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

Question 1

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.

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Question 2

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.

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Question 3

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

(a)

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

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(b)

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

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(c)

Explain how these two regression lines illustrate reverse causality concerns.

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Question 4

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 .

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Question 5

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.

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Question 6

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.

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Question 7

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.

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Question 8

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 .

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Question 9

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.

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Question 10

Regression models and their use in extrapolation and interpolation.

(a)

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.

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(b)

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

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