35.10 Example: Food digestibility
A study evaluated various food for sheep (Moir 1961). One combination of variables assessed is shown in Fig. 33.6.
The RQ is:
Does the digestible energy requirement of feed increase with dry matter digestibility percentage (and if so, what is the relationship)?
In this study, x is the dry matter weight digestibility percentage, and y is the digestible energy. The data are shown in Fig. 35.16. Using the software output (Fig. 35.17 (jamovi); Fig. 35.18 (SPSS)), the values of the slope and y-intercept in the sample are b0=−0.193 and b1=0.047. The regression equation is
^y=−0.193+0.047x.
FIGURE 35.16: The digestibility data set
The slope means that when the dry matter weight digestibility increases by 1 percentage point, the digestible energy increases, on average, by 0.047 Cal/gram.
Each sample will produce slightly different sample slopes, so we can test to see if the slope in the population is non-zero due to sampling variation, using a hypothesis test:
- H0: β1=0;
- H1: β1>0.
The parameter is β1, the population slope for the regression equation predicting digestible energy from dry matter weight.
The alternative hypothesis is one-tailed, based on the RQ.
From the software output, t=39.322, which is huge; the two-tailed P-value is P<0.001. Since we have a one-tailed alternative hypothesis, the P-value is less than 0.001/2=0.0005. There is very strong evidence that the digestible energy increases as the dry matter weight digestibility increases.
The approximate 95% CI for the population slope β1 is
0.047±(2×0.001), or from 0.045 to 0.049.

FIGURE 35.17: jamovi output for the sheep-feed data

FIGURE 35.18: SPSS output for the sheep-feed data
The results are statistically valid. We write:
The sample presents very strong evidence (t=39.322; one-tailed P<0.0005) of a relationship between dry matter weight digestibility and the digestible energy (slope: 0.047; n=36; 95% CI from −0.045 to −0.049) in the population.
Click on the hotspots in the following image, to see what the areas under the normal curve mean.