Question

linear model help

Answer 1

x y

20 399
18 323
5 26
13 170
2 3
15 220
2 5

Answer 2

Use the dataset and its analyses to determine whether a linear model is the best fit. Explain your reasoning.

Answer 3

2.
Find the correlation coefficient using L1 as the x values. (1 point)
• 0.9617
• 0.9807*
• 20.64
• –57.44
3. Find the coefficient of determination. What percent of the variation is explained by the LSRL?
(1 point)
• 4%
• 98%
• 2%
• 96%*

Answer 4

You don't have the equation of the line, but if you want to find the equation of the line of best fit, then:

\(\large\color{black}{ \displaystyle \hat{y}=\left(r\times \frac{S_y}{S_x}\right)x+\left(\bar{y} -r\times \frac{S_y}{S_x}\times \bar{x}\right)}\)

Answer 5

\(\hat{y}\) is the notation for the predicated value of y
(or notation for line of best fit in statistics)

\(S_y\) and \(S_x\) are the deviations in y and x respectively
(Doesn't matter if both are sample or population deviations)

\(\bar{y}\) and \(\bar{x}\) are the mans of y list and x list respectively.

Answer 6

O-O

Answer 7

I think that it is better if you draw a scatter plot using data you provided, in order to guess the law between such data. Furthermore might be necessary drawing some graphs using logarithmic scales

Answer 8

Those are the notations I explained.
Now want to go through finding each of the components of the line of best fit?

Answer 9

yes please, do you want me to make a scatterplot

Answer 10

For your question, I will add that if you had some line:
y=mx+b

which you claimed to be the line of best fit, it must satisfy two conditions to be truly best fit.
[1] Sum of residuals is 0.
[2] Goes through the point \(\left(\bar{x},\bar{y}\right)\)

Answer 11

if I am saying some too complicated stuff then I apologize.

Answer 12

I graphed it and it looked weird, like a wet noodle..

Answer 13

we can draw a scatter plot placing the \(x\) values on a horizontal line, and placing the \(y\) value on a vertical scale

Answer 14

ok

Answer 15

\(y\) values*

Answer 16

please what graph do you get?

Answer 17

let me graph again

Answer 18

ok!

Answer 19

please tag me when you have finished

Answer 20

@Michele_Laino

Answer 21

ok! It looks like that we have a quadratic function, between \(x\) and \(y\), such that \(y=0\) when \(x=0\). So, we can conjecture this function:
\[\Large y = A{x^2}\]
which is not helpful.
nevertheless, if we take the logarithm of both sides, we get:
\[\Large \log y = \log A + 2\log x\]
which represent a linear dependence between \(\log y\) and \(x\)

Answer 22

so, you have to apply the formulas of linear fit, in order to compute the value of the constant \(\log A\). Of course, in order to do such analytical fit, you have to consider the subsequent ordered pairs: \((x,\log y)\)

Answer 23

the logarithm can be taken with respect to the base \(10\)

Answer 24

ok

Answer 25

@Michele_Laino what do i do now, im stuck lol

Answer 26

you have to draw a scatter plot, placing the \(x\) values along a horizontal scale, and placing the \( \log y\) values along a vertical scale

Answer 27

please try, then tag me when you have finished

Answer 28

@mathmate

Answer 29

wait what are the log y values? im sorry

Answer 30

@Michele_Laino

Answer 31

for example, If I consider the first ordered pair:
\((20,399)\)
you have to graph this pair: \((20, \log 399)=(20, 2.6)\)
so your scatter plot has to contains this pair \((20, 2.6)\)
Please do the same with the remaining ordered pairs

Answer 32

ok so i got the points
(20,2.6)
(18,2.5)
(5,1.4)
(13,2.3)
(2,0.5)
(15,2.3)
(2,0.7)

do i put those in a scatterplot

Answer 33

@Michele_Laino

Answer 34

please can you redo your scatter plot using these oredered pairs:
\[\begin{gathered}
\left( {20,2.600} \right) \hfill \\
\left( {18,2.509} \right) \hfill \\
\left( {5,1.415} \right) \hfill \\
\left( {13,2.230} \right) \hfill \\
\left( {2,0.477} \right) \hfill \\
\left( {15,2.342} \right) \hfill \\
\left( {2,0.699} \right) \hfill \\
\end{gathered} \]

Answer 35

ordered*

Answer 36

i got sort of the same graph

Answer 37

ok! Now you have to draw a straight line, which passes in vicinity of the most number of points

Answer 38

ok

Answer 39

thats going to be slightly difficult lol

Answer 40

I try to draw such straight line, please wait...

Answer 41

here is the straight line:

Answer 42

ohh ok.

Answer 43

now we have to search for its intercept, with the vertical axis

Answer 44

ok

Answer 45

please write the requested value:

Answer 46

0.76?

Answer 47

I think that it is greater than 0.76

Answer 48

thats what the green dot is on T-T

Answer 49

oh, im bad at this hold on a tick

Answer 50

0.86?

Answer 51

I think it is \(0.805\), I have used a proportion

Answer 52

so we can write this:
\[\Large \log A = 0.805\]
so, what is \(A=...?\)

Answer 53

is that the 0.805

Answer 54

yes!

Answer 55

it is the vertical intercept of the straight line

Answer 56

so does that mean that the linear model is the best fit?

Answer 57

linear model is the best fit for the logarithmic expression, since we have conjectured this realtion:
\(\huge y=Ax^2\)

Answer 58

and \(\large A=...?\) please what is the value of \(\Large A\) if \(\Large \log A=0.805\)?

Answer 59

0.805? lol

Answer 60

is a and log a the same?

Answer 61

we have to do this computation:
\[\Large \log A = 0.805 \Rightarrow A = {10^{0.805}} \cong 6.38\]

Answer 62

ohh so a=6.38?

Answer 63

correct!! :)

Answer 64

so that means the linear model is the best fit?

Answer 65

we can use the linear model for these variables: \(x\) and \( \log y\)
so we can write:
\[\Large y = 6.38{x^2}\]
nevertheless it is not very precise, since, as you can see the red line doesn't fit all points, for example the first two points

Answer 66

then linear model is the best fit for these variables:

\(\huge (x, \,\log y)\)

Answer 67

in confused again lol im sorry haha

Answer 68

because we used the data in the set, did we find weather a linear model was the best fit?

Answer 69

wait or because x,log y means that we did and its the linear best fit model.. im confused

Answer 70

no problem :)
as we can see, we have applied the best fit, (red line) to these ordered pairs \((x,\,\log y)\)
since we have understood that there is a quadratic relation between \(x\) and \(y\), from the first scatter plot

Answer 71

so that means yes that a linear model is the best fit?

Answer 72

yes! It is the best fit for this function:
\(\log y= \log A +2x\)

Answer 73

Or, in other words, data you provided can not be fitted using a linear model, nevertheless, the subsequent data \((x,\,\log y)\) can be fitted by a linear model