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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).since 1979 the divorce rate statistics (see Exercise 5 ) are given in Table 11 .(a) Find the line of best fit for this data (label 1980 year 11 so as to continue with Exercise 5). (b) Combine this with the result of Exercise 5 to obtain a piecewise linear function that describes the American divorce rate over the eighteen year period. (c) Does this indicate any significant change in people's behavior?$$\begin{array}{|l|l|l|l|l|l|l|}\hline \text { Year } & \mathbf{1 9 8 0} & \mathbf{1 9 8 1} &\mathbf{1 9 8 2} & \mathbf{1 9 8 3} & \mathbf{1 9 8 4} & \mathbf{1 9 8 5} & \mathbf{1 9 8 6} & \mathbf{1 9 8 7} \\\hline \text { Rate } & 5.2 & 5.3 & 5.1 & 5.0 & 4.9 & 5.0 & 4.8 & 4.8 \\\hline\end{array}$$

a)(b) $f(x)=\left\{\begin{array}{cc}0.207 x+3.44 & x \leq 10 \\ -0.067857 x+5.996 & x \geq 11\end{array}\right.$(c) decreased.

Algebra

Chapter 1

Functions and their Applications

Section 8

Regression

Functions

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University of Michigan - Ann Arbor

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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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02:53

Sometimes when one plots a…

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The divorce rate (number o…

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Ages of couples again Has …

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The scatter plot below sug…

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Consider the values for th…

Alright, here we are going to be using linear regression and our knowledge of piecewise functions to form piecewise function that best describes the data here. This data is giving us information on the divorce rate. During between the period of 1980 and 1987 were also given that between the years of 1970 1979 we had a function describing that portion of the data equal 2.2 oh seven x plus 3.44 That would be for the 1st 10 years of the data collection. Now we're given another set of data and we need to first determine its line of best fit. To do that, you can use your graphing calculator plugging the information in you'll get out. A linear regression to be Y is equal to negative 0.67 x plus 5.99 And this year let's keep in mind as well that this is for the years of 1980 through 1987. Alright, which gives us in the terms of us actually typing in the data 1980 is our year 11. Okay, so this would be our line of best fit for the data that was just now given to us. But now we want to form a function that describes this set of data and whatever data they had previously given us to give us the regression result of y equals 0.2 oh seven x plus 3.44 To do this, we know we have f of X, which is equal to a function of our first one, where we have a 0.2 oh seven x plus 3.44 and that one stands if X, which is a number of years, is less than or equal to 10. We know that because it was collected between the years one and 10, so it must include your tent and anything below it. Our second piece is negative 0.67 x plus 5.99 and that one would be if X is that greater than or equal to 11 because it occurs. It begins at 1980 extents in 1987. So anything greater than or equal to EUR 11 this year would be our piecewise function, describing the entire set of data and what we can interpret from this is that people's behaviour changed a little bit. You can see based on our coefficients. So within our first one, we have 10.207 So that's telling us that as each year goes by, our divorce rate is increasing by 0.2 of seven. But that only is between the years of 1970 1979. Once we get to 1980 through 87 r coefficient changes from positive to negative, indicating that the divorce rate went from increasing to decreasing. So it looks as though divorce rates are decreasing and that would be the interpretation that we can get from people's behavior.

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