Single Variable Calculus, Chapter 1, 1.2, Section 1.2, Problem 21

It is tabulated in the report of National Health Interview Survey the ulcer rate per 100 population for various family incomes.


$
\begin{array}{|c|c|}
\hline
&\text{Ulcer rate}\\
\text{Income} & (\text{per 100 population)}\\
\hline
\$4,000 &14.1\\
\$6,000 &13.0\\
\$12,000& 13.4\\
\$16,000 &12.0\\
\$20,000 &12.4\\
\$30,000 &10.5\\
\$45,000 &9.4\\
\$60,000 &8.2\\
\hline
\end{array}
$





a.) Decide whether a linear model is appropriate by making a scatter plot of the given data





Linear model is appropriate because the graph resembles a straight line with negative slope.




b.) Use the first and last data points to find and graph a linear model.


To find the linear model, we will use point slope form
$y = mx + b$
where:
$x$ = income
$y$= ulcer rate

We let the first point as the first data while the second point as the last data.


$
\begin{equation}
\begin{aligned}
y - y_1 &= \displaystyle \frac{y_2-y_1}{x_2 - x_1} (x-x_1)\\
y - 14.1 &= \displaystyle \frac{8.2-14.1}{60000-4000}(x-4000)\\
y - 14.1 &= -0.000105 x + 0.4214\\
y &= -0.000105 x + 14.5214
\end{aligned}
\end{equation}
$




c.) Find and graph the least squares regression line.





d.) Estimate the ulcer rate for an income of \$25,000 by using the linear model in part (c)

Referring to the graph in part(c) the Ulcer Rate for Income of \$25,000 is $\approx$ 12%.


e.) Use the model to predict the ulcer rate of someone with an income of \$80,000


Referring to the graph in part(c) the Ulcer Rate for Income of \$80,000 is $\approx$ 6.05%.



f.) Is it reasonable to apply the model to someone with an income of \$200,000?

No, we only use linear model. 200,000 is very much larger than 60,000, results won't be accurate anymore.