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Problem 23 Hard Difficulty

The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey.

(a) Make a scatter plot of these data and decide whether a linear model is appropriate.
(b) Find and graph a linear model using the first and last data points.
(c) Find and graph the least squares regression line.
(d) Use the linear model in part (c) to estimate the ulcer rate for an income of \$25,000 dollars.
(e) According to the model, how likely is someone with an income of \$80,000 to suffer from peptic ulcers?
(f) Do you think it would be reasonable to apply the model to someone with an income of \$200,000?

Answer

a. A linear model does seem appropriate.
(b) Using the points (4000,14.1) and $(60,000,8.2),$ we obtain
\[
\begin{array}{l}
y-14.1=\frac{8.2-14.1}{60,000-4000}(x-4000) \text { or, equivalently } \\
y \approx-0.000105357 x+14.521429
\end{array}
\].
(c) Using a computing device, we obtain the least squares regression line $y=-0.0000997855 x+13.950764$
The following commands and screens illustrate how to find the least squares regression line on a $\mathrm{TI}-84$ Plus.
(d) When $x=25,000, y \approx 11.456 ;$ or about 11.5 per 100 population.
(e) When $x=80,000, y \approx 5.968 ;$ or about a $6 \%$ chance.
(f) When $x=200,000, y$ is negative, so the model does not apply.

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Video Transcript

in this problem, we're going to make a scatter plot and do some linear regression for the data about peptic ulcer rates, and this is a great problem to do on a graphing calculator. So the first thing we want to do is go into the stat menu and go into the edit function and type in all of our data into list one endless, too. Once we have the data typed in, we can make the scatter plot so we go to second function y equals to take us to the stat plot menu. We go into the menu for plot one, turn it on and make sure it is a scatter plot. And it's using list one endless, too those air, the default options. But just in case you have made changes previously with your calculator, you want to change them back to these. All right now, the calculator can select a good viewing window for us for this set of data. If we go to zoom and then down to number nine zoom staff. So now we have a look at our scatter plot. Do we think that a linear model is appropriate? I would say it looks like these points are going down in a fairly linear fashion, so I think so. So the next thing we want to do is part B. We want to use the first and last data points to find a linear model, and then we'll graph it. So here we have the first and last data points and what we want to do is use those points to find the slope. And right now, I have expressed that as an exact answer, but at some point will probably want to approximate that. And then next we can use point slope form using our slope and one of our points. It doesn't matter which one. I chose to use the first point as my Y one and my wife and my ex one. And then we can simplify that equation. And now we have it in slope intercept form. And at this point, I have put in some approximate values for the slope in the Y intercept. And we're going to graft that on the calculator and see how it looks compared to the scatter plot. So back to the calculator, go into the y equals menu and weaken type in that function. Negative 0.105 x plus 13.7 All right, well, press graph, and we see that it's a fairly decent line going through the points, although there are lots of points above the line and none below. So it's not the best. We could get a better linear model if we use the least squares regression line, which is what we're going to do for part C. The least squares regression line takes into account all of the points, not just two of the points. So how we can get this is we go back into the stat menu and now we go over to calculate, and we go down to choice number four linear regression. We press enter, and we're going to use list one and list, too. We don't need to worry about frequency list. We do want to graft this. So we're going to store this regression equation in why one? And the way to find that is to go to variables the A. R s over to why variables choose function and choose why one. Okay, now we can calculate, and here we get the values for the least squares regression line. And if we press y equals, we see that that was pasted into our Y one menu are y equals menu. Now the compress graph. And there we see the least squares regression line, and we see that it is a better fit than the linear model we made by just using two points. Okay, now we're onto Part D, and we're gonna estimate the also rate for an income of $25,000. So what we can do is we can go into our table set menu, and we can make sure that our table set is on independent ask, and this allows us to type in whatever value we want for X, and it will calculate the Y value for us. We could instead use the graph. But I think for parts D, E and F, to be most efficient will use the table. So now we go into the table, which is second graf, and we can type whatever number we want. Don't worry. If they're already numbers there from some previous use of your calculator, you can just type right over these numbers. So $25,000 for the income gives us a peptic ulcer rate of 11.456 and then for part E $80,000 income. Type that into the table a swell, and we get the rate of 5.9679 per 100 people. And then finally, for part F do we think it would be reasonable to apply the model to someone with an income of $200,000? Well, it's possible that as this income goes up and the rate goes down, we could end up in the negative area, and so that wouldn't make sense. Let's see what happens if we type in $200,000. Yes, that's exactly what happened. So it doesn't make sense to extend the model for this value because it's not sensible to have a negative ulcer rate that doesn't make any sense.