Advanced Placement Statistics Section 4 Advanced Placement Statistics Section 4.1: Transforming to Achieve Linearity EQ: How do you determine what type of regression equation best models your data?

Introduction to Nonlinear Transformations Up to this point in the course, all of our data has been modeled by lines.

However many relationships are nonlinear.

Sometimes things aren’t what they appear. Objects thrown in the air parabolic Population growth exponential Sometimes things aren’t what they appear.

Transforming Data --- apply a function to straighten nonlinear patterns into linear patterns How to determine if a line is the BEST Model? RECALL: Correlation Coefficient Scatterplot Coefficient of Determination Residual Plot

Part I: Exponential Model CALCULATOR ACTIVITY: Part I: Exponential Model Using L1 for year (after 1970) and L2 number of transistors, create a scatterplot for the data on p. 271 in your text book. Note: Cut x-axis off [2nd ZOOM] AXES OFF

Your scatterplot shows the growth in number of transistors on a computer chip from 1970 to 2000. Explanatory Variable is year after 1970. Response Variable is number of transistors on computer chips. It appears that the pattern of growth is nonlinear.

Obtain regression statistics. Create a residual plot for this data

Day 18 Agenda: Quiz 3.1 & 3.2 --- 30 minutes Only half of you have signed up for the AP Stat Exam. Reminder that that needs to be done as soon as possible. The link is on the calendar page on the blog.

The day in the life of a high school student…

Discuss why it appears that the pattern of growth is non-linear. Pattern in the residual plot Non linear trend Low Coefficient of Determination Calculate the natural log of NUMBER OF TRANSISITORS and place it in L3. Ln(L2)  L3

Create the scatterplot for L1 and L3. Since logs were taken for the response variable only, we will “back transform” this LSRL to an exponential model.

Calculate the Linear Regression Equation: Use LinReg L1, L3 to calculate the LSRL of Ln(# of transistors) on yrs since 1970 .

0.997 0.995 Paste it in Y1 then graph. 7.41 0.332 r = ______ 7.41 0.332 pred Ln(# of trans) = ___ + ___ (yrs since 1970) 0.997 0.995 r = ______ r2 = ______

See p 273 and compare the Minitab output with your output. Less of a pattern in the residual plot Linear trend High Coefficient of Determination Would you consider this linear model a good fit for this transformed data? Explain.

Using the transformed model to make predictions. Step 1: Evaluate the LSRL for the year 2003. pred Ln (# of transistors) = 7.41 + 0.332 (33) pred Ln (# of transistors) = 18.366

Why do we use base e? Natural logs, ln, are in base e. Step 2: Substitute value into the exponential function pred (# of transistors) = e18.366 Why do we use base e? Natural logs, ln, are in base e. Common logs, log, are in base 10. pred (# of transistors) ≈ 95,000,000

Step 3: Sketch this exponential model over the original scatterplot to see how well it “fits”. Y1 = e7.41 + 0.332x

Part II: Power Model Using L1 for DIST and L2 for period of revolutions create a scatterplot for the data on p. 282 in your text book. Your scatterplot shows the increase in distance from the sun as the period of the revolutions also increases. Explanatory Variable is distance from the sun. Response Variable is period of the revolutions. It appears that the pattern of growth is nonlinear.

Would you consider this linear model a good fit for this data? Non linear trend Pattern in the residual plot But a high Coefficient of Determination Would you consider this linear model a good fit for this data?

Calculate the natural log of DIST and place it L3 and the natural log of period of revolutions and place it L4. Create a scatterplot of these transformations.

Calculate the Linear Regression Equation: Since logs were taken for both the explanatory and response variables, we will “back transform” this LSRL to a power model. Calculate the Linear Regression Equation: 0.000254 1.50 pred Ln(per of rev) = _____ + ____(ln(dist from sun)) 0.999 r = _______ 0.999 r2 = ______

Would you consider this linear model a good fit for this transformed data? Although curved pattern is apparent, size of residuals is VERY SMALL.[see p. 284]

Rewrite the LSRL as a Power Equation: Step 1: Use your model to predict the period of revolutions for Xena, a new planet discovered in our solar system July 31, 2005. Xena is an average of 102.15 astronomical units from our sun. Step 2: Substitute value into the exponential function pred Ln(per of rev) = 0.000254 + 1.50 (Ln(Dist from Sun)) pred (per of rev) = e 6.9399 = 1032.67 per of rev

Step 3: Sketch this model over your original scatterplot to see how well it fits. Y1 = e 0.000254 + 1.50 (Ln(X)) NOTE Ln(X)

BAT on AP Exam: interpret a residual plot, discuss coefficient of determination, and use a nonlinear model to make predictions Assignment: Complete WS problems #1 p. 276 #5 Perform Exp Reg and Power Reg to determine which is best model Use your model to answer g) p. 285 #11 Perform Exp Reg and Power Reg to determine which is best model. Use your model to answer d)