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Sometimes when one plots a set of data, it appears to be piece wise linear, causing one to ponder an explanation for the change in pattern at the comer(s).For the United States, the numbers of cyclists (not motorcyclists) killed in accidents with automobiles is given in Table $10 .$ (a) Is there any obvious place where the data breaks into two straight lines? (b) Fit a piecewise linear function to the data. (c) Does this information say anything about the safety of riding a bicycle? $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline \text { Year } & \mathbf{1 9 6 0} & \mathbf{1 9 6 5} & \mathbf{1 9 7 0} & \mathbf{1 9 7 5} & \mathbf{1 9 8 0} & \mathbf{1 9 8 2} & \mathbf{1 9 8 4} & \mathbf{1 9 8 5} \\\hline \text { Number } & 460 & 680 & 780 & 1000 & 1200 & 1100 & 1100 & 1100 \\\hline\end{array}$$

(a) It changes after $1980(x=21)$(b) $f(x)=\left\{\begin{array}{cl}36 x+428 & \text { if } x \leq 21 \\ 1100 & \text { if } x \geq 23\end{array}\right.$(c) Riding became safer (perhaps because of helmet usage, or some other rea-son.

Algebra

Chapter 1

Functions and their Applications

Section 8

Regression

Functions

Harvey Mudd College

Baylor University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Sometimes when one plots a…

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A data set consists of ten…

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Consider the following tab…

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A data set consists of nin…

Okay, So here we're going to be combining linear regression with piecewise functions to determine the two functions. The two linear functions which best describe the data we have here, which is a data set which is explaining the number of bicycle accidents that resulted in death by automobile between the years 1960 and 1985. We first want to take a look at where any break might occur in this data. If you were to graph a scatter plot of this, you would get a decent idea of it. Or we could just look at the table we have here. You can see that from 1960 to 19 80 the number of deaths is increasing. And then we get to 1982. It drops down to 1100 where it remains through 1985. So it looks as though we have a break that occurs at between 1982 1980. I'm sorry. At 1980 that appears to be where that break is. We have 1980 then we go to 1982 and all of a sudden our numbers shift from 1982 onward. We have a constant number of deaths at 1100 whereas prior to that we were constantly increasing the number of deaths by year. To develop a piecewise function of this, we simply take these two breaks and find two linear equations, one for each. So in putting into your graphing calculator, you can do the data up and through the year 1980 finding a linear regression for it. That would give you a linear regression of why is equal to 36 x plus 4 28 and then if you did the same thing. But for 1982 onward, you would get that y is equal to 1100. To form the piecewise function of this. We want to make sure we have it in the correct notation we can do where we have FX being equal to a function of 36 X plus 4 28. And that is if X, which is our year, is less than or equal to 21 and our next one is F of X is equal to 1100 if our year is greater than or equal to 23 now. If we were looking to interpret this, which we often are when it comes to economic data. Let's take a look at what the data might tell us about bicycle safety. So we're looking at the number of deaths to remind you, and we see that it's constantly increasing our first function of our piecewise function that we have here because our coefficient 36 is positive right here. That's telling us that as each additional year passes, the number of deaths is likely to increase by 36 more until we get to 1982 and we remain at 1100. So it appears as a bicycle safety may be getting better. Say it's becoming safer to ride a bike. This could be, you know, say helmet usage was it becomes more encouraged during this time. The cultural shift potentially between you know what you should do and shouldn't do when you're using a bicycle or automobile safety. Maybe crosswalks were implemented more seriously. We don't necessarily know the exact reason based upon the data, but we could likely conclude that it got safer to ride a bike somewhere in there. And that is why our numbers stopped increasing and instead remained at a constant year after year

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