A gardener is investigating what effect the volume of weed killer used 𝑥 has on the number of weeds 𝑦 in her garden. She collects data and then fits a linear regression model of the form 𝑦 equals 𝑎 plus 𝑏𝑥 to the data. Would you expect the regression coefficient 𝑏 to be positive or negative in this context?
Since we know that the gardener is using a linear regression model, to calculate a line in the form 𝑦 equals 𝑎 plus 𝑏𝑥, this is a line of best fit that tells us how 𝑦 is changing in relationship to 𝑥. We know the 𝑥-variable is the volume of weed killer used and that the 𝑦-variable represents the number of weeds in the garden.
We can think that as the volume of weed killer goes up, the less weeds should be in the garden. The more 𝑥 we have, the less 𝑦 we should have. This would correlate to a negative slope. And in the line 𝑦 equals 𝑎 plus 𝑏𝑥, the 𝑏-variable represents the slope, the changes in 𝑦 over the changes in 𝑥. Since we expect the more weed killer you use, the less the number of weeds will be, we can say that we expect the regression coefficient 𝑏 to be negative.