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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 $1 \le x \le 5$ 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.

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Question 2

A researcher fits a power-law model $y = ax^b$ 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: $\bar{X} = 50$ , $\bar{Y} = 20$
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 $y = ax + b$ .

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2.

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

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3.

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 $0 \le t \le 20$ .

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Question 5

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.

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Question 6

A quadratic trend for a species population over years $t$ is given by $P(t) = -0.1t^2 + 2t + 100$ 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 $\ln y = 0.3x + 2.$

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 $y = ax + b$ 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 $P = 1.8t + 50,$ 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.

1.

A regression model estimated using data up to the year 2020 is $g(t) = 0.02t + 0.5$ , 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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2.

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

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