Question

apter 3 8. The average concentration of PCB's in lake trout is shown in the table as a function of age. The EPA has classified PCB's as Year PCB Conc 1 a probable human carcinogen. 2 19 4 4.2 a) Use regression to find a power equation to model the 6 7.2 data. Round the numbers in your equation to 2 decimal 3 118 places. b) Use your equation to estimate the age of a fish PCB Concentration in Lake Trout with a PCB concentration of 15 ppm, rounded to the nearest tenth. c) Use your equation to estimate the age of a fish with a PCB concentration of 6 ppm, rounded to the nearest tenth. 16 12 10 PCB Concentration (ppm) 6 2 2 7 Age

Answer 1

a)

It is asked to make Power Regression. They are of General Form: y = ax^b

We can further Break This Equation as Follows for the Calculation Purpose:

y = ax^b

$\ln y=\ln a+b \ln x$

Since no Specific Method is Specified, So we will use Spreadsheet to calculate this.

We will Enter the Given Data ( i.e. x and y), then we will calculate ln(x) and ln(y).

Then Using Excel, we will find the Coefficients a and b.

Step 1) Enter the Data and Calculate natural Log with the Formula as shown below

А B C 1 X Y 1 2 1 32 1.9 D In(x) =LN(A2) =LN(A3) =LN(A4) =LN(A5) =LN(A6) E In(y) =LN(B2) =LN(B3) =LN(B4) =LN(B5) =LN(B6) 4 4 4.2 5 6 7.2 11.8 6 8

We will get the Result as Follows:

A B с D E 1 X In(x) In(y) |У 1 2 1 0 0 3 2 1.9 4 4 4.2 0.69314718 0.64185389 1.38629436) 1.43508453 1.79175947| 1.97408103 2.07944154 2.46809953 5 LD 7.2 6 00 11.8

Step 2 : Now go to the Data Tab > Analysis Column > Data Analysis and select Regression

? Х OK Cancel Data Analysis Analysis Tools Histogram Moving Average Random Number Generation Rank and Percentile Regression Sampling t-Test: Paired Two Sample for Means t-Test: Two-Sample Assuming Equal variances t-Test: Two-Sample Assuming Unequal Variances z-Test: Two Sample for Means Help

Select the Input and Output Data Range:

Regression ? X Input Input Y Range: OK $E$1:$E$6 MH Cancel Input X Range: $D$1:$D$6 Help Labels Constant is Zero 95 % Confidence Level: $D$90 Output options Output Range: New Worksheet Ply: O New Workbook Residuals Residuals Standardized Residuals Residual Plots Line Fit Plots Normal Probability Normal Probability Plots

We will get the Following Result

SUMMARY OUTPUT Regression Statistics Multiple R0.99488187 R Square 0.98978994 Adjusted Rs 0.98638659 Standard Err 0.1160942 Observation 5 ANOVA df Regression Residual Total SS MS F gnificance F 1 3.919739 3.919739 290.8279 0.000439 3 0.040434 0.013478 4 3.960173 Intercept In(x) Coefficients andard Err t Stat P-value Lower 95%Upper 95%ower 95.09pper 95.0% -0.0907486 0.096865 -0.93686 0.417944 -0.39902 0.217519 -0.39902 0.217519 1.17178307 0.068711 17.05368 0.000439 0.953113 1.390454 0.953113 1.390454

So, the First Row of Coefficients Column is a (-0.09) and the Second Row of Coefficients Column is b (1.17)

Now, we can substitute it back in the regression equation.

ln (y) = ln (-0.09) + 1.17 ln (x)

y = e^{-0.09+ 1.17 ln(x)}

y = e^-0.09(x) ^1.17

$\bold{y = 0.91 (x)^{1.17}}$

b) Given: PCB Concentration = 15

Just Susbtitute y=15 in above equation and solve for x

15 = 0.91 (x)^1.17

x = 10.97 years. $\approx$ 11 years

c) Given: PCB Concentration = 6

Just Susbtitute y=6 in above equation and solve for x

6 = 0.91 (x)^1.17

x = 5.01 years $\approx$ 5 years.

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