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IQ and grades Exercise 3 (page 158$)$ included the plot shown below of school grade point average$(\mathrm{GPA})$ against 1$Q$ test score for 78 seventh-grade students. (GPA was recorded on a 12 -point scale with $\mathrm{A}+=12, \mathrm{A}=11, \mathrm{A}-=10, \mathrm{B}+=9, \ldots, \mathrm{D}-=1$ and $\mathrm{F}=0 .$ . Calculation shows that the mean and standard deviation of the 1 $\mathrm{Q}$ scores are $\overline{x}=108.9$ and $s_{x}=13.17 .$ For the GPAs, these values are $\overline{y}=$ 7.447 and $s_{y}=2.10 .$ The correlation between 1 $\mathrm{Q}$ and GPA is $r=0.6337 .$ (a) Find the equation of the least-squares line for predicting GPA from 1$Q$ . Show your work.(b) What percent of the observed variation in these students' GPAs can be explained by the linear relationship between GPA and IQ?(c) One student has an IQ of 103 but a very low GPA of 0.53 . Find and interpret the residual for thisstudent.

A) $\hat{y}=-3.5519+0.1010 x$ B) 40.16$\%$ C) RESIDUAL$=-6.3211$

Intro Stats / AP Statistics

Chapter 3

Describing Relationships

Section 2

Least-Squares Regression

Sampling and Data

Linear Regression and Correlation

Temple University

University of North Carolina at Chapel Hill

Boston College

Lectures

0:00

05:52

The following table contai…

13:11

A study of children's…

02:40

Data Analysis: IQ Scores a…

so the problem gives us that the mean of it was equal to 108.9. The standard deviation of 13.17 the mean of G P A is equal to 7.447 a standard deviation of 2.1. Now, given the correlation is 0.6337 So first we need to find the least squares regression line for predicting GP A Given I keel, the key phrases least squares regression line. This tells us it's going to be a y equals MX plus being now we need to find em and be These are the two equations you should have written down for this. So I am or slow is equal to correlation time standard deviation y over standard deviation of acts and be our interceptor is equal to the mean of y minus slow times the mean of backs. So let's start with finding the swell first. So we have our that's 0.6337 and we have the standard deviation of y and x school were given in the problem which one was wide and which one was acts but you and if we didn't know that we can tell like you is going to be ex, cause that's the given and the one we're trying to solve for his g p A. So that's gonna be the why. So the standard deviation for G p A. Will go on top. That's the two point wine when a bottom will be the 13.1 Savannah and this is 0.101 now for B. We'll take the lean of why so 7.447 My minus the slow, which we just found is point y know, wine times the mean of X which is 108.9 and this is equal to negative three point by by 19 So now we can make our equation. Why is equal to em? So 0.101 packs minus three point by 519 and that is the end of part A. Now, for part B, we're going to calculates r squared. All we have to do is this. Take are from square it R squared is equal to 2.6337 squared, which is point for 016 Normally, you want this written as a percentage so 40.16% now for parts CIA. So a student has an I Q of 100 free and a G p a of 1000.5 free buying the residue on. Explain what it means. This is something else. You should have written down. Residue. U'll is equal to observe minus expected A. So we have our observed value. That's the 0.5 free. We're gonna be looking at the Wye Valley use of the GPS values. So we have a 0.5 free. We just need to find the expected. And the way we'll get are expected. It is by plugging in this 100 free into the equation. We laid it so I'll do it over here. Why is equal 2.1 a wine Times 100 free, minus free 1000.5519 And this is equal to six 0.851 wine. So now we have our observed and expected. So point by free minus six 0.8511 and we now know that the residue u'll is equal to negative 6.3211 So what does residue mean? It means that our equation was about 6.32 off the actual valley Ill. So you would use this to determine how good of an equation this is the and we are done.

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