# Use exponential regression to find a function that models the data. begin{array}{|c|c|} hline x & 1 & 2 & 3 & 4 & 5 hline f(x) & 14 & 7.1 & 3.4 & 1.8 & 0.8 hline end{array} Question
Exponential models Use exponential regression to find a function that models the data. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{1}&{2}&{3}&{4}&{5}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{14}&{7.1}&{3.4}&{1.8}&{0.8}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ 2020-11-12
Using a graphing calculator, the exponential model for the given table is: $$\displaystyle{f{{\left({x}\right)}}}\ \approx\ {28.98}{\left({0.49}\right)}^{{{x}}}$$ ExpReg $$\displaystyle{y}={a}\ \cdot\ {b}^{{{x}}}$$
$$\displaystyle{a}={28.97771528}$$
$$\displaystyle{b}={0.4918059809}$$
$$\displaystyle{r}^{{{2}}}={0.9988155367}$$
$$\displaystyle{r}=\ -{0.9994075929}$$

### Relevant Questions For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{1}&{2}&{3}&{4}&{5}&{6}&{7}&{8}&{9}&{10}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{409.4}&{260.7}&{170.4}&{110.6}&{74}&{44.7}&{32.4}&{19.5}&{12.7}&{8.1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ 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. The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.
Source:
Centers for Disease Control and Prevention.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{amp, Cases}\backslash{h}{l}\in{e}{1995}&{a}\mp,\ {295}\backslash{h}{l}\in{e}{1997}&{a}\mp,\ {166}\backslash{h}{l}\in{e}{1999}&{a}\mp,\ {109}\backslash{h}{l}\in{e}{2001}&{a}\mp,\ {115}\backslash{h}{l}\in{e}{2003}&{a}\mp,\ {94}\backslash{h}{l}\in{e}{2005}&{a}\mp,\ {107}\backslash{h}{l}\in{e}{2007}&{a}\mp,\ {79}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
a) Plot the data on a graphing calculator, letting $$\displaystyle{t}={0}$$ correspond to the year 1995.
b) Using the regression feature on your calculator, find a quadratic, a cubic, and an exponential function that models this data.
c) Plot the three functions with the data on the same coordinate axes. Which function or functions best capture the behavior of the data over the years plotted?
d) Find the number of cases predicted by all three functions for 20152015. Which of these are realistic? Explain. The table gives the number of active Twitter users worldwide, semiannually from 2010 to 2016. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Years since}&\text{January 1, 2010}&\text{Twitter user}&\text{(millions)}\backslash{h}{l}\in{e}{0}&{30}&{3.5}&{232}\backslash{h}{l}\in{e}{0.5}&{49}&{4.0}&{255}\backslash{h}{l}\in{e}{1.0}&{68}&{4.5}&{284}\backslash{h}{l}\in{e}{1.5}&{101}&{5.0}&{302}\backslash{h}{l}\in{e}{2.0}&{138}&{5.5}&{307}\backslash{h}{l}\in{e}{2.5}&{167}&{6.0}&{310}\backslash{h}{l}\in{e}{3.0}&{204}&{6.5}&{317}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Use a calculator or computer to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form $$\displaystyle\hat{{{y}}}={a}\ \cdot\ {x}^{{{p}}}$$. Here are data on the eight planets of our solar system. Distance from the sun is measured in astronomical units (AU), the average distance Earth is from the sun. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\ \text{ Planet }\ &\ \text{ Distance from sun }\ &\text{(astronomical units) }\ &\ \text{ Period of revolution }\ &\text{(Earth years) }\ \backslash{h}{l}\in{e}\ \text{ Mercury }\ &{0.387}&{0.241}\backslash{h}{l}\in{e}\ \text{ Venus }\ &{0.723}&{0.615}\backslash{h}{l}\in{e}\ \text{ Earth }\ &{1.000}&{1.000}\backslash{h}{l}\in{e}\ \text{ Mars }\ &{1.524}&{1.881}\backslash{h}{l}\in{e}\ \text{ Jupiter }\ &{5.203}&{11.862}\backslash{h}{l}\in{e}\ \text{ Saturn }\ &{9.539}&{29.456}\backslash{h}{l}\in{e}\ \text{ Uranus }\ &{19.191}&{84.070}\backslash{h}{l}\in{e}\ \text{ Neptune }\ &{30.061}&{164.810}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Calculate and interpret the residual for Neptune. The following table shows the approximate average household income in the United States in 1990, 1995, and 2003. ($$\displaystyle{t}={0}$$ represents 1990.)
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{t(Year)}&{0}&{5}&{13}\backslash{h}{l}\in{e}\text{H(Household Income in}\ \{1},{000}{)}&{30}&{35}&{43}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Which of the following kinds of models would best fit the given data?
Explain your choice of model. ( a, b, c, and m are constants.)
a) Linear: $$\displaystyle{H}{\left({t}\right)}={m}{b}\ +\ {b}$$
b) Quadratic: $$\displaystyle{H}{\left({t}\right)}={a}{t}^{{{2}}}\ +\ {b}{t}\ +\ {c}$$
c) Exponential: $$\displaystyle{H}{\left({t}\right)}={A}{b}^{{{t}}}$$ The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{Population}\backslash{h}{l}\in{e}{1960}&{94.092}\backslash{h}{l}\in{e}{1965}&{98.883}\backslash{h}{l}\in{e}{1970}&{104.345}\backslash{h}{l}\in{e}{1975}&{111.573}\backslash{h}{l}\in{e}{1980}&{116.807}\backslash{h}{l}\in{e}{1985}&{120.754}\backslash{h}{l}\in{e}{1990}&{123.537}\backslash{h}{l}\in{e}{1995}&{125.327}\backslash{h}{l}\in{e}{2000}&{126.776}\backslash{h}{l}\in{e}{2005}&{127.715}\backslash{h}{l}\in{e}{2010}&{127.579}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose $$\displaystyle{t}={0}$$ to correspond to 1960 or 1980.] The table gives the midyear population of Norway, in thousands, from 1960 to 2010.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Year}&\text{Population}\backslash{h}{l}\in{e}{1960}&{3581}\backslash{h}{l}\in{e}{1965}&{3723}\backslash{h}{l}\in{e}{1970}&{3877}\backslash{h}{l}\in{e}{1975}&{4007}\backslash{h}{l}\in{e}{1980}&{4086}\backslash{h}{l}\in{e}{1985}&{4152}\backslash{h}{l}\in{e}{1990}&{4242}\backslash{h}{l}\in{e}{1995}&{4359}\backslash{h}{l}\in{e}{2000}&{4492}\backslash{h}{l}\in{e}{2005}&{4625}\backslash{h}{l}\in{e}{2010}&{4891}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 3500 from each of the population figures. Then, after obtaining a model from your calculator, add 3500 to get your final model. It might be helpful to choose $$\displaystyle{t}={0}$$ to correspond to 1960.] Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided $$\displaystyle{95}\%$$ confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance. $$\begin{array}{|l|l|l|}\hline t(\text{in years})&m(\text{amoun of radioactive material})\\\hline0&\\\hline5730\\\hline11460\\\hline17190\\\hline\end{array}$$