Descriptive Methods in Regression and Correlation

In this section we introduced a descriptive measure of the utility of the regression equation for making predictions.

a. Identify the term and symbol for the descriptive measure.

b. Provide an interpretation.

Fill in the blank. 

The graph of a linear equation with one independent variable is a _____.

Consider the linear equation y=b₀+b₁x.

a. Identify and give the geometric interpretation of b₀ .

b. Identify and give the geometric interpretation of b₁ .

Answer true or false to each statement, and explain your answers.

a. The graph of a linear equation slopes upward unless the slope is 0 .

b. The value of the y-intercept has no effect on the direction that the graph of a linear equation slopes.

In Exercise 4.5, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

Given equation is,

$y=4x+3$

In Exercise 4.6, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation.

Given equation is,

$y=2x-1$

In Exercise 4.7, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing      the equation.

c. use two points to graph the equation. 

given equation is,

$y=-7x+6$

In Exercise 4.8, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

Given equation is,

$y=-4x-8$

In Exercise 4.9, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

Given equation is,

$y=0.5x-2$

In Exercise 4.10, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

$y=-0.75x-5$

In Exercise 4.11, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

Given equation is,

$y=2$

In Exercise 4.12, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

Given equation is,

$y=-3x$

In Exercise 4.13, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

$y=1.5x$

We identify the y-intercepts $5$ and slopes $2$ respectively, without graphing the equation-

(a) determine whether it slopes is upward, slopes downward, or is horizontal, without graphing the equation.

(b) find its equation.

(c) use two points to graph the equation.

We identify the y-intercepts $-3$ and slopes $4$ respectively, without graphing the equation-

(a) determine whether it slopes is upward, slopes downward, or is horizontal, without graphing the equation.

(b) find its equation.

(c) use two points to graph the equation.

We identify the y-intercepts $-2$ and slopes $-3$ respectively, without graphing the equation-

(a) determine whether it slopes is upward, slopes downward, or is horizontal, without graphing the equation.

(b) find its equation.

(c) use two points to graph the equation.

We identify the y-intercepts  $0.4$ and slopes $1$ respectively, without graphing the equation-

(a) determine whether it slopes is upward, slopes downward, or is horizontal, without graphing the equation.

(b) find its equation.

(c) use two points to graph the equation.

The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you let $y$ denote Fahrenheit temperature and $x$ denote Celsius temperature, you can express the relationship between those two scales with the linear equation $y=32+1.8x$
a. Determine $b_0$  and $b_1$
b. Find the Fahrenheit temperatures corresponding to the Celsius temperatures $-40°$, $0°$, $20°$, and $100°$.
c. Graph the linear equation $y=32+1.8x$, using the four points found in part (b).
d. Apply the graph obtained in part (c) to estimate visually the Fahrenheit temperature corresponding to a Celsius temperature of $28$. Then calculate that temperature exactly by using the linear equation $y=32+1.8x$

A ball is thrown straight up in the air with an initial velocity of $64$ feet per second (ft/sec). According to the laws of physics, if you let $y$denote the velocity of the ball after $x$ seconds, $y=64-32x$.
a. Determine $b_0$ and $b_1$ for this linear equation.
b. Determine the velocity of the ball after $1, 2, 3, 4$ second.
c. Graph the linear equation $y=64-32x$, using the four points obtained in part (b).
d. Use the graph from part (c) to estimate visually the velocity of the ball after $1.5$ sec. Then calculate that velocity exactly by using the linear equation $y=64-32x$.

Measuring Temperature. The two most commonly used scales for measuring temperature are the Fahrenheit and Celsius scales. If you let $y$ denote Fahrenheit temperature and $x$ denote Celsius temperature, you can express the relationship between those two scales with the linear equation is $y=32+1.8x$.

a. Find the y-intercept and slope of the specified linear equation.
b. Explain what the y-intercept and slope represent in terms of the graph of the equation.
c. Explain what the  y-intercept and slope represent in terms relating to the application.

