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:

$H_0$ : $\rho=0$ ;
$H_1$ : $\rho\ne0$ : 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

References

Chitwood DE, Devinny JS. Treatment of mixed hydrogen sulfide and organic vapors in a rock medium biofilter. Water Environment Research. Water Environment Federation; 2001;73(4):426–35.

Devore JL, Berk KN. Modern mathematical statistics with applications. Thomson Higher Education; 2007.