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 1 3 1 6 3 6 1 4 4
(a) Find the equation of the line of best fit.
y hat = + x
(ii) Graph this equation on a scatter diagram. (Do this on paper. Your instructor may ask you to turn in this work.)
(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
Incorrect: Your answer is incorrect.
3
Incorrect: Your answer is incorrect.
5
Incorrect: Your answer is incorrect.
7
Incorrect: Your answer is incorrect.
9
Incorrect: Your answer is incorrect.
(c) Find the value of e for each of the points above (e = y - y hat).
y e y e
3 3
1 6
3 1
1 4
6 4
(d) Find the variance se2 of those points about the line of best fit.
To find the equation of the line of best fit, we can use linear regression. Let's calculate the equation step by step.
(a) Find the equation of the line of best fit.
To find the equation of the line of best fit, we need to find the values of the slope and y-intercept. We can use the formula:
slope (b) = [(n * ∑(xy)) - (∑x * ∑y)] / [(n * ∑(x^2)) - (∑x)^2]
y-intercept (a) = (mean of y) - (slope * mean of x)
First, let's calculate the required summations:
∑x = 1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9 = 50
∑y = 3 + 1 + 3 + 1 + 6 + 3 + 6 + 1 + 4 + 4 = 32
∑(xy) = (1*3) + (1*1) + (3*3) + (3*1) + (5*6) + (5*3) + (7*6) + (7*1) + (9*4) + (9*4) = 199
∑(x^2) = (1^2) + (1^2) + (3^2) + (3^2) + (5^2) + (5^2) + (7^2) + (7^2) + (9^2) + (9^2) = 285
Now let's substitute these values into the formulas:
slope (b) = [(10 * 199) - (50 * 32)] / [(10 * 285) - (50)^2] = -0.24 (rounded to two decimal places)
mean of x = (1 + 3 + 5 + 7 + 9) / 5 = 5
mean of y = 32 / 10 = 3.2
y-intercept (a) = 3.2 - (-0.24 * 5) = 4.4
Therefore, the equation of the line of best fit is:
y hat = 4.4 - 0.24x
(ii) Graph this equation on a scatter diagram. (Do this on paper. Your instructor may ask you to turn in this work.)
(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.
To find y hat for each x value, substitute the x values into the equation of the line of best fit:
For x = 1: y hat = 4.4 - 0.24 * 1 = 4.16
For x = 3: y hat = 4.4 - 0.24 * 3 = 3.88
For x = 5: y hat = 4.4 - 0.24 * 5 = 3.6
For x = 7: y hat = 4.4 - 0.24 * 7 = 3.32
For x = 9: y hat = 4.4 - 0.24 * 9 = 3.04
Therefore, the y hat values for the given x values are: 4.16, 3.88, 3.6, 3.32, and 3.04.
(c) Find the value of e for each of the points above (e = y - y hat).
To find e, subtract y hat from the corresponding y values:
For x = 1: e = 3 - 4.16 = -1.16
For x = 3: e = 1 - 3.88 = -2.88
For x = 5: e = 3 - 3.6 = -0.6
For x = 7: e = 1 - 3.32 = -2.32
For x = 9: e = 4 - 3.04 = 0.96
Therefore, the values of e for the given points are: -1.16, -2.88, -0.6, -2.32, and 0.96.
(d) Find the variance se^2 of those points about the line of best fit.
To find the variance, we need to calculate the sum of squared errors (SSE) and divide it by the number of data points (n).
First, calculate the SSE:
SSE = (∑(e^2)) = (-1.16^2) + (-2.88^2) + (-0.6^2) + (-2.32^2) + (0.96^2)
= 1.3456 + 8.2944 + 0.36 + 5.3824 + 0.9216
= 16.302
The number of data points (n) is 5.
Variance (s^2) = SSE / (n-2) = 16.302 / (5-2) = 5.434
Therefore, the variance se^2 of those points about the line of best fit is 5.434.
To answer these questions, we will need to find the equation of the line of best fit and then use it to calculate the values requested.
(a) To find the equation of the line of best fit, we can use linear regression. This is done by finding the slope and intercept that minimize the sum of the squared differences between the observed values and the predicted values on the line.
