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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).The numbers of federal officials convicted of corruption in the period 1977 to1986 is given in Table $9 .$ The graph is easily visualized as two straight lines:one from 1977 to $1982,$ the other from 1983 through 1987 . Find the lines of best fit for the data as so divided.$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline \text { Year } & \mathbf{1 9 7 7} & \mathbf{1 9 7 8} & \mathbf{1 9 7 9} & \mathbf{1 9 8 0} & \mathbf{1 9 8 1} & \mathbf{1 9 8 2} & \mathbf{1 9 8 3} & \mathbf{1 9 8 4} & \mathbf{1 9 8 5} & \mathbf{1 9 8 6} \\\hline \text { Number } & 94 & 91 & 115 & 131 & 159 & 147 & 424 & 429 & 470 & 523 \\\hline\end{array}$$

Algebra

Chapter 1

Functions and their Applications

Section 8

Regression

Functions

Missouri State University

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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For each of the following …

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Draw a scatter diagram for…

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For each of the data sets,…

Okay, here we are going to be using linear regression to determine the lines of best fit for the set of data here, which describes the number of federal officials who were corrupted or who were convicted of corruption during this time period. You can see that there's a natural break that occurs within this data between 1977 and 1980 to our values range between 90 and 1 60 or so and all of a sudden 1983 that number jumps up to 4 24 and from there remains high, getting all the way up to a maximum of 5 23 in the year 1986 so we can see that one straight line wouldn't necessarily fit this data very well. But it looks as though two straight lines to separate ones for each of these breaks would best explain this relationship. So to start, let's find out the linear regression for the years 1977 through 1982 to capture that first characteristic of the data. To do this, you can use your graphing calculator in putting it into your stack tables your L one and L two for a T. I 84 which is what I'm using, and then we'll go ahead and calculate the linear regression using our graphing calculators. Doing that, we'll find that our first linear regression is why is equal to 13.857 x plus 74 30.333 That would be our line of best fit for that first piece of our data. And if you're curious at all with this one, it did give us an are equal to 0.93 and an R squared equal to 0.86 Now let's do the same thing. But for our second piece of data here we get a line of best fit where y is equal to 33.8 x plus 1 74.2. And that gives us the second line of best fit for the number of officials convicted of corruption between the years 1983 and 1986. And again, just for curiosity, we have an R equal to 0.95 and in R squared, which is equal to 0.9 oh,

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