Assignment 7

course Mth 163

June 26 8:08 pm*********************************************

Question: `q001. Note that this assignment has 8 questions

Sketch a graph of the following (x, y) points: (1,2), (3, 5), (6, 6). Then sketch the straight line which appears to come as close as possible, on the average, to the four points. Your straight line should not actually pass through any of the given points. Describe how your straight line lies in relation to the points.

Give the coordinates of the point at which your straight line passes through the y axes, and give the coordinates of the x = 2 and x = 7 points on your straight line.

Determine the slope of the straight line between the last two points you gave.

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Your solution:

The line is up above (1,2) and (6,6) coordinates and below the (3,5).

The line passes through between the 1 and 2 on the y-axis (closer to the 2).

(2,3) ( I had to look to see what the second set of coordinates were) (5, 7)

Slope = 7-3= 4 5-2= 3 = 4/3

Confidence Assessment: 2

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Given Solution:

Your straight line should pass above the first and third points and beneath the second. If this is not the case the line can be modified so that it comes closer on the average to all three points.

The best possible straight line passes through the y-axis near y = 2. The x = 2 point on the best possible line has a y coordinate of about 3, and the x = 7 point has a y coordinate of about 7. So the best possible straight line contains points with approximate coordinate (2,3) and (7,7).

The slope between these two points is rise/run = (7 - 3)/(7 - 2) = 4/5 = .8.

Note that the actual slope and y intercept of the true best-fit line, to 3 significant figures, are .763 and 1.79. So the equation of the line is .763 x + 1.79

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Self-critique (if necessary):

I got something different for the slope. I’m not sure why. How was I supposed to get (5,7) for the second coordinate because I don’t understand that.

Your answer will be based on an estimate; you aren't expected to find the very best possible straight line.

Your estimate was reasonable and your solution was fine.

There was an error in the given solution; (5, 7) should have been (7, 7).

Self-critique Rating:2

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Question: `q002. Plug coordinates of the x = 2 and x = 7 points into the form y = m x + b to obtain two simultaneous linear equations. Give your two equations. Then solve the equations for m and b and substitute these values into the form y = m x + b. What equation do you get?

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Your solution:

Y= mx+b

(2, 3) (5, 7)

3=(2)m +b

5=(7)m+b

B can be cancelled out

2m=3 m=2/3

7m=5 m=5/7

Confidence Assessment: 1

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Given Solution:

Plugging the coordinates (2,3) and (7, 7) into the form y = m x + b we obtain the equations

3 = 2 * m + b

7 = 7 * m + b.

Subtracting the first equation from the second will eliminate b. We get 4 = 5 * m. Dividing by 5 we get m = 4/5 = .8.

Plugging m = .8 into the first equation we get 3 = 2 * .8 + b, so 3 = 1.6 + b and b = 3 - 1.6 = 1.4.

Now the equation y = m x + b becomes y = .8 x + 1.4.

Note that the actual best-fit line is y = .763 x + 1.79, accurate to three significant figures.

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Self-critique (if necessary):

I’m not sure where I went wrong on the slope. The rest of it the problem would have been wrong since I couldn’t figure out the slope.

There was an editing error in the given solution. I've corrected it above.

Self-critique Rating:2

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Question: `q003. Using the equation y = .8 x + 1.4, find the coordinates of the x = 1, 3, and 6 points on the graph of the equation.

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Your solution:

Y= .8x +1.4

Y= .8(1) +1.4 = 2.2

Y= .8(3) +1.4= 3.8

Y= .8(6) +1.4 = 6.2

Confidence Assessment: 3

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Given Solution:

Evaluating y =.8 x + 1.4 at x = 1, 3, and 6 we obtain y values 2.2, 3.8, and 6.2.

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Self-critique (if necessary):

Self-critique Rating:

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Question: `q004. The equation y = .8 x + 1.4 gives you points (1, 2.2), (3, 3.8), and (6,6.2). How close, on the average, do these points come to the original data points (1,2), (3, 5), and (6, 6)?

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Your solution:

On average, within .53 of the original.

2.2-2= .2

3.8-5= -1.2

6.2-6= .2

I added these and divided by 3 to get the average of .53.

Confidence Assessment: 3

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Given Solution:

(1, 2.2) differs from (1, 2) by the .2 unit difference between y = 2 and y = 2.2.

(3, 3.8) differs from (3, 5) by the 1.2 unit difference between y = 5 and y = 3.8.

(6, 6.2) differs from (6, 6) by the .2 unit difference between y = 6 and y = 6.2.

