2.6 Competing function model validation
Define the term "rate of change".
What are the three main types of function models?
What is a residual plot?
When might it be more appropriate to have an underestimate or overestimate for any given interval?
Given the following sets of data, identify whether a linear, quadratic, or exponential model would be most appropriate:
| x | -7 | -6 | -5 | -4 | -3 | -2 | -1 |
|---|---|---|---|---|---|---|---|
| y | 11 | 6 | 3 | 2 | 3 | 6 | 11 |
| x | -2 | -5 | 0 | 2 | 5 | 8 |
|---|---|---|---|---|---|---|
| y | 1 | 4 | 0 | -1 | -4 | -7 |
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 0 | -3 | -4 | -3 | 0 |
| x | -3 | -2 | -1 | 0 | 1 | \, 2 | \, 3 |
|---|---|---|---|---|---|---|---|
| y | \, \, \dfrac{1}{64} | \, \, \dfrac{1}{16} | \, \, \dfrac{1}{4} | 1 | 4 | 16 | 64 |
Construct and plot the graph based on the following data sets.
| x | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| y | 12 | 0 | -4 | 0 | 12 |
| x | -6 | -4 | -2 | 0 |
|---|---|---|---|---|
| y | -3 | -2 | -1 | 0 |
| x | - 6 | - 4 | - 2 | 0 | 2 |
|---|---|---|---|---|---|
| y | -7 | 5 | 9 | 5 | -7 |
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | \dfrac{1}{8} | \dfrac{1}{4} | \dfrac{1}{2} | 1 | 2 | 4 | 8 |
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 8 | 4 | 2 | 1 | \dfrac{1}{2} | \dfrac{1}{4} | \dfrac{1}{8} |
| x | -10 | -5 | 0 | 5 |
|---|---|---|---|---|
| y | 2 | 1 | 0 | -1 |
Given the following residuals after running a linear regression on a data set:
| x | -7 | -6 | -5 | -4 | -3 | -2 | -1 |
|---|---|---|---|---|---|---|---|
| \text{residual} | 1 | -1 | 0 | 2 | -2 | 1 | -1 |
Does the linear model is a good fit? Explain.
A model predicted a value of 10 for a given input, but the actual value was 7. What is the error of this prediction?
What is the error in the model if the predicted value is 150 and the actual value is 145?
When would you prefer to overestimate a prediction? Provide an example.
When would you prefer to underestimate a prediction? Provide an example.
The table below shows the number of hours studied and the corresponding test scores of a group of students.
| \text{Hours Studied} | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Test Score} | 70 | 80 | 88 | 94 | 99 |
Can this data be modelled with a linear, quadratic, or exponential function? Justify your answer.
The scatter plot below shows the relationship between a car's speed and its fuel consumption.
Based on the scatter plot, would a linear, quadratic, or exponential function be most appropriate to model this data? Justify your answer.
A company is trying to predict their sales for the next year. They have data from the past 5 years. Would it be more beneficial for the company to overestimate or underestimate their sales? Explain your reasoning.
Consider the following data set:
| x | -7 | -6 | -5 | -4 | -3 | -2 | -1 |
|---|---|---|---|---|---|---|---|
| y | 49 | 36 | 25 | 16 | 9 | 4 | 1 |
A student claims that a linear model would be the best fit for this data. Do you agree or disagree? Justify your response.
Suppose a model is consistently overestimating its predictions. What kind of impact could this have in a real-world context? Provide an example to support your response.