Date post: | 17-Jan-2018 |
Category: |
Documents |
Upload: | jayson-nelson |
View: | 223 times |
Download: | 0 times |
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