let x represents the number of hours studying...

eddy2017

eddy2017

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hi, i am asking the forum tutors to help me understand quickly how to set up an answer to questions like the one i am posting here.
a student recorded her grades on assessments in respect to the number of hours she studied for each assessment. which of the following equations is the linear equation of the function representing the data below. . let x represents the number of hours studying and y represent the assessment grade.
the answer given( without any explanation, of course) y=10x+50

i want to know how to get to this solution and understand the process.
thanks in advance for your help and for the forum.

Hours studying Assessment grade
1 --------------------60
1.5-------------------65
2.5-------------------75
3.5-------------------80
4.5 -------------------95
Ok, this is what i can do
i think i can take the table above and put in a x, y axis on a coordinate plane, where the x axis is going to represent the hours she spent studying, and the y axis is going to be the grades she got.
i have this already set up. what is the next step?.
eddy
 
eddy2017 said:
hi, i am asking the forum tutors to help me understand quickly how to set up an answer to questions like the one i am posting here.
a student recorded her grades on assessments in respect to the number of hours she studied for each assessment. which of the following equations is the linear equation of the function representing the data below. . let x represents the number of hours studying and y represent the assessment grade.
the answer given( without any explanation, of course) y=10x+50

i want to know how to get to this solution and understand the process.
thanks in advance for your help and for the forum.

Hours studying Assessment grade
1 --------------------60
1.5-------------------65
2.5-------------------75
3.5-------------------80
4.5 -------------------95
Ok, this is what i can do
i think i can take the table above and put in a x, y axis on a coordinate plane, where the x axis is going to represent the hours she spent studying, and the y axis is going to be the grades she got.
i have this already set up. what is the next step?.
eddy
Click to expand...
You can plot the points to see what the graph looks like (e.g. a line). But if you are given the type of function that models the data (linear) and asked to derive the equation, plotting, IMO, is not necessary, especially for a simple linear function.
To figure out the equation you need to know its form: y = mx + b.
m and b are unknowns. To find them set up 2 equations using 2 data points and solve them.
 
lev888 said:
You can plot the points to see what the graph looks like (e.g. a line). But if you are given the type of function that models the data (linear) and asked to derive the equation, plotting, IMO, is not necessary, especially for a simple linear function.
To figure out the equation you need to know its form: y = mx + b.
m and b are unknowns. To find them set up 2 equations using 2 data points and solve them.
Click to expand...
okay, thanks. i will give it a shot and show you later.
 
I agree with lev's answer, but there are two points to consider.

In a real application, you would never know in advance that the relationship is linear. In that case, the kind of graphing you discussed is necessary.

Furthermore, in a real application, your graph might show that a line was approximately a good fit but not an exact one. In a slightly more advanced course, you will find how to find the best linear approximation.
 
eddy2017 said:
hi, i am asking the forum tutors to help me understand quickly how to set up an answer to questions like the one i am posting here.
a student recorded her grades on assessments in respect to the number of hours she studied for each assessment. which of the following equations is the linear equation of the function representing the data below. . let x represents the number of hours studying and y represent the assessment grade.
the answer given( without any explanation, of course) y=10x+50

i want to know how to get to this solution and understand the process.
thanks in advance for your help and for the forum.

Hours studying Assessment grade
1 --------------------60
1.5-------------------65
2.5-------------------75
3.5-------------------80
4.5 -------------------95
Ok, this is what i can do
i think i can take the table above and put in a x, y axis on a coordinate plane, where the x axis is going to represent the hours she spent studying, and the y axis is going to be the grades she got.
i have this already set up. what is the next step?.
eddy
Click to expand...
You could use a spreadsheet (like Excel) get the equation for best-fit-curve.
 
lev888 said:
You can plot the points to see what the graph looks like (e.g. a line). But if you are given the type of function that models the data (linear) and asked to derive the equation, plotting, IMO, is not necessary, especially for a simple linear function.
To figure out the equation you need to know its form: y = mx + b.
m and b are unknowns. To find them set up 2 equations using 2 data points and solve them.
Click to expand...
i plotted two points
(1,60)
(4.5, 95)
i used these two points to find the slope (m)
m= y2-y1/x2-x1
this gives me m =10
now that i know the slope, i need b now.
how can i find b ( i know is the y-intercept)
you asked me to set up two equations using two data points. i'm stuck here. how can i set em up.
i see it is a straight line, but i don't see how to set up the two equations. give me a hint, pls.
 

