4.3 Fitting Linear Models to Data

According to the data from the table, what temperature can we predict it is if we counted 20 chirps in 15 seconds?

Find the least squares regression line using the cricket-chirp data in the table.

Calculate the correlation coefficient for cricket-chirp data in the table.

Use the model we created using technology in the example to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?

Describe what it means if there is a model breakdown when using a linear model.

What is extrapolation when using a linear model?

Explain how to interpret the absolute value of a correlation coefficient.

A regression was run to determine whether there is a relationship between the diameter of a tree ( $x$ , in inches) and the tree’s age ( $y$ , in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.

$r = −0. \text{89}$

$r = −0.39$

The U.S. import of wine (in hectoliters) for several years is given in the table. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters?

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Graph $f \left(x\right) = - 2 x - 10.$ Pick a set of five ordered pairs using inputs $x = −2 , \text{1} , \text{5} , \text{6} , 9$ and use linear regression to verify the function.

Find to the nearest tenth and interpret the x-intercept.

$\left(\text{25} 00 , 2000\right) , \left(\text{265} 0 , 2001\right) , \left(3000 , 2003\right) , \left(\text{35} 00 , 2006\right) , \left(\text{42} 00 , 2010\right)$

Use linear regression to determine a function $y ,$ where the year depends on the population. Round to three decimal places of accuracy.

$\left(\text{46} , \text{25} 0\right) , \left(\text{48} , \text{3} 05\right) , \left(50 , \text{35} 0\right) , \left(\text{52} , \text{39} 0\right) , \left(\text{54} , \text{41} 0\right) .$

Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds.

$\left(\text{46} , \text{25} 0\right) , \left(\text{48} , \text{225}\right) , \left(50 , \text{2} 05\right) , \left(\text{52} , \text{18} 0\right) , \left(\text{54} , \text{165}\right) .$

Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds.

Determine whether the algebraic equation is linear. $2 x + 3 y = 7$

Determine whether the function is increasing or decreasing.

$f \left(x\right) = 7 x - 2$

Given each set of information, find a linear equation that satisfies the given conditions, if possible.

Passes through $\left(\text{7} , \text{5}\right)$ and $\left(\text{3} , \text{17}\right)$

Does the following table represent a linear function? If so, find the linear equation that models the data.

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}2 x - 6 y = 12 \\ - x + 3 y = 1\end{matrix}$

Find the x- and y- intercepts of the given equation

$7 x + 9 y = - 63$

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Line 1: Passes through $\left(5 , 11\right)$ and $\left(10 , 1\right)$

Line 2: Passes through $\left(−1 , 3\right)$ and $\left(−5 , 11\right)$

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Write an equation for a line perpendicular to $f \left(x\right) = 5 x - 1$ and passing through the point (5, 20).

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

A car rental company offers two plans for renting a car.

Plan A: 25 dollars per day and 10 cents per mile

Plan B: 50 dollars per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

A town’s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.

Find the linear function y, where y depends on $x ,$ the number of years since 1980.

Consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.

Assume the population is changing linearly.

1. ⓐHow much did the population grow between the year 2004 and 2012?
2. ⓑWhat is the average population growth per year?
3. ⓒFind an equation for the population, P, of the school t years after 2004.

Consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.

What does your model predict the moose population to be in 2020?

If these trends were to continue, what would be the median home value in Pima Central in 2015?

In what year will the percentage exceed 12%?

Based on the set of data given in the table, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.

$\left(\text{3},\text{6} 00 , 2000\right) ; \left(\text{4}, 000 , 2001\right) ; \left(\text{4},\text{7} 00 , 2003\right) ; \left(\text{6}, 000 , 2006\right)$

Predict when the population will hit 12,000.

$\left(\text{3},\text{6} 00 , 2000\right) ; \left(\text{4}, 000 , 2001\right) ; \left(\text{4},\text{7} 00 , 2003\right) ; \left(\text{6}, 000 , 2006\right)$

According to the model, what is the population in 2014?

Determine whether the following algebraic equation can be written as a linear function. $2 x + 3 y = 7$

Determine whether the following function is increasing or decreasing. $f \left(x\right) = 7 x + 9$

Find a linear equation, that has an x intercept at (–4, 0) and a y-intercept at (0, –6), if possible.

