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Lines: Slopes, y-intercepts and their interpretations

Graphing Review

Label the following on the graph:

x-axis

y-axis

The origin

(label with its coordinates)

Quadrants I, II, III, IV

Plot points in the rectangular coordinate system

1. Plot the following points and label them using ordered pairs. Also, state the quadrant or axis on

which each point is located.

a. (5,3)

b. (1, 4)

c. (0,2)

d. 2

2 , 53

e. ( 0.25,1.75)

f. 9

,02

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Slope Notes

In this section we will are learning about linear equations. One component of linear equations is the rate of

change of lines, which measures the steepness of the line. We refer to this as the slope of the line. To do

this, we use two points on the line and find the ratio of the vertical change (called the rise) to the

corresponding horizontal change (called the run) as we move from one point to another.

The Slope Formula: (we reserve the letter m to mean the slope of a line)

The slope of a line can be described in several ways:

run

rise

changehorizontal

changeverticalmslope

Given two different points on a line 11, yxP and 22 , yxQ we can find an expression for the slope of the

line passing through the two points. First find the rise and the run.

The slope of the line PQ is given by:

m SLOPE = RUN

RISE=

RISE =____________ RUN = _______________

2. Find the slope of the line that passes through the points:

a.) 7,3&3,2 b.) 2,1 & 2, 3

What is slope?

We experience real life examples of slope every

day when we walk or drive.

As we move up a hill, it takes more energy to get

ourselves moving and to keep moving. The

steeper the hill, the more energy it takes.

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A line with a positive slope rises / falls from left to right. (Choose the best answer)

A line with a negative slope rises / falls from left to right. (Choose the best answer)

The slope of a horizontal line is ___________. The slope of a vertical line is ___________.

Let’s learn how this relates to Statistics, by learning the language! Please note that SLOPE is NOT

the same NUMBER as CORRELATION (we will get to this later)

Draw an example

Draw an example

What can you say about how correlation and slope are related? _______________________________

3. The graph below describes John’s savings account balance. Use the graph to answer the following:

_________________________________ occurs when each variable in the

function moves in the same direction.

As the values of x increase, the values of y _____________.

Moving from left to right, trace the line with your finger. Notice that the line

increases.

_________________________________ occurs when the two variables of a

function move in opposite directions. As the values of x increase, the values of y

_____________.

Moving from left to right, trace the line with your finger. Notice how the line

decreases.

Questions:

a. Does John’s Savings account balance have a positive

correlation, or negative correlation?

b. Find the slope, include the label as part of your

answer

c. Write a sentence describing an overview of John’s

savings per month

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4. In 1998, Jane bought a house for $144,000. In 2009, the house is worth $245,000. Find the average

annual rate of change in dollars per year of the value of the house. Round your answer to the nearest

cent. Let x represent number of years after 1990.

a. Write the data provided into two ordered pairs.

b. Why did we let x represent number of years after 1990?

c. Find the slope, including units, and write a sentence interpreting the slope.

Equations of Lines

Graph an equation by plotting points

Definitions

The __________________________________________________ is the set of points whose

coordinates ,x y in the x/y-plane satisfy the equation.

A ___________________________________________ has the form (also called standard form)

Ax By C

where A, B, and C are real numbers and A and B cannot both be zero.

If time is involved, it will always be

your x coordinate!

Step 1: Find at least two ordered pairs that satisfy the equation.

(Substitute values for x and find corresponding values for y.

Occasionally if it’s easier, substitute values for y and find

corresponding values for x.)

Step 2: Plot the points from step 1 in the x/y-plane.

Step 3: Don’t forget to connect the points with a smooth line (or curve if the

equation is not linear). Use arrows on the ends.

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5. Complete the given ordered pairs for the given equation

2 8y x

a. (0, )

b. ( , 0)

c. (3, )

d. ( , 7)

6. Make a table of values for the given equation using x = 0, 1, 2, 3, and 4.

4 1.5P x

x P (x,P)

7. Graph the equation 10 2P D . Find at least three solutions and label them on your graph

using ordered pairs.