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eddy2017 said:
i plotted two points
(1,60)
(4.5, 95)
i used these two points to find the slope (m)
m= y2-y1/x2-x1
this gives me m =10
now that i know the slope, i need b now.
how can i find b ( i know is the y-intercept)
you asked me to set up two equations using two data points. i'm stuck here. how can i set em up.
Click to expand...
Plug in m and one data point into y = mx + b and solve for b.
 
which makes y=10x+50 a true statement
check
60=10(1)+50
60=60
so that is the linear equation representing the function.
 
Last edited:
eddy2017 said:
thanks, Doc Khan. i wish i knew how to do it.
Click to expand...
It is not real mathematics or even real thought. Excel will give you a line of best linear fit through a purely mechanical process. It will indeed be mathematically correct, but it may be nonsense. A non-linear model may be a much better approximation. In other words, Excel will give you the best linear approximation, but it does not tell you whether a non-linear approximation will provide an even better approximation.

Modern tools like Excel relieve us of a lot drudgery, but they too often induce people to stop thinking critically. I use Excel a lot, but It does not substitute for thought. The combination of your graphing method and common sense will permit you to outthink Excel.
 
JeffM said:
It is not real mathematics or even real thought. Excel will give you a line of best linear fit through a purely mechanical process. It will indeed be mathematically correct, but it may be nonsense. A non-linear model may be a much better approximation. In other words, Excel will give you the best linear approximation, but it does not tell you whether a non-linear approximation will provide an even better approximation.

Modern tools like Excel relieve us of a lot drudgery, but they too often induce people to stop thinking critically. I use Excel a lot, but It does not substitute for thought. The combination of your graphing method and common sense will permit you to outthink Excel.
Click to expand...
Thanks for the comment. Really interesting!.
 
JeffM said:
It is not real mathematics or even real thought. Excel will give you a line of best linear fit through a purely mechanical process. It will indeed be mathematically correct, but it may be nonsense. A non-linear model may be a much better approximation. In other words, Excel will give you the best linear approximation, but it does not tell you whether a non-linear approximation will provide an even better approximation.

Modern tools like Excel relieve us of a lot drudgery, but they too often induce people to stop thinking critically. I use Excel a lot, but It does not substitute for thought. The combination of your graphing method and common sense will permit you to outthink Excel.
Click to expand...
I beg to differ. Excel can calculate "best fit" curve using any reasonable function. Exce is not a thinker - but an obedient butler. You can calculate R^2 values and "estimate" good -ness" of fit.
 
Subhotosh Khan said:
I beg to differ. Excel can calculate "best fit" curve using any reasonable function. Exce is not a thinker - but an obedient butler. You can calculate R^2 values and "estimate" good -ness" of fit.
Click to expand...
Oh, I agree that Excel (especially when extended with the data analysis package) can do much more than linear approximations, but to use those tools effectively, you need to understand basic statistics.

It is easy to get Excel to develop a linear function using least squares (either including the origin or not) and conclude that the relationship is linear. After all, Excel does the least squares computations far more reliably than I ever would and will even draw a lovely graph suitable for export to a powerpoint presentation. Being easy does not preclude being wrong. Excel will say “this is the linear relationship you asked for“ even if you give it data generated by a sine function. That is not math in my opinion.

I said that I use Excel a lot. I do so because it does computations correctly and virtually instantaneously. But it cannot help me find whether a linear or exponential function best fits the data unless I ask it to do more than fit a linear approximation. It cannot help me determine whether I should use two independent variables rather than one unless I think to ask questions that address that issue. And it cannot tell me whether the best mathematical fit makes any sense no matter what questions I ask.

So, yes, to answer Eddy’s original question about finding the linear relationship (given that the problem was constructed to give a perfect linear fit), you could get Excel to answer it correctly. But the point I am making is that in most situations in real life, asking for a linear equation may be the wrong question. And if the data is not linear, Excel will say “mathematically, this is the best linear fit” if you pose your question that way.
 
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