Does the table represent a linear function? If so, find a linear equation that models the data.

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

$\begin{matrix}y = \frac{3}{4} x - 9 \\ - 4 x - 3 y = 8\end{matrix}$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

Find the x- and y-intercepts of the equation $2 x + 7 y = - 14.$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

Write an equation for a line perpendicular to $f \left(x\right) = 4 x + 3$ and passing through the point $\left(8 , 10\right) .$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

A car rental company offers two plans for renting a car.

Plan A: $25 per day and $0.10 per mile

Plan B: $40 per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

A town’s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.

In what year will the percentage drop below 4%?

$\left(4 , 500 , 2000\right) ; \left(4 , 700 , 2001\right) ; \left(5 , 200 , 2003\right) ; \left(5 , 800 , 2006\right)$

Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.

$\left(4 , 500 , 2000\right) ; \left(4 , 700 , 2001\right) ; \left(5 , 200 , 2003\right) ; \left(5 , 800 , 2006\right)$

What is the correlation coefficient for this model?

According to the data from the table, what temperature can we predict it is if we counted 20 chirps in 15 seconds?

Find the least squares regression line using the cricket-chirp data in the table.

Calculate the correlation coefficient for cricket-chirp data in the table.

Use the model we created using technology in the example to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?

Describe what it means if there is a model breakdown when using a linear model.

What is extrapolation when using a linear model?

Explain how to interpret the absolute value of a correlation coefficient.

A regression was run to determine whether there is a relationship between the diameter of a tree ( $x ,$ in inches) and the tree’s age ( $y ,$ in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.

Draw a scatter plot for the data provided. Does the data appear to be linearly related?

Draw a scatter plot for the data provided. Does the data appear to be linearly related?

Draw a scatter plot for the data provided. Does the data appear to be linearly related?

For the following data, draw a scatter plot. If we wanted to know when the temperature would reach 28 °F, would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer.

$r = - 0. \text{89}$

$r = - 0.39$

The U.S. import of wine (in hectoliters) for several years is given in the table. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters?

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

Graph $f \left(x\right) = - 2 x - 10$. Pick a set of 5 ordered pairs using inputs $x = −\text{2} , \textrm{ }\text{1} , \textrm{ }\text{5} , \textrm{ }\text{6} , \textrm{ }\text{9}$ and use linear regression to verify the function.

$\left(\text{46} , \textrm{ }\text{1} , \text{6} 00\right) , \left(\text{48} , \textrm{ }\text{1} , \text{55} 0\right) , \left(\text{5} 0 , \textrm{ }\text{1} , \text{5} 0 \text{5}\right) , \left(\text{52} , \textrm{ }\text{1} , \text{54} 0\right) , \left(\text{54} , \textrm{ }\text{1} , \text{495}\right)$.

Find to the nearest tenth and interpret the x-intercept.

$(\text{2500},\textrm{ }\text{2000}),\textrm{ }(\text{2650},\textrm{ }\text{2001}),\textrm{ }(\text{3000},\textrm{ }\text{2003}),\textrm{ }(\text{3500},\textrm{ }\text{2006}),\textrm{ }(\text{4200},\textrm{ }\text{2010})$

Use linear regression to determine a function $y ,$ where the year depends on the population. Round to three decimal places of accuracy.

$\left(\text{46} , \textrm{ }\text{25} 0\right) , \left(\text{48} , \textrm{ }\text{3} 0 \text{5}\right) , \left(\text{5} 0 , \textrm{ }\text{35} 0\right) , \left(\text{52} , \textrm{ }\text{39} 0\right) , \left(\text{54} , \textrm{ }\text{41} 0\right)$.

Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds .

$(\text{46},\textrm{ }\text{250}),\textrm{ }(\text{48},\textrm{ }\text{225}),\textrm{ }(\text{50},\textrm{ }\text{205}),\textrm{ }(\text{52},\textrm{ }\text{180}),\textrm{ }(\text{54},\textrm{ }\text{165}) .$

Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds .

Determine whether the algebraic equation is linear. $2 x + 3 y = 7$

Determine whether the function is increasing or decreasing.

