Question

@Mathematics Hours Studying and Test Grades

Time (hours) Grade
1 66
2.5 70
3 81
3.5 92
4 97

Jennifer believes that she can predict her test scores based on how long she studies. She asks 5 of her friends how long they studied and what grade they made on the test and then preforms a regression to model the data. Her data is shown in the table. Which function best models the data?
A)y = 59.576(1.108)x
B)y = 63.461(1.105)x
C)y = 2.833x2 - 3.833x + 67
D)y = 7.733x2 - 24.4x + 82.667

Answer 1

I'm gonna make a video about this, I think my students might benefit from it. Give me like 10 minutes. :)

Answer 2

ok

Answer 3

It is done! :D

Note: I changed the numbers up a little, but it's the same thing.
http://www.youtube.com/watch?v=nNHfu6XeVS4

Download geogebra here: http://www.geogebra.org/cms/en/installers

Hope this helps!

Answer 4

@mathtecher1729 you changing the numbers to the problem I need help on isn't going to benefit me,, but thanks !

Answer 5

You should now be able to do any linear regression problem for any numbers. (including the original ones you posted). :)

Answer 6

@sabrina, what have you learnt about error functions?

The general approach to this problem is the following:

Suppose you have a set of data points \( D = \{ (x_i, y_i) | \ i=1,2, ..., n \} \).

We want to know if a function f where y = f(x) is a good approximation for these data points. The way to measure the "fit" of the function f to the set D is to measure the error between the prediction of the function f on a data point \( x_i \), \( f(x_i) \) and the observed value \( y_i \). The standard error function in this situation is this:
\[ E(f,D) = \sum_1^n (f(x_i) - y_i)^2 \]

You have a number of such functions f. The formal way to answer your question is to measure E(f,D) for each function f. The f which has the minimum value of E is the best fit among the possibilities given.

Answer 7

James,

I'm gonna go out on a limb and say that your explanation might be slightly more than what Sabrinay was bargaining for. I've taught a class where questions similar to hers were asked, and we basically showed students how to use the calc to do the nitty-gritty calculations.

The goal of the lesson was to understand the general idea that "the line of best fit... fits the data best, and since it's a straight line, we can use y = mx + b to represent it, and solve for various values if needed."

:-p

Answer 8

What James is doing is exactly what you need to do to solve this problem..I might even go a step further and look at
\[\sum_{i=1}^n |f(x_i) - y_i|\]

Answer 9

doing a regression does not help since none of the above functions is the 'best' fitting line under either criterion

Answer 10

I'm not familiar with this material at all @JamesJ,, but I have seen heard that the E looking symbol ends with whatever it begins with if i'm not mistaking. I need help doing THIS EXACT problem and not any "nitty gritty tricks," that @mathteacher1729 is trying to display regardless of his teaching experience .

Answer 11

using the lest squares (or abs) one of the functions is better than the others

Answer 12

*least

Answer 13

In other words, using the error function that I have given and/or the one Zarkon suggests, find the error for each function given as a possible answer.

Then determine which has the minimum error.

I'd definitely use Excel or another spreadsheet program if you've got it.

Answer 14

Sabrinay, I was trying to show you how to use your own computer to solve this problem. (inputting your own data points instead of the ones I made up, then finding the line of best fit).

James and Zarkon are trying to give you a college lecture on regression analysis. Interesting, but perhaps not appropriate at this point in time. :)

Excel will also do line of best fit.

Answer 15

and once he finds the 'line' (quadratic/exponential reg) of best fit he will notice that it matches none of the 4 functions he is given ....then what does he do?

Answer 16

Exactly. None of the solutions given is THE best fit regression line. So that being the case, which is the best fit.

And even if one of them were the best fit regression line, it could be that one of the quadratic approximation functions is a better fit.

Now if the question had asked "which of the following is the best fit regression line?", then I think the approach MT is proposing would be the right one.

But given that that is not the question, it appears to be asking for a more general approach.

Answer 17

Thanks everyone,, but I think I'm just going to drop this whole problem because I don't expect any of you to teach or go through working out the whole problem for me to gain an better understanding at this point. I do appreciate your attempts though.

Answer 18

I'll do it, because it's actually a fun exercise and I've haven't done this for a long time.

Answer 19

Ok,, if you think it's an fun exercise and your willing then I would be more than glad for you assist me. Begin when you're ready .

Answer 20

Those charts are separate or intertwined ?

Answer 21

are the first two functions lines or exponential functions...is the x in the exponent?

Answer 22

by the way they are written I'm betting exponential

Answer 23

scrap that. I found an error. Give me a minute.

Answer 24

Good point. Sabriyna: what are the first two functions. I thought they were straight lines, but are they power functions or exponentials? Can you write them out exactly?

Answer 25

For example, the first function, is
\[A. y = 59.576e^{1.108x} \]
or
\[B. y = 59.576x^{1.108} \]
or something else?

Answer 26

I'm thinking \[59.576(1.108)^x\]

Answer 27

got it. Sabriyna, please confirm asap.

Answer 28

\[a\cdot b^x\] is the standard form of an exponential regression

Answer 29

it _has_ been a while

Answer 30

one can do the following '

\[y=ab^x\]
\[ln(y)=ln(a)+x\ln(b)\]

perform a linear regression using the normal equations

\[(A'A)^{-1}A'w\]

Answer 31

Right.

Answer 32

I think I understand linear regression part,, but I'm confused on the exponential regression part and I think that the answer will be rounded to D)

Answer 33

What is the form of the functions A and B. Please write them out again exactly.

Answer 34

And quickly, because I want to move on from this problem, as I'm sure you do too.

Answer 35

Ok. I'm going to assume it is what Zarkon is suggesting. The first function is
\[ 59.576(1.108)^x \]
Likewise for the second function .

Answer 36

That being the case, here's the data and the value of each of the trial functions:

Answer 37

And here's the calculation of the errors:

Answer 38

With both forms of error function, the third function, option C, has the smallest error.

You should replicate all of this analysis and check carefully for errors.

Answer 39

That's what I got.

Answer 40

Good.

Answer 41

If I ever teach linear regression again, I'm going to begin with error functions.

Answer 42

I hope sabrinay25 remembers to give you a medal for all that work you did.