Question Type 3: Forecasting another value given a linear fit Exercises

In a calibration experiment, the linear fit between temperature $x$ (°C) and resistance $R$ (Ω) is $R=200+1.5x$ valid for $0 \le x \le 200$. Find $R$ at $150\,^\circ\text{C}$.

The linear model for a bacteria culture growth over time $t$ (hours) is $N(t)=100+25t$, based on observations from $t=0$ to $t=12$. Estimate $N(20)$ and comment on the prediction’s reliability.

The linear regression model for the relationship between hours studied $x$ and exam score $y$ is given by $\text{Score} = 40 + 4.5x$ based on data for $x$ in the range 2 to 8. Forecast the score for a student who studies 10 hours and comment on the reliability of this prediction.

A study of car speed $x$ (km/h) versus fuel consumption $y$ (L/100 km) yields the linear fit $\text{Consumption} = 5.2 + 0.03x,$ for speeds between 40 and 120 km/h. Predict the consumption at 140 km/h and discuss the reliability of this forecast.