course mth 158
If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution:
If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
020. `* 20
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Question: * 4.2.8 / 2.6.8 (was 2.5.6). graph like basic stretched cubic centered around (20,20)
How well does the graph appear to indicate a linear relation?
Describe any significant deviation of the data from its best-fit linear approximation.
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Your solution:
It appears to be a cube function. According to the way the book reads, the graph does not appear to indicate a linear relation.
I am not sure how to do the best-fit linear approximation. I drew a 45 degree line from (0, 0) and examined the points. I really do not understand the best fit linear approximation and cannot answer the question. From looking at the data, I see that some of the points are above the line and some are below, but that still doesn’t help me with understanding it.
The 45-degree line is the best possible linear fit.
Basically the best linear fit is the straight line that comes as close as possible, on the average, to the points of the graph. There are some technicalities about how 'closeness' is defined, but we can safely save the technicalities for a later course.
confidence rating: 0
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Given Solution:
* * The graph is curved and in fact changes its concavity. The data points will lie first above the best-fit straight line, then as the straight line passes through the data set the data points will lie below this line. **
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Self-critique (if necessary):
The given solution says the graph is curved and changes its concavity, but does that mean it is a linear relation?
'Linear' means 'straight-line', so if the graph curves it's not linear.
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Self-critique Rating: 0
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Question: * 4.2.22 / 3.1.90. Sales S vs. advertising expenditures A. 335 339 337 343 341 350 351 vs. 20, 22, 22.5, 24, 24, 27, 28.3 in thousands of dollars.
Does the given table describe a function? Why or why not?
What two points on your straight line did you pick and what is the resulting equation?
What is the meaning of the slope of this line?
Give your equation as a function and give the domain of the function.
What are the predicted sales if the expenditures are $25,000?
a) No, the table does not describe a function based on the ordered pairs. There are 2 of the same numbers for A {24, 24} with different results for S {343, 341}.
b) I’m not exactly sure how to answer this question. To find the slope of the line, I picked the change in y (339-335) and the change in x (22-20) to get the slope of the line 2/1
c) You will get a rise of 2 and a run of 1, meaning for every unit that you move 1 unit to the right; you will move 2 units up the y axis to obtain the point. (I think the question is asking for more of a specific answer like for every x dollars spent in advertising, the sales increase of decrease by x, but when I tried to solve it that way, I noticed that the expenditures were not constant. They varied going from a difference of 2, .5, 1.5, 0, 3, and 1.3. Then the difference of the sales varied going from a difference of +4, -1, +5, -2, +9, +1. Since the rate of change was not constant, I could not come up with a solution to interpret the slope)
d) y – y1=m(x-x1)
y – 339 = 2(x – 22)
y = 2x – 44 + 339
y = 2x + 295 m = 2 y intercept is 295
Expressed as a function we have
f(x) = 2x + 2925
e) D {x| 20 <= x <= 28.3}
f) f(x) = 2x + 2925
f(x) = 2(25000) + 2925
= Approximate sales would be $529,250
The table does not describe a function because ordered pairs that have the same first element and a different second element. Specifically 24,000 is paired with both 343,000 and 341,000.
I picked the points (20000,335000) (27000,350000).
INSTRUCTOR COMMENT: These are data points, not points on the best-fit straight line graph. You should have sketched your line then picked two points on the line, and the line will almost never pass through data points.
STUDENT SOLUTION CONTINUED:
The slope between these points is
slope = (350000-335000)/(27000-20000) = 15000/7000 = 15/7 = 2.143 approximately.
Our equation, using this slope and the first chosen point, is therefore
y-335000=2.143(x-20000)
y- 335 = 2.143x-42857.143
y= 2.143x+29214.857 equation of the line
Expressed as a function we have
f(x) = 2.143x+292142.857.
Predicted sales for expenditure $25000 will be
f(25000) = 2.143(25000) + 292142.857
= 53575 + 292142.857
= 345717.857
We therefore have predicted sales
f(25000)= $345,717.86
INSTRUCTOR COMMENT: Excellent solution, except for the fact that you used data points and not points on the best-fit line.
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Self-critique (if necessary):
I did the same thing the student did with the pairs. I didn’t use the best-fit line (because I’m not sure how to do that yet). My answers were different than the given solution, but I’m not sure they are wrong since we didn’t use the same points. I think I did it right except for the interpretation of the slope of the line…I’m not sure if I did that right or not.
Your solution was good, and perfectly consistent with the data. An eyeball estimate of the best-fit line is all that's asked for here.
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Self-critique Rating: 2
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Question: * extra problem (was 2.5.12). Sketch a graph of y vs. x for x = 5, 10, 15, 20, 25; y = 2, 4, 7, 11, 18. Fit a good straight line to the data and pick two points on this line. Use these points to find an estimated equation for your line. **** What two points did you select on the line you graphed, and what is the equation of the line through those points? **** What is the equation of the best-fit line and how well does the line fit the data? Describe any systematic deviation of the line from the best-fit line. ****
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Your solution:
I tried to do this problem. I used graphing paper to make sure the line was straight. I plotted the points (5, 2) (10, 4) (15, 7) (20, 11) (25, 18) then I drew a straight line from (4, 0) straight through the data. (My graph was on the scale of 1 unit both x and y axis) There were 2 points above the line (5, 2) and (25, 18) the rest were below the line.
