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The divorce rate (number of divorces per thousand couples) in the United States for the decade of the 1970 's is given Table 7 . (a) Find and (b) plot the line of best fit and (c) determine the regression coefficient. (d) What would this line predict about the divorce rate in $1987 ?$ See Exercise 11 for a continuation. $$\begin{array}{|c|l|l|l|l|l|l|l|l|l|l|}\hline \text { Year } & \mathbf{1 9 7 0} & \mathbf{1 9 7 1} & \mathbf{1 9 7 2} & \mathbf{1 9 7 3} & \mathbf{1 9 7 4} & \mathbf{1 9 7 5} & \mathbf{1 9 7 6} & \mathbf{1 9 7 7} & \mathbf{1 9 7 8} & \mathbf{1 9 7 9} \\\hline \text { Rate } & 3.5 & 3.7 & 4.1 & 4.4 & 4.6 & 4.9 & 5.0 & 5.0 & 5.2 & 5.4 \\\hline\end{array}$$

a)b)(c) 0.976633(d) 7.2

Algebra

Chapter 1

Functions and their Applications

Section 8

Regression

Functions

Campbell University

McMaster University

University of Michigan - Ann Arbor

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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02:07

The divorce rate for a gi…

01:12

Table 4 shows the marriage…

01:45

The marriage and divorce r…

02:27

Marriage rate. The marriag…

01:32

DIVORCE RATE The divorce r…

01:29

The marriage rate for a gi…

Okay, So here we are taking a look at the data set which is describing the divorce rate during the 19 seventies, the rate being the number of couples to get divorced per 1000 couples. What we are trying to do here is developed the line of best fit. That is the linear regression that best describes the data doing that. We'll be using our graphing calculators. I am using 80 I 84 at the bottom right of your screen. You can see the steps that I've taken that's helpful to you at all. Um, but yeah, let's begin to start. I've simply plotted these points on the graph you see here just a basic scatter plot just to get a feel for what the data looks like when it's graft. But to graph to actually produce this linear regression, I've got into stat edit and put in my data into my l one l two going to Stack Falcon, calculating that linear regression our calculator finds for us that why is equal to 0.2 oh seven x plus 3.44 where X is our year and why being the rate the divorce rate. Um, from this, if we wanted to graph this, we can see that our intercept is 3.44 So that will sit somewhere right in here. And we have a slope of about point 207 so we could sketch a graph. That probably fits the data pretty well. Maybe. Look, something like that. That would be our why function or are linear function that we just found. Now, if we wanted to determine how well this actually fits the data, we want to look at our coefficient of correlation, and that's going to show up directly under your regression results on your calculator, you should have an r squared value and in our value r squared being r coefficient of determination and ours are coefficient of correlation. With that, we can see that are coefficient of determination is 0.95 and R coefficient of correlation 0.97 can add one more decimal place make more accurate than real. Um, this is telling us that around 95% of our of the variation in our y so in the divorce rate is explained by the information included in the status that so that's pretty good. Um, here, if we wanted to say we wanted to predict the divorce rate in year, uh, let's see, say, 1987. So if we wanted to predict that, we need to determine 1987 which year that would fall in our data set. So we started at 1970 one. Oops. Starting at 1971. That was our first year. We had, um, a total of 10 years. Sorry. Uh, so from 1970 to 1979 that was our data set, which was 10 years. So we wanted 1987 minus 1970. It was a 17, but we'll add one. We're at 18. So 18 years later. So this is gonna be the X value that we include in our function We just found. So let's go ahead and plug that in to determine what the divorce rate is predicted to be during the year 1987. Doing that we have y is equal to 0.2 oh, seven times 18 plus 3, +44 Calculating that we can get two Oh, seven times 18 us. Okay, here we go. So we get the predicted divorce rate is equal to 7.166 We can round that we'd like to about 7.2.

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