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Table 5 shows the gross national product (GNP) of the United States in billions of dollars for the years 1978 through 1983 . (a) Find the line of best fit.(b) Plot the line. (c) From the equation of the line, predict the gross national product in 1987 . (The actual figure was 4488.5 .)

a)b)c) $(\mathrm{c}) 4267.7$

Algebra

Chapter 1

Functions and their Applications

Section 8

Regression

Functions

Missouri State University

Campbell University

McMaster 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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05:54

Okay, so here we are, working with funding lines of best fit using regression methods. There are definitely very complicated ways of doing this by hand. However, the easiest and best way to ensure that it's done accurately is to do it using your graphing calculator here. Other programs such as Excel stay to use. All these sorts of data analysis programs can certainly do it too. But with a simple, linear regression that we're working with now your graphing calculator can handle it just fine. I am using a T I 84 plus just for reference because as I go, I'll start referencing the buttons that I press and the methods that I used to get to the results. So to start, we have this data set right here showing gross national product by here. Um, we're gonna start by inputting this data into our stat edit table in your graphic calculator. So I do this by hitting my staff button and then number one, which is edit entering into your l one and L two columns. Start with L one, and in this case, instead of doing it by year, we're just gonna do it as 1234561 being the first year in our data set. So you can see at the top of the table. I've listed this one through 6, 1978 being one so filling your l one out through six and then l two is going to be your g r g n p values. You can go ahead and start inputting those into your l two all the way down, obviously to 3000 311. After you've done that, you can hit your stat button again, scroll over to kill Kirk and scroll down, then to your fourth option, which is the linear regression option scroll down to calculate it will produce it. The values for a Y equals a X plus B equation. In this case, it gives us the values of why is equal to for a we have 234.57 Obviously the X remains plus it gives us an intercept value of 1927.133 and this would be our line of best fit this situation now, I determined to do a linear regression because I first graft all of this. I did a scatter plot, and it looked like it followed a relatively linear regression. So this here is the linear regression or the line of best fit for this particular data set. If we wanted to determine or predict, say, the gross national product in a year like 1987 we can certainly do that. This would be a year 10 if it were included in our dataset, so we just need to plug 10 in for X. Doing that, we have to 34 point oh, 57 times 10 plus 1927.133 In doing that, we get a Y value equal to 4267.70 That would be the predicted gross national product for the year 1987. Uh, just in case you're curious, the actual figure during that year was 404,488 0.5. You can see we're actually not too far off. Our model did a relatively decent job of predicting it, but determining the true fit of the model that is beyond this particular topic. Yeah,

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