Fall 2015

Looking Back In Chapters 7 & 8, we worked with LINEAR REGRESSION We learned how to: Create a scatterplot Describe a scatterplot Determine the linear regression equation Create a residuals plot Attempt 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 = And Does the r-value support our description?

Step 3 – Plot Residuals Is the residual plot random Or Can 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 will not cover all curved data!!!!

Exponential Models Exponential modes are often useful for modeling relationships where the variables grow or shrink by a percentage of a current amount Examples: Compound interest Population growth

CW1: Complete the table for

CW1: Solution

Consider a Linear Model

Compare the 1 st Differences Linear y = x Exponential

CW 2 - Practice! For each table - identify if the function is linear or exponential.

CW 2 - Solution! a) Linear: increases by 5 each time b) Exponential – multiply by 2 each time c) Exponential – divide by 3 each time (multiply by 1/3) d) Linear – Subtract 3 each time (add negative 3)

CW 3 – Words Based 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

So Linear – add or subtract the same value each time Exponential – multiply or divide by the same value each time

CW 4 – What Does it Mean?

CW 4 – Solution a)675 b)-75 c)Predicted = residual = actual – predicted 12 = x – x =

Warning You 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 association Between two quantitative variables Remember!!!!!! If we see a curved relationship, it’s not appropriate to calculate r or even use the term “correlation”.

CW5 Complete 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 year What equation would you use? 10% by year What equation would you use? Year10 per year 10% per year

CW5 – Method 10 people per year What equation would you use? 10% by year What equation would you use? Year10 per year 10% per year

CW5 – Solution 10 people per year What equation would you use? 10% by year What equation would you use? Year10 per year10% per year

CW 9.1 WS – Complete the Table How much can you complete in 10 minutes!

Can we identify the type of function just from looking at the equation?

Linear y = a + b x

Exponential Explanatory Variable is an exponent

Guidelines Check for: Conditions Residual plots Can only predict direction of the regression equation Don’t predict x’s from y’s Avoid A ssumption 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 Model Power models can have A positive exponent (such as those that model changes in area relative to linear measurements) Or A negative exponent(such as those modeling gas volume relative to its pressure). We will work more with the Power Model next class.