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**AP Statistics Section 4.1 B Power Law Models**

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Exponential growth occurs when a variable is multiplied by a fixed number in each equal time period. In other words, exponential growth increases by a fixed percent of the previous total in each equal time period.

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**Example 2: You won a lottery**

Example 2: You won a lottery. You have a choice of (a) getting $100,000 each day for the month of November or (b) getting $0.01 on November 1st, $0.02 on November 2nd, $0.04 on November 3rd, etc until November 30th. What is your choice?

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**For choice (a), at the end of the month you would have**

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**For choice (b), you would have**

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**This example illustrates the power of exponential growth.**

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**Example 3: Here is the number of bacteria present after a given number of hours. Graph the data.**

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**The curve is increasing exponentially**

The curve is increasing exponentially. How can we tell if an exponential model is appropriate? For equal increments in x, calculate the ratio of a y-value divided by the previous y-value. All such ratios should be equal for a perfectly exponential function.

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**Exponential growth can be modeled by the equation ________, where a and b are constants.**

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**To help us “straighten” this model we will use a logarithm transformation.**

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**Take the ln of both sides**

Take the ln of both sides. The ln of a product equals the sum of the ln’s . The ln of a power equals the power times the ln. Notice that this form is the equation of a line.

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**For a logarithmic transformation we will take the log of the y-values and leave the x-values alone.**

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**Graph hours (x) in L1 vs. ln y in L3.**

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Calculate the LSL on the transformed points and construct a residual plot to verify the validity of our model.

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**Determine r2 and interpret it in the context of the problem.**

99.9% of the variation in the natural log of the number of bacteria is accounted for by the linear relationship with the number of hours

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**Interpret the slope and y-intercept in the context of the problem.**

The slope tells us that an increase in one hour will cause an increase of in the natural log of the number of bacteria. The y-intercept tell us that at hour 0, the natural log of the number of bacteria will be -.0047

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**Solve the LSL for and predict how many bacteria will be present at 3**

Solve the LSL for and predict how many bacteria will be present at 3.75 hours.

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**Example 4: In 1965, Gordon Moore, a founder of Intel Corp**

Example 4: In 1965, Gordon Moore, a founder of Intel Corp., predicted that the number of transistors in an integrated circuit chip would double every 18 months (i.e exponential growth). Here is some data from a stats package that was run on the data of the number of transistors versus time since The data was straightened out by using a logarithmic transformation on the number of transistors. x = # of years since 1970, y = ln (transistors). We use “years since” so that the values are smaller and don’t create overflow problems with the calculator.

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**Predictor Coef R-Sq = 99.5% Constant 7.4078 Yrs since1970 0.3316**

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Write the LSL.

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Interpret in context. 99.5% of the variation in ln(#transistors) is accounted for by the linear relationship with the # of years since 1970.

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**Predict the number of transistors in the year 2003.**

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