ap statistics section 4.1 b power law models
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AP Statistics Section 4.1 B Power Law Models. 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. - PowerPoint PPT PresentationTRANSCRIPT
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AP Statistics Section 4.1 BPower 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. 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
000,000,3$10000030
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For choice (b), you would have
trp )1( 30)11(01.
24.418,737,10$
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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. 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.
39.13.1
4.3 29.1
2.4
3.1 3.1
8.1
4.2
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Exponential growth can be modeled by the equation
________, where a and b are constants.
xbay
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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.
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.
)ln(ln xbay
)lnlnln xbay
)lnlnln bxay
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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 ourmodel.
)(#5860.0047.)ln(# hoursbacteria
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Determine r2 and interpret it in the context of the problem.
999.2 r
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 .5860 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 3.75 hours.
y
xy 5860.0047.ˆln
xey 5860.0047.ˆ
xeey 5860.0047.ˆ
96.8ˆ y
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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 1970. 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 7.4078Yrs since1970 0.3316
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Write the LSL.
1970) since years(3316.4078.7s)transistorln(#
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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.
2r
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Predict the number of transistors in the year 2003.
)33(3316.4078.7ˆ ey
78.853,231,93ˆ y