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HW- pg. 276 (4.5) 4.1 & 4.2 Quiz TUESDAY 10-22-13 www.westex.org HS, Teacher Websites 10-17-13 Warm up—AP Stats The following table shows monthly premiums for a 10-year term life insurance policy worth $1,000,000: Age 40 45 50 55 60 65 Monthly Premium $29 $46 $68 $106 $157 $257 Use your calc. to turn each fraction into a decimal rounded to the nearest hundredth. 46 68 106 157 257 = ____ ____ ____ ____ ____ 29 46 157 68 106 Notice anything? Name _________________________ AP Stats 4 More about Relationships between Two Variables 4.1 Transforming to Achieve Linearity Day 2 Date _______ Objectives Explain how linear growth differs from exponential growth. Use a logarithmic transformation to linearize a data set that can be modeled by an exponential model. Exponential Growth In linear growth, a fixed increment is __________ to the variable in each equal time period. _______________ growth occurs when a variable is ____________ by a fixed number in each equal time period. It is typical that with exponential growth the increase appears slow for a long period and then seem to explode. Linear growth differs from exponential growth in that successive terms are related by ____________ in the linear case and _______________ in the _______________ case. Linear growth – increase by a fixed __________. Exponential growth – increase by a fixed __________. If an exponential model of the form y = a·bx describes the relationship between x and y, we can use ______________ to _______________ the data to produce a linear relationship. Algebraic Properties of Logarithms logb x = y Rules 1. 2. 3. if and only if (iff) by = x for logs: logb (MN) = _________________________ logb (M/N) = _________________________ logb Xp = _________________________ These properties hold for all positive values of base b except b = 1. Base 10 or base e (2.71828…) are used most frequently. Log10 x = ______ loge x = ______ Example 4.5 Moore’s law and computer chips-Exponential growth Gordon Moore, one of the founders of Intel Corporation, predicted in 1965 that the number of transistors on an integrated circuit chip would double every 18 months. This is “Moore’s law,” one way to measure the revolution in computing. Here are the data on the dates and number of transistors for Intel microprocessors: Processor Date Transistors Processor Date Transistors 4004 1971 2250 486DX 1989 1,180,000 8008 1972 2,500 Pentium 1993 3,100,000 8080 1974 5,000 Pentium II 1997 7,500,000 8086 1978 29,000 Pentium III 1999 24,000,000 286 1982 120,000 Pentium 4 2000 42,000,000 386 1985 275,000 USE YEARS SINCE 1970 for x values Find LSRL. Write equation ___________________________________________________ Define variables. ________________________________________________ r = ______ r2 = ______ Interpret r2 in context of problem. ___________________________________ _______________________________________________________________________ Do you think an exponential model might describe the relationship between years since 1970 and # of transistors? y = abx is the basic exponential growth model. We want this equation to be _______________ so it will look linear. To transform an exponential model we take the log of both sides of the equation. We will do this ONCE to see that taking the log of both sides of the exponential equation linearizes the data. y = abx log y = log(abx) = __________________ = __________________ Notice that the y intercept is __________ and the slope is __________ so we have a straight line! After this ONE time we will just remember that when we are using an exponential model the result is you take the log of just the _____ value, but not the _____ value. *For a power model you take the log of y & the log of x to linearize.* (will see this tomorrow) Transform the data by taking the log of y (Transistors). Apply least-squares regression to the transformed data. Find LSRL. Write equation ___________________________________________________ Define variables. ________________________________________________ r = ______ r2 = ______ Interpret r2 in context of problem. ___________________________________ _______________________________________________________________________ To do prediction, we need to _______ the logarithm transformation to return to the original units of measurement. (Take __________) Predict transistors for Itanium 2 in 2003 ________________________________________