I don’t know if I did that correct, but from that point on, I don’t know how to answer the questions. If I picked 2 points on the line and gave an equation for those 2 points I could say:
(12, 6) (17, 11) and give the slope of the line which is
11-6/17-12 = 5/5 = 1
then do the point slope
y – y1 = m(x – x1) to get
y – 6 = 1(x – 12)
y = x – 12 + 6
y = x -6
then solve for x by letting y = 0
x – 6 = 0
x = 6
but that doesn’t make any sense on this graph because looking at it….y would not be -6 when x = 6
To look at the graph, when x = 6, y would approximately = 1.5
What am I doing wrong?
According to your model, when x = 6 the value of y would be y = x - 6 = 6 - 6 = 0.
The data point (5, 2) isn't that far from (6, 0). Depending on the scatter of the data points, it might or might not be a reasonable result. However, in any case, your model makes perfectly good sense.
confidence rating: 0
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Given Solution:
* * STUDENT SOLUTION WITH INSTRUCTOR COMMENT:
I chose the points (5,2) and (10,4)
The slope between these points is
slope = rise / run = (4-2)/(10-5) = 2/5
so the equation is
y-4 = 2/5(x - 10), which we solve for y to get
y = 2/5 x.
INTRUCTOR COMMENT:
This fits the first two data points, but these are not appropriate points to select. The data set curves, with increasing slope as we move to the right.
You need to sketch the best-fit line, as best you can see it, and pick two points on that line. The best-fit line is not likely to pass through any of the data points, and you should never use data points to determine the equation of the best-fit line.
Make an accurate sketch of the data points. Sketch your best-fit straight line, the straight line that comes as close as possible on the average to the points. Extend the line slightly beyond the data set.
Estimate the y coordinates of the x = 1 and x = 20 points of this line. Find the equation of the straight line through these points.
The coordinates of your points should be reasonably close to (1, 5.5) and (20, 30), though because it's a little difficult to judge exactly where the line should be you are unlikely to obtain these exact results. The equation will be reasonably close to y = .8 x - 3. **
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Self-critique (if necessary):
I really need help on this. I will take this problem to the tutor on Thursday and see if I can get a grasp on understanding these problems.
It looks to me as though you pretty much have it.
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Self-critique Rating: 0
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Question: * extra problem (was 2.5.18). Individuals with incomes 15k, 20k, 25k, 30, 35, 40, 45k, 50k, 55k, 60k, 65k, 70k (meaning $15,000, $20,000, etc; 'k' means 'thousand') have respective loan amounts 40.6, 54.1, 67.7, 81.2, 94.8, 108.3, 121.9, 135.4, 149, 162.5, 176.1, 189.6 k. Sketch a graph of loan amount vs. income, fit a good straight line to the data and use two points on your line to estimate the equation of the line.
I am not sure if I did this correctly, but I sketched a graph of the given amounts and drew a line from each given numbers on the y axis to the corresponding x numbers. Then I picked a point on the x axis and drew a line straight up until it intersected with a sloping line. From that point, I drew a line to the point on the y axis that was horizontal to that point. I came up with:
(35, 108.3) (45, 67.7)
From those two points, I came up with a slope of -4.06 using the y1 – y2/ x1 – x2
Then I did the following:
y – y1 = m(x – x1)
y – 108.3 = -4.06(x – 35)
y = -4.06x + 250.4
But that didn’t really make sense because if you let y = 250.4 you just end up with 0, so I am not sure how to do it.
ERRONEOUS STUDENT SOLUTION WITH INSTRUCTOR COMMENT:
Using the points (15,000, 40,600) and (20,000 , 67,700) we obtain
slope = rise / run = (67,700 - 40,600) / (20,000 - 15,000) = 271/50
This gives us the equation
y - 40,600= (271/50) * (x - 15,000), which we solve for y to obtain
y = (271/50) x - 40,700.
INSTRUCTOR COMMENT: You followed most of the correct steps to get the equation of the line from your two chosen points. However I think the x = 20,000 value is y = 54,100, not 67,700; the latter corresponds to x = 25,000. So your equation won't be likely to fit the data very well.
Another reason that your equation is not likely to be a very good fit is that you used two data points, which is inappropriate; and in addition you used two data points near the beginning of the data list. If you were going to use two data points you would need to use two typical points much further apart.
{]In any case to solve this problem you need to sketch the best-fit line, as best you can see it, and pick two points on that line. The best-fit line is not likely to pass through any of the data points, and you should never use data points to determine the equation of the best-fit line.
Make an accurate sketch of the data points. Sketch your best-fit straight line, the straight line that comes as close as possible on the average to the points. Extend the line slightly beyond the data set.
Estimate the y coordinates of the x = 10,000 and x = 75,000 points of this line. Find the equation of the straight line through these points.
The coordinates of your points should be reasonably close to (5000, 19000) and (75000, 277,000). It's fairly easy to locate this line, which does closely follow the data points, though due to errors in estimating you are unlikely to obtain these exact results. The equation will be reasonably close to y = 2.7 x - 700 .
If we let y = 42,000 we can solve for x:
42,000 = 2.7 x - 700 so
2.7 x = 42,700 and
x = 42,700 / 2.7 = 15,800 approx..
Your solution will differ slightly due to differences in your estimates of the line and the two points on the line. **
**** What is the equation of the line of best fit? **** How well does the line fit the scatter diagram of the data? Describe any systematic deviation of the line from the best-fit line. **** What is your interpretation of the slope of this line? **** What loan amount would correspond to annual income of $42,000?
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Self-critique (if necessary):
I am really confused. I need to know how to do the best line fit.
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&#Good responses. See my notes and let me know if you have questions. &#