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

Question 1

Skill question

The question tests forecasting using a quadratic regression model and evaluating the validity of extrapolation.

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

[4]

Question 2

Skill question

A researcher fits a power-law model $y = a x^{b}$ to a set of data by transforming it into the linear form $ln y = ln a + b ln x$ . From the data, they determine that $b = - 1.2$ and $ln a = 2$ .

Predict the value of $y$ when $x = 0.5$ and discuss why extrapolating this model to $x > 100$ might be misleading.

[5]

Question 3

Skill question

For two variables $X$ and $Y$ , the following summary statistics are given:

Sample means: $\overset{ˉ}{X} = 50$ , $\overset{ˉ}{Y} = 20$
Sample standard deviations: $s_{X} = 10$ , $s_{Y} = 5$
Pearson product-moment correlation coefficient: $r = 0.6$

1.

Find the equation of the regression line of $Y$ on $X$ in the form $y = a x + b$ .

[4]

2.

Find the equation of the regression line of $X$ on $Y$ in the form $x = cy + d$ .

[4]

3.

Explain how these two regression lines illustrate reverse causality concerns.

[2]

Question 4

Skill question

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 $0 \leq t \leq 20$ .

[5]

Question 5

Skill question

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

[4]

Question 6

Skill question

A quadratic trend for a species population over years $t$ is given by $P (t) = - 0.1 t^{2} + 2 t + 100$ for $0 \leq t \leq 10$ .

(a) Use this model to predict $P (20)$ .

(b) Discuss why this extrapolation might yield unrealistic results.

[4]

Question 7

Skill question

A researcher fits an exponential growth model by regressing $ln y$ on $x$ and obtains $ln y = 0.3 x + 2.$

Predict $y$ at $x = 10$ and indicate why extrapolating this model might misrepresent the data.

[4]

Question 8

Skill question

The question requires fitting a least squares regression line to a small dataset and performing a prediction via extrapolation.

Given the data points $(1, 3)$ , $(2, 5)$ , and $(4, 9)$ , find the equation of the least squares regression line $y = a x + b$ and use it to predict the value of $y$ at $x = 6$ .

[6]

Question 9

Skill question

Data for the population $P$ (in thousands) of a town from year 2000 to 2010 fit the regression line $P = 1.8 t + 50,$ where $t$ is the number of years since 2000.

Predict the population in 2020 and discuss potential issues with this extrapolation.

[4]

Question 10

Skill question

Regression models and their use in extrapolation and interpolation.

1.

A regression model estimated using data up to the year 2020 is $g (t) = 0.02 t + 0.5$ , where $g$ is the GDP growth rate and $t$ is the number of years since 2000.

Calculate the predicted growth rate in 2050.

[2]

2.

Explain the dangers of forecasting the GDP growth rate so far into the future using this model.

[2]

IB Mathematics Applications & Interpretation Question Type 4: Using Bivariate Data to Plot Scatter Diagrams Exercises