$f \left(x\right) = 7 x - 2$

Given each set of information, find a linear equation that satisfies the given conditions, if possible.

Passes through $\left(\text{7} , \text{5}\right)$ and $\left(\text{3} , \text{17}\right)$

Does the following table represent a linear function? If so, find the linear equation that models the data.

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}2 x - 6 y = 12 \\ - x + 3 y = 1\end{matrix}$

Find the x- and y- intercepts of the given equation

$7 x + 9 y = −63$

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

1. Line 1: Passes through $\left(5 , 11\right)$ and $\left(10 , 1\right)$
2. Line 2: Passes through $\left(−1 , 3\right)$ and $\left(−5 , 11\right)$

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

Write an equation for a line perpendicular to $f \left(x\right) = 5 x - 1$ and passing through the point (5, 20).

Use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

A car rental company offers two plans for renting a car.

1. Plan A: 25 dollars per day and 10 cents per mile
2. Plan B: 50 dollars per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

A town’s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.

Consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.

Assume the population is changing linearly.

1. ⓐ How much did the population grow between the year 2004 and 2012?
2. ⓑ What is the average population growth per year?
3. ⓒ Find an equation for the population, P, of the school t years after 2004.

Consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.

What does your model predict the moose population to be in 2020?

If these trends were to continue, what would be the median home value in Pima Central in 2015?

In what year will the percentage exceed 12%?

Based on the set of data given in the table, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.

$(\text{3},\text{600},\textrm{ }\text{2000});\textrm{ }(\text{4},\text{000},\textrm{ }\text{2001});\textrm{ }(\text{4},\text{700},\textrm{ }\text{2003});\textrm{ }(\text{6},\text{000},\textrm{ }\text{2006})$

Predict when the population will hit 12,000.

$(\text{3},\text{600},\textrm{ }\text{2000});\textrm{ }(\text{4},\text{000},\textrm{ }\text{2001});\textrm{ }(\text{4},\text{700},\textrm{ }\text{2003});\textrm{ }(\text{6},\text{000},\textrm{ }\text{2006})$

According to the model, what is the population in 2014?

Determine whether the following algebraic equation can be written as a linear function. $2 x + 3 y = 7$

Determine whether the following function is increasing or decreasing. $f \left(x\right) = 7 x + 9$

Given the following set of information, find a linear equation satisfying the conditions, if possible.

x intercept at (–4, 0) and y-intercept at (0, –6)

Does the table represent a linear function? If so, find a linear equation that models the data.

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

$\begin{matrix}\begin{matrix} \\ y = \frac{3}{4} x - 9\end{matrix} \\ - 4 x - 3 y = 8\end{matrix}$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

Find the x- and y-intercepts of the equation $2 x + 7 y = - 14.$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

Write an equation for a line perpendicular to $f \left(x\right) = 4 x + 3$ and passing through the point $\left(8 , 10\right) .$

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

A car rental company offers two plans for renting a car.

1. Plan A: $25 per day and $0.10 per mile
2. Plan B: $40 per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

Determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

A town’s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.

Find the linear function $y$, where $y$ depends on $x$, the number of years since 1980.

In 2004, a school population was 1250. By 2012 the population had dropped to 875. Assume the population is changing linearly.

1. ⓐ How much did the population drop between the year 2004 and 2012?
2. ⓑ What is the average population decline per year?
3. ⓒ Find an equation for the population, P, of the school t years after 2004.

In what year will the percentage drop below 4%?

$(\text{4},\text{500},\textrm{ }\text{2000});\textrm{ }(\text{4},\text{700},\textrm{ }\text{2001});\textrm{ }(\text{5},\text{200},\textrm{ }\text{2003});\textrm{ }(\text{5},\text{800},\textrm{ }\text{2006})$

Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.

$(\text{4},\text{500},\textrm{ }\text{2000});\textrm{ }(\text{4},\text{700},\textrm{ }\text{2001});\textrm{ }(\text{5},\text{200},\textrm{ }\text{2003});\textrm{ }(\text{5},\text{800},\textrm{ }\text{2006})$

What is the correlation coefficient for this model?