Exercises for Question Type 1: Determining best fit model for a given set of data

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Exercises for Question Type 1: Determining best fit model for a given set of data - IB

Question 1

Level: SL/HL | Paper: Paper 2 | Subject: Mathematics: Applications and Interpretation

A set of data (x, y) is given by: (1, 2), (2, 4), (3, 8), (4, 16) Investigate which model form—a power model y = a x^{b} or an exponential model y = A B^{x} —fits the data better. Calculate the sum of squared residuals (SSR) for both models to support your conclusion.

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Question 2

A data set (x, y) consists of the following points: (10, 20), (20, 80), (30, 180), (40, 320) Assuming a power model y = a x^{b}, use linearization to find estimates of a and b .

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Question 3

The data below are recorded for x and y : (1, 2), (2, 5), (3, 10), (4, 17). Compare the models (a) quadratic y = a x^{2} + b x + c, (b) exponential y = A B^{x}, by computing and comparing their sum of squared residuals (SSR) values. Which model fits better?

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Question 4

A scatter plot of data (x, y) shows that as x doubles, y roughly quadruples. Propose a model of the form y = a x^{b}, explain your reasoning, and describe briefly how you would estimate a and b .

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Question 5

Given that the sum of squared residuals (SSR) for three candidate models fitted to a data set are SSR_{linear} = 150, SSR_{quadratic} = 120, and SSR_{exponential} = 130 . State which model provides the best fit and justify your answer.

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Question 6

t

$t$ (hours)	0	1	2	3
$y$	100	80	64	51.2

(a)

Determine the value of A and the value of k .

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(b)

Calculate the half-life of the radioactive material.

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Question 7

A bacterial culture grows according to an exponential law. The measured population at times t (hours) is given: (0, 100), (1, 180), (2, 324), (3, 583) Find the exponential model y = A B^{t} that best fits the data.

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Question 8

Statistics - Model Selection and Sum of Squared Residuals (SSR)

The following SSR (sum of squared residuals) values are obtained for four candidate models fitted to the same data: SSR_{linear} = 500, SSR_{quadratic} = 200, SSR_{cubic} = 180, SSR_{exponential} = 190 Select the best model and justify your choice, mentioning any trade-offs.

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Question 9

The data set below is collected for variables x and y : (1, 3), (2, 5), (3, 7), (4, 9) . (a) Fit a linear model y = m x + b to these points and compute the sum of squared residuals (SSR). (b) Fit an exponential model y = A B^{x} by linearizing ln y vs x and compute the SSR. (c) Determine which model is more appropriate.

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Question 10

After fitting a linear model and a cubic model to a given data set, you obtain S S R_{linear} = 50 and S S R_{cubic} = 0 . Should you choose the cubic model? Explain any additional considerations beyond S S R .

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Question 11

You fit both an exponential model and a cubic model to a data set. The sum of squared residuals (S S R) values are S S R_{exp} = 120 (with 2 parameters) and S S R_{cubic} = 110 (with 4 parameters). Discuss which model you would choose, referring to S S R and model complexity.

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IB Mathematics Applications & Interpretation Question Type 1: Determining Best Fit Model for a Given Set of Data Exercises