1. Calculate the mean of x and y:
- Mean of x: (1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9)/10 = 5
- Mean of y: (3 + 1 + 3 + 1 + 6 + 3 + 6 + 1 + 4 + 4)/10 = 3.3
2. Calculate the deviations from the mean for each x and y value:
- Deviations for x: (1 - 5), (1 - 5), (3 - 5), (3 - 5), (5 - 5), (5 - 5), (7 - 5), (7 - 5), (9 - 5), (9 - 5)
= -4, -4, -2, -2, 0, 0, 2, 2, 4, 4
- Deviations for y: (3 - 3.3), (1 - 3.3), (3 - 3.3), (1 - 3.3), (6 - 3.3), (3 - 3.3), (6 - 3.3), (1 - 3.3), (4 - 3.3), (4 - 3.3)
= -0.3, -2.3, -0.3, -2.3, 2.7, -0.3, 2.7, -2.3, 0.7, 0.7
3. Calculate the product of the deviations for each point:
- Product of deviations: (-4)(-0.3), (-4)(-2.3), (-2)(-0.3), (-2)(-2.3), (0)(2.7), (0)(-0.3), (2)(2.7), (2)(-2.3), (4)(0.7), (4)(0.7)
= 1.2, 9.2, 0.6, 4.6, 0, 0, 5.4, -4.6, 2.8, 2.8
4. Calculate the sum of the squared deviations of x:
- Sum of squared deviations of x: (-4)^2 + (-4)^2 + (-2)^2 + (-2)^2 + (0)^2 + (0)^2 + (2)^2 + (2)^2 + (4)^2 + (4)^2
= 16 + 16 + 4 + 4 + 0 + 0 + 4 + 4 + 16 + 16 = 80
5. Calculate the sum of the product of deviations:
- Sum of product of deviations: 1.2 + 9.2 + 0.6 + 4.6 + 0 + 0 + 5.4 + (-4.6) + 2.8 + 2.8
= 22
6. Calculate the slope (m):
- Slope (m) = Sum of product of deviations / Sum of squared deviations of x
= 22 / 80 = 0.275
7. Calculate the intercept (c):
- Intercept (c) = Mean of y - Slope * Mean of x
= 3.3 - 0.275 * 5 = 1.725
Therefore, the equation of the line of best fit is y hat = 0.275x + 1.725.
(ii) To graph this equation on a scatter diagram, you can plot the given points (x, y) from the table and then draw the line of best fit using the equation above.
(b) To find the ordinates y hat for the points on the line of best fit:
- Substitute the given x values (1, 3, 5, 7, 9) into the equation y hat = 0.275x + 1.725.
- Calculate the corresponding y hat values for each x value.
For x = 1:
- y hat = 0.275 * 1 + 1.725 = 2.0
For x = 3:
- y hat = 0.275 * 3 + 1.725 = 2.550
For x = 5:
- y hat = 0.275 * 5 + 1.725 = 3.300
For x = 7:
- y hat = 0.275 * 7 + 1.725 = 4.050
For x = 9:
- y hat = 0.275 * 9 + 1.725 = 4.800
Therefore, the ordinates (y hat) for the points with x values 1, 3, 5, 7, and 9 are 2.00, 2.55, 3.30, 4.05, and 4.80 respectively.
(c) To find the value of e for each of the points above (e = y - y hat):
- Subtract the corresponding y hat values from the given y values.
- Calculate the difference e for each y value.
For y = 3 and y hat = 2.00:
- e = 3 - 2.00 = 1.00
For y = 1 and y hat = 2.55:
- e = 1 - 2.55 = -1.55
For y = 3 and y hat = 3.30:
- e = 3 - 3.30 = -0.30
For y = 1 and y hat = 4.05:
- e = 1 - 4.05 = -3.05
For y = 6 and y hat = 4.80:
- e = 6 - 4.80 = 1.20
For y = 3 and y hat = 2.55:
- e = 3 - 2.55 = 0.45
For y = 6 and y hat = 3.30:
- e = 6 - 3.30 = 2.70
For y = 1 and y hat = 4.05:
- e = 1 - 4.05 = -3.05
For y = 4 and y hat = 4.80:
- e = 4 - 4.80 = -0.80
For y = 4 and y hat = 4.80:
- e = 4 - 4.80 = -0.80
Therefore, the values of e for each of the given points are 1.00, -1.55, -0.30, -3.05, and 1.20 respectively.
(d) To find the variance se^2 of those points about the line of best fit:
- Calculate the squares of the differences e for each point.
- Calculate the sum of the squared differences.
- Divide the sum of squared differences by the number of points minus 2.
For e = 1.00:
- (1.00)^2 = 1.00
For e = -1.55:
- (-1.55)^2 = 2.40
For e = -0.30:
- (-0.30)^2 = 0.09
For e = -3.05:
- (-3.05)^2 = 9.30
For e = 1.20:
- (1.20)^2 = 1.44
Sum of squared differences = 1.00 + 2.40 + 0.09 + 9.30 + 1.44 = 14.23
Number of points minus 2 = 5 - 2 = 3
Variance (se^2) = Sum of squared differences / (Number of points minus 2)
= 14.23 / 3
= 4.74
Therefore, the variance (se^2) of those points about the line of best fit is 4.74.