Air-Conditioning Repairs. Richard's Heating and Cooling in Prescott, Arizona, charges $\backslash (55$ per hour plus a $\backslash )30$ service charge. Let$x$ denote the number of hours required for a job, and let $y$ denote the total cost to the customer.. The linear equation is$y=30+55x$.

a. Find the y-intercept and slope of the specified linear equation.
b. Explain what the y-intercept and slope represent in terms of the graph of the equation.
c. Explain what the  y-intercept and slope represent in terms relating to the application.

Rental-Car Costs. During one month, the Avis Rent Car rate for renting a Buick LeSabre in Mobile, Alabama, was $68.22$ per day plus $0.25$ per mile. For a one day rental, let $x$ denote the number of miles driven and let $y$ denote the total cost, in dollars..  The linear equation is $y=68.22+0.25x$

a. Find the y-intercept and slope of the specified linear equation.
b. Explain what the y-intercept and slope represent in terms of the graph of the equation.
c. Explain what the y-intercept and slope represent in terms relating to the application.

In Exercise 4.14, we give linear equations. For each equation,

a. find the $y$-intercept and slope.

b. determine whether the line slopes upward, slopes downward, or is horizontal, without graphing the equation.

c. use two points to graph the equation. 

$y=-3$

A law of physics. Refer to exercise $4.26$. The linear equation is $y=64-32x$.

a. Find the $y$-intercept and slope of the specified linear equation.

b. Explain what the $y$-intercept and slope represent in terms of the graph of the equation.

c. Explain what the $y$-intercept and slope represent in terms relating to the application.

We identify the y-intercepts $0$and slopes $-0.5$ respectively, without graphing the equation-

(a) determine whether it slopes is upward, slopes downward, or is horizontal, without graphing the equation.

(b) find its equation.

(c) use two points to graph the equation.

Hooke's Law. According to Hooke's law for springs, developed by Robert Hooke ($1635-1703$), the force exerted by a spring that has been compressed to a length $x$ is given by the formula $F=-k\left(x-x_0\right)$, where $x_0$ is the natural length of the spring and $k$ is a constant, called the spring constant. A certain spring exerts a force of $32 lb$ when compressed to a length of $2 ft$ and a force of $16 lb$ when compressed to a length of $3 ft$. For this spring, find the following

a. The linear equation that relates the force exerted to the length compressed

b. The spring constant

c. The natural length of the spring

In Exercises 4.20, we identify the $y$-intercepts and slopes, respectively, of lines. For each line.

a. determine whether it slopes upward, slopes downward, or is horizontal, without graphing the equation.

b. find its equation.

c. use two points to graph the equation.

$-1.5$ and $0$

In Exercises 4.21, we identify the $y$-intercepts and slopes, respectively, of lines. For each line.

a. determine whether it slopes upward, slopes downward, or is horizontal, without graphing the equation.

b. find its equation.

c. use two points to graph the equation.

$3$ and $0$

In Exercises 4.22, we identify the $y$-intercepts and slopes, respectively, of lines. For each line.

a. determine whether it slopes upward, slopes downward, or is horizontal, without graphing the equation.

b. find its equation.

c. use two points to graph the equation.

$0$ and $3$

Rental-Car Costs. During one month. the Avis Rent-A-Car rate for renting a Buick LeSabre in Mobile. Alabama, was \(68.22 per day plus 25\) per mile. For a 1-day rental, let x denote the number of miles driven and let y denote the total cost, in dollars.

a. Find the equation that expresses y in terms of x.

b. Determine b₀ and b₁ .

c. Construct a table similar to Table 4.1 on page 158 for the x-values 50,100 ,and 250 miles.

d. Draw the graph of the equation that you determined in part (a) by plotting the points from part (c) and connecting them with a line.

e. Apply the graph from part (d) to estimate visually the cost of driving the car 150 miles. Then calculate that cost exactly by using the equation from part (a).

