiency associated with the inlet temperature?

The population parameter is $\rho$, the correlation between the removal efficiency and inlet temperature.

A scatterplot of $n=32$ samples (Fig. 34.8) suggests an approximately linear relationship (Chitwood and Devinny 2001; Devore and Berk 2007). The output (jamovi: Fig. 34.9; SPSS: Fig. 34.10) shows that the sample correlation coefficient is $r=0.891$, and so $R^2=(0.891{)}^2=79.4$%. This means that about 79.4% of the variation in removal efficiency can be explained by knowing the inlet temperature.

FIGURE 34.8: The relationship between removal efficiency and inlet temperature

To test if a relationship exists in the population, write:

• $H_0$: $\rho =0$;
• $H_1$: $\rho \ne 0$: Two-tailed (as implied by the RQ).

The software output (jamovi: Fig. 34.9; SPSS: Fig. 34.10) shows that $P<0.001$ (which is what $P=0.000$ in SPSS means). We conclude:

The sample presents very strong evidence (two-tailed $P<0.001$) that removal efficiency depends on the inlet temperature ($r=0.891$; $n=32$) in the population.

The relationship is approximately linear and there is no obvious non-constant variance, and the sample size is larger than 25, so the hypothesis test results will be statistically valid.

FIGURE 34.9: jamovi output for the removal-efficiency data

FIGURE 34.10: SPSS output for the removal-efficiency data