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Now we want to be able to come up with the equation of the line. We usually want an equation of the form

___________________________________ which is called _______________________________

8. Find the equation of the line of the line passing through (2,6) and (5,12)

9. Find the equation of the line of the line passing through (0,4) and (2,8)

Steps:

1. Find the slope (use two points)

2. Rewrite the slope intercept form with the slope

3. Pick one point (x,y) and plug into equation to find b

4. Write your final answer for the equation of the line

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“Scatterplots, Linear Relationships and Correlation/Association”

10. Jake is driving towards Los Angeles at a constant rate of speed. After 3 hours he notices that he is 172

miles away and after 4 hours he notices that he is 110 miles away.

a. Write the information above as two ordered pairs, with time being the independent variable (first

variable) and the distance from Los Angeles being the dependent variable (second variable).

b. Using your ordered pairs from part (a), write a linear equation that models the distance Jake is away

from Los Angeles, D, as a function of the time he has been driving, t.

c. What is the slope of your linear equation including units? Write a sentence to interpret the slope in

context of the information.

d. What is the y-intercept of your linear equation? Write a sentence to interpret the y-intercept.

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Scatterplots may be the most common and most effective display for data.

In a scatterplot, you can see ______________________________________________ and even the

occasional extraordinary value(s) sitting apart from the others, called _______________________.

Scatterplots are the best way to start observing the relationship and the ideal way to picture associations

between _____________________________________

How do we decide which one goes on the x-axis, and the y-axis?

– An ____________________________________ explains or may influence changes in a response variable,

and is the _____________________________

– A ____________________________ measures an outcome of a study, and is the __________________

– Sometimes there is no distinction, so it doesn’t matter which one we call which.

Identify the explanatory and response variables in the following:

How the price of a package of meat is related to the weight of the package?

a. Explanatory:_____________________________

b. Response:_______________________________

There are four things we want to know when looking at scatterplots:

1. _______________________________

2. _______________________________

3. _______________________________

4. _______________________________

What does each dot represent?

(feet)

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A positive relationship will rise/fall to the right, in a line (circle one)

A negative relationship will rise/fall to the right, in a line (circle one)

If it’s not a positive or negative, can it still have a strong relationship?

For us, we are looking at two forms:

1. ___________________________

2. ___________________________

What is the form of the following?

Stronger Relationship Weaker Relationship

FORM: When we talk about the form of a relationship, we are talking

about the _____________________________________.

The Direction can be

________________________________________________________

STRENGTH: Finish this statement

The stronger the relationship…

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Summary Analysis Paragraph:

What’s an outlier?

The following scatterplot describes the

association between a driver’s age in

years and the maximum distance the

driver can read a sign from in feet.

Write a summary analysis paragraph on

the given data.

Be sure to include the direction, form,

strength and outliers. What does the

data suggest?

(feet)

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“Linear Relationships, and Correlation”

We need some way to measure how strong the potential linear relationship is.

What does “r” stand for? __________________________________

Why is the r value important?

Which one has a stronger linear association?

1

1

ni i

i x y

x x y y

s sr

n

We’re not going to ever calculate r by hand in

this class. How will we find it?

Plot 1

Plot 2

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Things on “r” correlation coefficient:

1. Is “r” the same number as slope? _____________________

2. Does “r” have units? Like miles/hour, like slope did? ______________

3. Is the correlation coefficient heavily influenced by outliers? ____________

4. Does correlation coefficient measure the strength of a LINEAR relationship? _______

5. Does correlation coefficient measure the strength of a NON-LINEAR relationship? _______

6. Does an “r” value close to 1 mean that a linear model is the best fit? ________

r = 0.869

7. Correlation is always between ___________ and ___________

Value of r Strength of relationship

-1.0 to -0.8 or 0.8 to 1.0 Strong Negative/Positive

-0.8 to -0.6 or 0.6 to 0.8 Moderate Negative/Positive

-0.6 to -0.4 or 0.4 to 0.6 Weak Negative/Positive

-0.4 to 0.4 No linear association or correlation

Note: These measurements of the ‘r-value’ are not set or defined to these boundaries. Other

factors may have influence on the strength of a linear relationship. However, this rule of

thumb is what we will be using in this module.

Strong

Negative

Moderate

Negative

Weak

Negative

No Linear Association

Weak

Positive

Moderate

Positive

Strong

Positive

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Linear Modeling and Predictions

Using the linear equation above, find the predicted max distance each age can read:

Age 57

Age 93

Age 8

Predictions

If we have a linear equation to describe the pattern of a scatterplot, we can use the line to predict a data

point that is not there if ____________________________________________________________.

When can we not predict or when will predictions have a potentially large amount of error?

