Chapter 9What’s My Curve?

Fall 2015

Looking BackIn Chapters 7 & 8, we worked with

LINEAR REGRESSIONWe learned how to:

Create a scatterplotDescribe a scatterplotDetermine the linear regression equationCreate a residuals plotAttempt to justify that a linear model fit the

data

Consider:Penguin Dive Duration and Heart Rate

Step 1 – LOOK: Describe this scatterplot.

Step 2 – Find Correlation Coefficient & Regression Equation

When we input the data into a calculator, we find:

r = -0.85

And

Does the r-value support our description?

Step 3 – Plot ResidualsIs the residual plot random

OrCan you see a curve?

What if Our Data Is Not Linear Enough?In Chapter 9, we will look at 2 curved models:

Note: These models willnot cover all curved data!!!!

ˆExponential: xy ab

ˆPower: by ax

Exponential ModelsExponential modes are often useful for

modeling relationships where the variables grow or shrink by a percentage of a current amount

Examples:Compound interestPopulation growth

ˆ xy ab

CW1: Complete the

table forˆ 4(3)xy

CW1: Solutionˆ 4(3)xy

Consider a Linear Model

Compare the 1st DifferencesLineary = 3 + 2x

Exponentialˆ 4(3)xy

CW 2 - Practice!For each table - identify if the

function is linear or exponential.

CW 2 - Solution!

a) Linear: increases by 5 each timeb) Exponential – multiply by 2 each timec) Exponential – divide by 3 each time (multiply by 1/3)d) Linear – Subtract 3 each time (add negative 3)

CW 3 – WordsBased on the description – identify

if the function is linear or exponential

CW 3 – Solution

a) L b) E c) L d) E e) E f) L g) E

SoLinear – add or subtract the same value each timeExponential – multiply or divide by the same value each time

CW 4 – What Does it Mean?

CW 4 – Solution

a)675b)-75c) Predicted = 1518.75 residual = actual – predicted 12 = x – 1518.75 x = 1530.75

WarningYou cannot find a perfect model!All models are wrong!Regression models are useful, but they

simplify the relationship and fail to fit every point exactly.

Don’t say “Correlation”.A correlation (r) measures the strength and

direction of:A linear associationBetween two quantitative variables

Remember!!!!!!If we see a curved relationship, it’s not

appropriate to calculate r or even use the term “correlation”.

CW5Complete the table of values to represent the

number of employees each year for 6 years when a company initially employees 50 people and grows by:

10 people per yearWhat equation would you use?

10% by yearWhat equation would you use?

Year 10per year

10% per year

0 50 50123456

CW5 – Method10 people per year

What equation would you use?

10% by yearWhat equation would you use?

Year 10 per year

10% per year

0 50 50123456

CW5 – Solution10 people per year

What equation would you use?

10% by yearWhat equation would you

use?

Year 10 per year

10% per year

0 50 501 60 552 70 60.53 80 66.554 90 73.215 100 80.536 110 88.58

CW9.1 WS – Complete the Table

How much can you complete in 10 minutes!

Can we identify the type of functionjust from looking at the equation?

Lineary = a + bx

ExponentialExplanatory Variable is an exponent

xy ab

Guidelines Check for:

Conditions Residual plots

Can only predict direction of the regression equation Don’t predict x’s from y’s

Avoid Assumption of causality and extrapolating beyond the data

Populations can’t grow exponentially indefinitely Note:

More advanced Statistics methods use transformations to linearize the relationship instead of fitting a curve to it, but in this course we simplify things for now by capitalizing on the power of the graphing calculator and computer software to keep the data in their original form and fit the curves to the relationship.

One drawback of this approach is the lack of a correlation coefficient (r) to help describe the strength of the relationship.

For now we’ll just have to trust what we see in the plot and residuals plot.

Power ModelPower models can have

A positive exponent (such as those that model changes in area relative to linear measurements)

OrA negative exponent(such as those modeling gas

volume relative to its pressure).

We will work more with the Power Model next class.

ˆ by ax