WARM UP: Use the Reciprocal Model to predict the y for an x = 8 1 20 2 11.1 3 7.7 4 5.9 6 11 2.2 16 1.5 x 1/y 1 .05 2 .09 3 .13 4 .17 6 .25 11 .45 16 .67 LinReg(x, 1/y)

• For Exponential Models take the LOG(y) and then perform a LinReg (x, LOG(y)). Use this model if there exist a common ratio from successive y-values. For Power Models take the LOG(x) and LOG(y) and then perform a LinReg (LOG(x), LOG(y)). Use this model if no common ratio exist from successive y-values.

Solve each equation for . (Perform the Inverse Log function of raising both sides as the exponent of base 10.) Use the Exponent property of xm+n = xm · xn Use the Exponent property of m Log x = Logxm

Choosing a Model Reciprocal: – A ratio of two variables exists for y Exponential or Power: -Perform a Stat, Calc #0=ExpReg and a Stat, Calc #A=PwrReg. Which ever model has the highest R2 will be the model you choose. Reciprocal: Exponential: Power:

Examples B A Minutes Bacteria Population 2 15 3 34 4 77 5 173 6 389 7 876 8 1971 Minutes Bacteria Population 2 33 3 104 4 226 5 418 6 690 7 1055 8 1521

A EXPONENTIAL FUNCTION So, Take the LOG of only the y variable and perform a Regression LinReg (L1=x , L3=LOG(y)). A Minutes Bacteria Population 2 15 3 34 4 77 5 173 6 389 7 876 8 1971 2.27 2.26 2.25

POWER FUNCTION So, Take the LOG of Both the x and y variables and perform a Regression LinReg (L3=LOG(x) , L4=LOG(y)). B Minutes Bacteria Population 2 33 3 104 4 226 5 418 6 690 7 1055 8 1521 3.15 2.17 1.44

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Use 2 digits for the year… 1980 = 80

Examine the Residuals of (x,y) Examine the Residuals of (x,y). Then use the Reciprocal Model to find a Regression Line to model the data. Use your model to predict y when x = 10. x y 2 71 4 38 6 26 8 20 12 14 16 10 LinReg(L1, L2) L1 L2 L3 = 1/y 2 71 .01408 4 38 .02632 6 26 .03846 8 20 .05 12 14 .07143 16 10 .1 .125 LinReg(L1, L3) R2 = 68.0 R2 = 99.9 Look at the Residuals y-hat = 16.1 when x = 10

Another way to Analyze Non-Linear Data is to perform both Examples B A Minutes Bacteria Population 2 15 3 34 4 77 5 173 6 389 7 876 8 1971 Minutes Bacteria Population 2 33 3 104 4 226 5 418 6 690 7 1055 8 1521 Another way to Analyze Non-Linear Data is to perform both Logarithmic Transformations on the Data and then compare the R2 and Residuals.

Plan B: Attack of the Logarithms

The Ladder of Powers -1 -1/2 x ½ 1 n Comment Model Name Power Ratios of two quantities (e.g., mph) often benefit from a reciprocal. POWER MODEL The reciprocal of the data -1 An uncommon re-expression, but sometimes useful. Reciprocal square root -1/2 Measurements that cannot be negative often benefit from a log re-expression. Exponential x Counts often benefit from a square root re-expression. Square Root ½ Data with positive and negative values and no bounds are less likely to benefit from re-expression. Linear (Raw Data) 1 Distributions that are skewed to the left. Power (nth) n Comment Model Name Power

Goals of Re-expression Goal 1: Make the distribution of a variable (as seen in its histogram, for example) more symmetric.

Solve each equation for . Warm – Up Solve each equation for . (Flip both sides.) (Perform the Inverse Log function of raising both sides as the exponent of base 10.) Use the Exponent property of xm+n = xm · xn

Solve each equation for . (Perform the Inverse Log function of raising both sides as the exponent of base 10.) Use the Exponent property of xm+n = xm · xn Use the Exponent property of m Log x = Logxm

Chapter 10 Straightening Relationships The relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first:

We can re-express fuel efficiency as gallons per hundred miles (a reciprocal) and eliminate the bend in the original scatterplot: