Common Error Variance (Mean squared error)

The plot of our population of data suggests that the college entrance test scores for each subpopulation have equal variance. We denote the value of this common variance as _σ_2.

That is, _σ_2 quantifies how much the responses (y) vary around the (unknown) mean population regression line μY=E(Y)=β0+β1x.
Why should we care about _σ_2? The answer to this question pertains to the most common use of an estimated regression line, namely predicting some future response.
Suppose you have two brands (A and B) of thermometers, and each brand offers a Celsius thermometer and a Fahrenheit thermometer. You measure the temperature in Celsius and Fahrenheit using each brand of thermometer on ten different days. Based on the resulting data, you obtain two estimated regression lines — one for brand A and one for brand B. You plan to use the estimated regression lines to predict the temperature in Fahrenheit based on the temperature in Celsius.
Will this thermometer brand (A) yield more precise future predictions …?

… or this one (B)?

> From <https://onlinecourses.science.psu.edu/stat501/node/254>