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Modeling Data An instructor gives regular 20 -point quizzes and 100 -point exams in a mathematics course. Average scores for six students, given as ordered pairs $(x, y),$ where $x$ is the average quiz score and $y$ is the average exam score, are $(18,87),(10,55),(19,96),(16,79),(13,76),$ and $(15,82) .$(a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data.(b) Use a graphing utility to plot the points and graph the regression line in the same viewing window.(c) Use the regression line to predict the average exam score for a student with an average quiz score of $17 .$(d) Interpret the meaning of the slope of the regression line.(e) The instructor adds 4 points to the average exam score of everyone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line.

$\begin{aligned} y &=4(7)+19 \\ &=68+17 \\ &=87 \end{aligned}$

Calculus 1 / AB

Chapter 0

Preparation for Calculus

Section 2

Linear Models and Rates of Change

Functions

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculus is the study of functions (as opposed to calculus, which is the study of change, and algebra, which is the study of operations and their application to solving equations). It is generally considered to be a part of mathematics that prepares students for calculus.

31:55

In mathematics, a function (or map) f from a set X to a set Y is a rule which assigns to each element x of X a unique element y of Y, the value of f at x, such that the following conditions are met: 1) For every x in X there is exactly one y in Y, the value of f at x; 2) If x and y are in X, then f(x) = y; 3) If x and y are in X, then f(x) = f(y) implies x = y; 4) For every x in X, there exists a y in Y such that f(x) = y.

01:50

Modeling Data An instructo…

01:16

An instructor gives regula…

04:59

DATA ANALYSIS: AVERAGE SC…

05:18

So problem 76 is about modeling data. Um, this case we're looking at his relationship between, uh, quiz scores and exam scores. Okay. And it asked you to use your, um, graphing utility in order. Thio do a graph step lot of it. The tea I 84. It means, uh, one of them that they have a sense of the l one. Get him the hell too. When you do your couch calculation for it, you're gonna d'oh Lynn rag. Meaning a linear regression. And when you do that, um, that would give you something in the form of X plus B between MX plus B. Um, you get the value A equals 3.97 approximately, and you get B equals, um, 18.9. So that tells you is that the equation? That of the line that best fits this data is about why equals or ex plus 19 from around, Mister. Now, mister bore around this to 19. Okay. Um, so we use the regression capabilities, we get the regression line, and when we grab that regression line, it's a little bit on the high side. We're here. Something like that. Um, So we wanted, then use this line. So this is a regression line. So this is that What they say is it's the least squares. What stories is Find the distance between each of these points and the line, the vertical distance, and it's squaring those distances. And it is finding out from that when the minimum the line that has the minimum distance to all these points. Right? Okay, so that's what the calculators doing for you. So when you grab this line, this is the line that comes closest to all the points. Okay? And so basically, what we want to do right now is to find out, used this mine in order. Find out, um, what you would get the exam score for student Who's quiz scores 17. Remember, X is the quest for and why is the exams OK? So basically, what we want to do is the exam scores been equal four times whenever the quiz for is which in this case 17. What's 19? So that's 68 1st 19 seven, the predicted exam score. If we have scored 17 which now makes sense, it's a little bit. It's around the same things you get for 18. Close around it up. 19. Wilmore 16 last. That's so that's a reasonable prediction for, um, Let me see. So, um, the other question is, What's that? What does this slope mean? This or Okay, so that is the rate of change, right? So that's how much the exam score changes based on how much the quiz score changes. All right, so that means that the exam scores are four times as high as the quis corpse. Um, finally, important. E um, it says what? Um what a big instructor added four points to everyone's score. Basically, what that would do would be to take all these points we had before and move them all up work. Okay, so what would happen is this rational? I would also move up for it had the same slow because the rate of change between the two lines is still the same rate change between them is the same. But the difference being all of these points or four times its size, so this 19 would turn into a 23 visits for once inside with the slope would remain unchanged, right

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