Use the following table to answer the questions. (Give your answers correct to two decimal places.)
x 1 1 3 3 5 5 7 7 9 9
y 3 2 6 1 3 3 3 2 5 3
(a) Find the equation of the line of best fit.
y hat = 2.6,.1x Correct: Your answer is correct. . + Correct: Your answer is correct. . x
(b) Find the ordinates y hat for the points on the line of best fit whose abscissas are x = 1, 3, 5, 7, and 9.
1 . .
3 . .
5 . .
7 . .
9 . .
(c) Find the value of e for each of the points above (e = y - y hat).
y e y e
3 . . 3
2 3
6 2
1 5
3 3
yhat = 3x
I got the first part the answers were 2.6 and .1x, I can not figure out the rest of it B & C.
There are some of rule you have to follow for particular abscissa.
(2, 4, 6, 8, 10), the abscissa is the half the ordinate
Rule: x = (1/2)y
y = 2x
(1,2 3,4,5): the abscissa is two less than the ordinate.
Rule: y-2 =x
y = x + 2
(1,3,5,7,9), the abscissa is one third of the ordinate.
Rule: x = (1/3)y
y = 3x
(0,1, 2, 3.....,10), abscissa
Rule: y-1 = 2x
y = 2x +1
b) yhat = 3x
So on B I got the answers, 0.5, 1.5,
2.5,3.5,4.5
And everything that I have tried on C is coming up wrong????
To find the equation of the line of best fit, we will use the method of least squares. This involves minimizing the sum of the squares of the differences between the actual y-values and the predicted y-values.
(a) To find the equation of the line of best fit, we can use the formula y hat = a + bx, where a is the y-intercept and b is the slope.
First, we need to find the slope b. The formula for the slope is given by:
b = (n∑(xy) - (∑x)(∑y)) / (n∑(x^2) - (∑x)^2)
where n is the number of data points,
To calculate b:
n = 5 pairs of (x, y) values
∑x = 1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9 = 50
∑y = 3 + 2 + 6 + 1 + 3 + 3 + 3 + 2 + 5 + 3 = 31
∑(xy) = (1)(3) + (1)(2) + (3)(6) + (3)(1) + (5)(3) + (5)(3) + (7)(3) + (7)(2) + (9)(5) + (9)(3) = 160
∑(x^2) = 1^2 + 1^2 + 3^2 + 3^2 + 5^2 + 5^2 + 7^2 + 7^2 + 9^2 + 9^2 = 275
b = (5(160) - (50)(31)) / (5(275) - (50)^2)
b = (800 - 1550) / (1375 - 2500)
b = (-750) / (-1125)
b = 0.67
Next, we need to find the y-intercept a. The formula for the y-intercept is given by:
a = (∑y - b(∑x)) / n
To calculate a:
a = (31 - 0.67(50)) / 5
a = (31 - 33.5) / 5
a = -2.5 / 5
a = -0.5
Therefore, the equation of the line of best fit is:
y hat = -0.5 + 0.67x
(b) To find the y hat values for the abscissas x = 1, 3, 5, 7, and 9, we substitute these values into the equation of the line of best fit:
For x = 1: y hat = -0.5 + 0.67(1) = 0.17
For x = 3: y hat = -0.5 + 0.67(3) = 1.51
For x = 5: y hat = -0.5 + 0.67(5) = 2.85
For x = 7: y hat = -0.5 + 0.67(7) = 4.19
For x = 9: y hat = -0.5 + 0.67(9) = 5.53
Therefore, the y hat values for the abscissas x = 1, 3, 5, 7, and 9 are:
1 -> 0.17
3 -> 1.51
5 -> 2.85
7 -> 4.19
9 -> 5.53
(c) To find the value of e for each point (e = y - y hat), substitute the given y-values and the corresponding y hat values from part (b) into the equation:
For y = 3: e = 3 - 0.17 = 2.83
For y = 2: e = 2 - 1.51 = 0.49
For y = 6: e = 6 - 2.85 = 3.15
For y = 1: e = 1 - 4.19 = -3.19
For y = 3: e = 3 - 5.53 = -2.53
Therefore, the values of e for the given points are:
(3, e = 2.83)
(2, e = 0.49)
(6, e = 3.15)
(1, e = -3.19)
(3, e = -2.53)