_______________________________________________________________________________.

Learning objective: For a linear relationship, use the least squares regression line to

summarize the overall pattern and to make predictions.

We will

use lines to make predictions

identify situations where predictions can be misleading

develop a measurement for identifying the best line to summarize the data

use technology to find the best line

interpret the parts of the equation of a line to make our summary of the data

more precise

The line we will use to describe this

scatterplot is:

D = -3A + 576

Where A is age of the driver in years, and D

is the maximum distance in feet they can read

the sign.

(feet)

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Which line is better?

What’s a way we can

determine which is

better?

When we compare the sum of the areas of the yellow squares, the line on the left has

an SSE of 57.8. The line on the right has a smaller SSE of 43.9.

So the line on the right fits or models the data ___________, but is it the best fit?

The estimate made from a model is the

predicted value (denoted as ).

Residual = observed – predicted

=___________________

=___________________

Residual

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We want lines of the form:_____________________________________

To find these, we would need the descriptive statistics from Minitab, and then we can calculate slope and y

intercept.

Example. From the example, find the least squares regression line

Write down what each of the variables mean:

y

x

sm r

s b y mx

(fee

t)

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What is the predicted fat content for a WHOPPER sandwich that has 31g of protein?

The actual fat content ends up being 37g. What is the residual for the WHOPPER sandwich? What

does this mean?

Let’s interpret the slope and y-intercept. Don’t forget your units!

Slope:

y-intercept:

Use the data on Burger King burgers, fish, chicken

sandwiches, nuggets and strips to answer the following

regression questions. Data taken from July 2015 from

https://www.bk.com/pdfs/nutrition.pdf:

Equation of the Regression Line:

5040302010

80

70

60

50

40

30

20

10

0

S 9.26952

R-Sq 69.2%

R-Sq(adj) 68.3%

Protein (grams)

To

tal

Fat

(gra

ms)

Burger King Analysis: Protein & Total FatTotal Fat = 3.798 + 1.100(Protein)

r=0.832

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Residual Plots, Coefficient of Deterination, Standard Error

To create a residual plot, we will take the _____________ and plot these errors as distances from a base

line described by the explanatory or x-variable.

Completed Residual Plot:

We do a residual plot to get another view of how our ______________________________fits the

______________________________

A linear model is a good fit for the data when there is ______________________________in the residual

plot, this reassures us that the linear model is ______________________________.

A ______________________________to the residual plot indicates that a linear model may not be our

best choice of a model to fit the data. In cases of outliers, we may need to remove these outliers and review

our ______________________________to make a further analysis.

Lets look at some examples and make some determinations on whether a linear model would be a good fit for

the data based off of the residual plot.

Recall that the error or residual is the distance from the data point

and the line of regression which is given by:

y – y ̂

Take these distances and plot them as vertical distances based on

the x-value.

Here we are showing the graph of the points with an attached line

which shows the distances. When we do our residual plots these

connected lines will not be present.

You may want to use lines to get used to marking the distances if you

need and then erase them afterward to get your completed residual

plot

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Example: A

Example: B

Coefficient of Determination (r2): Is the __________________________________ of the variation in

the response variable that is associated or explained by its linear relationship with the explanatory variable

when using the least-squares regression line.

r2 = explained variation / total variation

1. Suppose we computed our regression line and we are given an r=0.75, calculate r2 and interpret what

this means.

2. The explanatory variable is describing the number of tons of paper trash and the response variable is

the number of tons of total trash. What is r2 and interpret it in context of the given data. (r = 0.729)

20151050

60

50

40

30

20

10

0

Tons of Paper Trash

To

ns o

f To

tal Tra

sh

Scatterplot of TOTAL vs PAPER

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Standard Error of Regression (Se): Measures the __________________________________________ in

predictions when using the regression line. We can think of this as the ___________________________of

our data based on our ____________________ model. (Recall that standard deviation measures the average

distance our data is from the mean)

It is calculated using the formula:

where SSE stands for _______________________________________________________________

Example:

Let’s take another look at the prediction that we made earlier using the regression line equation:

D = -3A + 576 or Distance = (-3*Age) + 576

Technology gives se = 51.35.

1. Write a sentence to interpret the standard error.

2. Predict the maximum distance that a 60 year old driver can read a highway sign.

3. Using your prediction and standard error from above, how far would you expect an actual 60 year old

driver to be able to read a sign from?

(feet

)