Question Type 4: Using bivariate data to plot scatter diagrams Exercises

Given a linear regression model $y = 2x + 5$ fitted over $x$ values from 0 to 10, forecast the value of $y$ at $x = 15$ and comment on the reliability of this extrapolation.

A regression model estimated using data up to year 2020 is $g(t)=0.02t+0.5,$ where $g$ is GDP growth rate and $t$ is years since 2000. Predict the growth rate in 2050 and explain the dangers of forecasting so far into the future.

Data for the population $P$ (in thousands) of a town from year 2000 to 2010 fit the regression line $P = 1.8t + 50,$ where $t$ is years since 2000. Predict the population in 2020 and discuss potential issues with this extrapolation.

A quadratic trend for a species population over years $t$ is given by $P(t)=-0.1t^2+2t+100$ for $t$ from 0 to 10. Use this model to predict $P(20)$ and discuss why this extrapolation might yield unrealistic results.

Data for monthly sales (in thousands) over $x=1$ to $5$ fit the quadratic regression model $S(x)=0.5x^2 -1.2x +10.$ Forecast sales for month $x=8$ and discuss the validity of this extrapolation.

A researcher fits an exponential growth model by regressing $\ln y$ on $x$ and obtains $\ln y =0.3x +2.$ Predict $y$ at $x=10$ and indicate why extrapolating this model might misrepresent the data.

A researcher fits a power‐law model $y=ax^b$ by transforming to $\ln y=\ln a+b\ln x$. From data they find $b=-1.2$ and $\ln a=2$. Predict $y$ when $x=0.5$ and discuss why extrapolating this model to $x>100$ might be misleading.

Given the data points $(1,3)$, $(2,5)$ and $(4,9)$, fit the least squares regression line $y=ax+b$ and then predict $y$ at $x=6$ by extrapolation.

A logistic growth model for a population is given by $N(t)=\frac{1000}{1+e^{-0.4(t-10)}}.$ Compute $N(0)$ and $N(30)$, and comment on using this model outside the range $0\le t\le20$.

For two variables $X$ and $Y$, the sample means are $\bar X=50$, $\bar Y=20$, sample standard deviations $s_X=10$, $s_Y=5$, and correlation $r=0.6$. Find (a) the regression line of $Y$ on $X$, (b) the regression line of $X$ on $Y$, and (c) explain how these two regressions illustrate reverse causality concerns.