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 = + 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 Incorrect: Your answer is incorrect. . 3 Incorrect: Your answer is incorrect. .
2 Incorrect: Your answer is incorrect. . 3
6 Incorrect: Your answer is incorrect. . 2 Incorrect: Your answer is incorrect. .
1 5
3 3
(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. Linear regression involves finding the slope (m) and y-intercept (b) of the line that minimizes the sum of squared differences between the observed y-values and the predicted y-values.
To calculate the equation of the line of best fit, follow these steps:
1. Calculate the mean of x and y.
- Mean of x (x̄) = (1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9) / 10 = 5
- Mean of y (ȳ) = (3 + 2 + 6 + 1 + 3 + 3 + 3 + 2 + 5 + 3) / 10 = 3.1
2. Calculate the differences between each x-value and the mean of x (x - x̄) and each y-value and the mean of y (y - ȳ).
x - x̄: -4, -4, -2, -2, 0, 0, 2, 2, 4, 4
y - ȳ: -0.1, -1.1, 2.9, -2.1, -0.1, -0.1, -0.1, -1.1, 1.9, -0.1
3. Calculate the product of each (x - x̄) and (y - ȳ) value.
-4 * (-0.1) = 0.4
-4 * (-1.1) = 4.4
-2 * 2.9 = -5.8
-2 * (-2.1) = 4.2
0 * (-0.1) = 0
0 * (-0.1) = 0
2 * (-0.1) = -0.2
2 * (-1.1) = -2.2
4 * 1.9 = 7.6
4 * (-0.1) = -0.4
4. Calculate the sum of the products.
0.4 + 4.4 + (-5.8) + 4.2 + 0 + 0 + (-0.2) + (-2.2) + 7.6 + (-0.4) = 7
5. Calculate the sum of the squared differences between x-values and the mean of x.
(-4)^2 + (-4)^2 + (-2)^2 + (-2)^2 + 0^2 + 0^2 + 2^2 + 2^2 + 4^2 + 4^2 = 80
6. Calculate the slope (m) using the formula:
m = Σ[(x - x̄)(y - ȳ)] / Σ[(x - x̄)^2]
= 7 / 80
= 0.0875
7. Calculate the y-intercept (b) using the formula:
b = ȳ - m * x̄
= 3.1 - 0.0875 * 5
= 2.7125
Therefore, the equation of the line of best fit is: y hat = 0.0875x + 2.7125.
To find the ordinates y hat for the points with abscissas x = 1, 3, 5, 7, and 9, substitute these x-values into the equation of the line of best fit:
For x = 1:
y hat = 0.0875 * 1 + 2.7125 = 2.8 (rounded to two decimal places)
For x = 3:
y hat = 0.0875 * 3 + 2.7125 = 2.975 (rounded to two decimal places)
For x = 5:
y hat = 0.0875 * 5 + 2.7125 = 3.15 (rounded to two decimal places)
For x = 7:
y hat = 0.0875 * 7 + 2.7125 = 3.325 (rounded to two decimal places)
For x = 9:
y hat = 0.0875 * 9 + 2.7125 = 3.5 (rounded to two decimal places)
The values for y hat for the points on the line of best fit with abscissas x = 1, 3, 5, 7, and 9 are approximately 2.8, 2.975, 3.15, 3.325, and 3.5, respectively.
To find the value of e for each of the points, subtract the corresponding value of y hat from the actual y-values:
For x = 1: e = y - y hat = 3 - 2.8 = 0.2 (rounded to two decimal places)
For x = 3: e = y - y hat = 2 - 2.975 = -0.975 (rounded to two decimal places)
For x = 5: e = y - y hat = 6 - 3.15 = 2.85 (rounded to two decimal places)
For x = 7: e = y - y hat = 1 - 3.325 = -2.325 (rounded to two decimal places)
For x = 9: e = y - y hat = 3 - 3.5 = -0.5 (rounded to two decimal places)
The values of e for each of the points are approximately 0.2, -0.975, 2.85, -2.325, and -0.5, respectively.