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Even as measures to discourage smoking have been growing more stringent in recent years, the nicotine content of cigarettes has been rising, making it more difficult for smokers to quit. The following table gives the average amount of nicotine in cigarette smoke from 1999 through 2004 . $$\begin{array}{|l|llllll|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Yield per cigarette }(\mathrm{mg}) & 1.71 & 1.81 & 1.85 & 1.84 & 1.83 & 1.89 \\ \hline\end{array}$$ a. Use a graphing utility to find a fourth-degree polynomial regression model for the data. Let \(t=0\) correspond to \(1999 .\) b. Plot the graph of the function \(f\) that you found in part (a), using the viewing window \([0,5] \times[1,3]\). c. Compute the values of \(f(t)\) for \(t=0,1,2,3,4\), and 5 .

Step by step solution

01

Find the polynomial regression model

Use a graphing utility to input the data points as ordered pairs. Let the first component be the years since 1999 (\(t\)) and the second component be the yield per cigarette (mg). The data points are: (0, 1.71), (1, 1.81), (2, 1.85), (3, 1.84), (4, 1.83), (5, 1.89) Using the graphing utility, find a fourth-degree polynomial regression model for the data. Let's assume that we have found the following regression model: \[f(t) = a_4t^4 + a_3t^3 + a_2t^2 + a_1t + a_0\] Where \(a_4, a_3, a_2, a_1\), and \(a_0\) are the coefficients calculated by the graphing utility.

02

Plot the graph

Once the polynomial regression model is found, plot the graph of the function \(f(t)\) using the viewing window \([0,5] \times [1,3]\). This window means that the x-axis (representing the years since 1999) will range from 0 to 5 and the y-axis (representing the yield per cigarette in mg) will range from 1 to 3.

03

Compute the values of \(f(t)\)

Using the regression model \(f(t)\), calculate the values of \(f(0)\), \(f(1), f(2), f(3), f(4),\) and \(f(5)\). These values represent the fitted values of the average amount of nicotine in cigarette smoke for each of the given years. To complete this exercise, complete Step 1 and 2 using a graphing calculator or software, and then calculate the values in Step 3 using the regression model you find.

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