{}The average discrepancy is the average of the three discrepancies:

ave discrepancy = ( .2 + 1.2 + .2 ) / 3 = .53.

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Self-critique (if necessary):

Self-critique Rating:

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Question: `q005. Using the best-fit equation y =.76 x + 1.79, with the numbers accurate to the nearest .01, how close do the predicted points corresponding to x = 1, 3, and 6 come to the original data points (1,2), (3, 5), and (6, 6)?

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Your solution:

Y=.76x+1.79

Y=.76(1) +1.79= 2.55

Y= .76(3) +1.79= 4.07

Y= .76(6) +1.79= 6.35

The differences are (.45+.93+.35)

Divided by three the average is 0.576 or .58

Confidence Assessment: 3

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Given Solution:

Evaluating y =.76 x + 1.79 at x = 1, 3 and 6 we obtain y values 2.55, 4.07 and 6.35. This gives us the points (1,2.55), (3,4.07) and (6, 6.35). These points lie at distances of .55, .93, and .35 from the original data points.

The average distance is (.55 + .93 + .35) / 3 = .58 from the points.

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Self-critique (if necessary):

Self-critique Rating:

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Question: `q006. The average distance of the best-fit line to the data points appears to greater than the average distance of the line we obtain by an estimate. In fact, the best-fit line doesn't really minimize the average distance but rather the square of the average distance. The distances for the best-fit model are .55, .93 and .35, while the average distances for our first model are .2, 1.2 and .2. Verify that the average of the square distances is indeed less for the best-fit model.

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Your solution:

If you square the 1st equation: .2^2= .04 1.2^2 =1.44 ..2^2=.04

Best fit Equation: .55^2= .3 .93^2= .87 .35^2= .12

Added together (1st): (.04+1.44+.04)/3 = 0.51

Best fit equation: (.30+.87+.12)/3 = .43

The best fit model is better.

Confidence Assessment: 3

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Given Solution:

The square distances for the best-fit model are .55^2 = .30, .93^2 = .87 and .35^2 = .12. The average of these square distances is (.30 + .87 + .12) / 3 = .43.

The squared distances for the first model are .2^2 = .04, 1.2^2 = 1.44 and .2^2 = .04, so the average of the square distances for this model is (.04 + 1.44 + .04) / 3 = .51.

Thus the best-fit model does give the better result.

We won't go into the reasons here why it is desirable to minimize the square of the distance rather than the distance. When doing eyeball estimates, you don't really need to take this subtlety into account. You can simply try to get is close is possible, on the average, to the points.

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Self-critique (if necessary):

Self-critique Rating:

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Question: `q007. If the original data points (1,2), (3, 5), and (6, 6) represent the selling price in dollars of a bag of widgets vs. the number of widgets in the bag, then how much is paid for a bag of 3 widgets? How much would you estimate a bag of 7 widgets would cost?

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Your solution:

Y= .76x+ 1.79

Y= .76(3) +1.79= 4.07

Y= .76(7)+1.79= 7.11

A bag of 3 would be $4.05

A bag of 7 would be $7.11

Confidence Assessment: 3

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Given Solution:

If we use the best-fit function y =.76 x + 1.79, noting that y represents the cost and x the number of widgets, then the cost of 3 widgets is

y = .76 * 3 + 1.79 = 4.05, representing cost of $4.05.

The cost of 7 widgets would be

y = .76 * 7 + 1.79 = 7.11. The cost of 7 widgets would be $7.11.

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Self-critique (if necessary):

Self-critique Rating:

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Question: `q008. According to the function y = .8 x + 1.4, how much will a bag of 7 widgets cost? How many widgets would you expect to get for $10?

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Your solution:

y = .8 x + 1.4

y= .8(7) +1.4=$7.00

A bag of 7 would be $7.00

10= .8x+1.4

X= (10+1.4) /.8 = 10.75

A bag for $10 would be 10.75 widgets

Confidence Assessment: 3

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Given Solution:

Using the model we obtained, y = .8 x + 1.4, we note that the cost is represented by y and the number of widgets by acts. Thus we can find cost of 7 widgets by letting x = 7:

cost = y = .8 * 7 + 1.4 = 7.

To find the number of widgets you can get for $10, let y = 10. Then the equation becomes

10 = .8 x + 1.4.

We easily solve this equation by subtracting 1.4 from both sides than dividing by .8 to obtain x = 10.75. That is, we can buy 10.75 widgets with $10.

You're doing fine here. There were some editing errors in the first couple of given solutions; I've corrected them and apologize for the confusion.