The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. begin{array}{|c|c|} hline text{Year} & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 hline text{Sales, S} & 898.2 & 1024.1 & 1366.8 & 1447.4 & 1822.1 & 2006.9 & 2268.9 hline end{array} a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t=4 corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property b=e^{ln b} to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011.

Question
Exponential models
asked 2021-01-19
The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&{2004}&{2005}&{2006}&{2007}&{2008}&{2009}&{2010}\backslash{h}{l}\in{e}\text{Sales, S}&{898.2}&{1024.1}&{1366.8}&{1447.4}&{1822.1}&{2006.9}&{2268.9}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with \(\displaystyle{t}={4}\) corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property \(\displaystyle{b}={e}^{{{\ln{\ }}{b}}}\) to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be $2740 million. Do the predictions from the two models agree with this projection? Explain.

Answers (1)

2021-01-20
Step 1 Exponential Regression Model Step 2 a) A scatter plot of the given data is displayed below: image Step 3 b) Using the regression feature, we find that an exponential model for the data in base e is \(\displaystyle{f{{\left({x}\right)}}}={488.964}{e}^{{{0.158}{x}}}\) image Step 4 c) Using the regression feature, A logarithmic model of given data is given by \(\displaystyle{g{{\left({x}\right)}}}=\ -{1333.635}\ +\ {1515.088}\ {\ln{\ }}{\left({x}\right)}\) image Step 5 d) To predict sales in 2011 using the exponential model, find f(11) as follows: \(\displaystyle{f{{\left({11}\right)}}}={488.964}{e}^{{{0.158}{\left({11}\right)}}}\ \approx\ {2768.29}\) To predict sales in 2011 using the logarithmic model, find g(11) as follows: \(\displaystyle{g{{\left({11}\right)}}}=\ -{1333.635}\ +\ {1515.088}\ {\ln{\ }}{\left({11}\right)}\ \approx\ {2299.39}\) The exponential model closely agrees with the projected amount. The exponential model is only off by about 28.29 million while the logarithmic model is off by a significant 440 million.
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Relevant Questions

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Source:
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Case: Dr. Jung’s Diamonds Selection
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9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
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asked 2020-11-06
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asked 2021-02-09
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asked 2021-01-19
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1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs.
2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.
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