Air-Conditioning Repairs. Richard's Heating and Cooling in Prescott, Arizona, charges $\backslash (55$ per hour plus a $\backslash )30$ service charge. Let x denote the number of hours required for a job, and let y denote the total cost to the customer.

a. Find the equation that expresses y in terms of x.

b. Determine $b_0$ and $b_1$.

c. Construct a table similar to Table 4.1 on page 158 for the x-values $0.5,1$, and $2.25$ hours.

d. Draw the graph of the equation that you determined in part (a) by plotting the points from part (c) and connecting them with a line.

e. Apply the graph from part (d) to estimate visually the cost of a job that takes 1.75 hours. Then calculate that cost exactly by using the equation from part (a).

Road Grade. The grade of a road is defined as the distance it rises (or falls) to the distance it runs horizontally, usually expressed as a percentage. Consider a road with a positive grade, $g$. Suppose that you begin driving on that road at an altitude $a_0$

a. Find the linear equation that expresses the altitude, $a$, when you have driven a distance, $d$, along the road. (Hint: Draw a graph and apply the Pythagorean Theorem

b. Identify and interpret the $y$-intercept and slope of the linear equation in part (a).

c. Apply your results in parts (a) and (b) to a road with a $5\%$ grade and an initial altitude of $1 mile$. Express your answer for the slope to four decimal places.

d. For the road in part (c). what altitude will you reach after driving $10 miles$ along the road?

e. For the road in part (c), how far along the road must you drive to reach an altitude of $3 miles$?

Vertical Lines. In this section, we stated that any nonvertical line can be described by an equation of the form $y=b_0+b_{1}x$.

a. Explain in detail why a vertical line can't be expressed in this form.

b. What is the form of the equation of a vertical line?

c. Does a vertical line have a slope? Explain your answer.

For a data set consisting of 2 data points:

For each of the following sets of data points, determine the regression equation both without and with the use of formula 4.1 on page 165.

a. Find the regression equation for the data points, use the defining formulas in definition 4.4 to obtain  $s_{xx}and s_{xy}$

b. Graph the regression equation and the data points

Fill in the blanks.

a. In the context of regression, an_______ is a data point that lies far from the regression line, relative to the other data points.

b. In regression analysis, an_______ is a data point where removal causes the regression equation to change considerably:

For which of the following sets of data points can you reasonably determine a regression line? Explain your answer.

For which of the following sets of data points can you reasonably determine a regression line? Explain your answer.

(a) Plot the data points and the first linear equation on one graph and the data points and the second linear equation on another.

(b) Construct tables for $x,y,e,e^2,\hat{y}$

(c) Determine which line fits the data points better according to the least-square criterion.

(a) Plot the data points and the first linear equation on one graph and the data points and the second linear equation on another.

(b) Construct tables for $x,y,e,e^2,\hat{y}$

(c) Determine which line fits the data points better according to the least-square criterion.

a) Plot the data points and the first linear equation on one graph and the data points and the second linear equation on another.

(b) Construct tables for $x,y,e,e^2,\hat{y}$

(c) Determine which line fits the data points better according to the least-square criterion.

a) Plot the data points and the first linear equation on one graph and the data points and the second linear equation on another.

(b) Construct tables for $x,y,e,e^2,\hat{y}$

(c) Determine which line fits the data points better according to the least-square criterion.

For a data set consisting of two data points:

a. identify the regression line.

b. What is the sum of squared errors for the regression line? Explain your answer.

Regarding a scatterplot,

a. Identify one of its uses.

b. What property should it have to obtain a regression line for the data?

Regarding the criterion used to decide on the line that best fits a set of data points,

a. What is that criterion called?

b. Specifically, what is the criterion?

Regarding the line that best fits a set of data points,

a. What is that line called?

b. What is the equation of that line called?

Regarding the two variables under consideration in a regression analysis,

a. What is the dependent variable called?

b. What is the independent variable called? 

Using the regression equation to make predictions for the values of the predictor variable outside the range of the observed values of the predictor variable is called _____.

a. Find the regression equation for the data points, use the defining formulas in definition 4.4 to obtain  $s_{xx}and s_{xy}$

b. Graph the regression equation and the data points