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 . .
3 . .
5 . .
7 . .
9 . .

a) yhat= 2.6 + .1x

To find the equation of the line of best fit, we can use the method of least squares. This involves finding the equation of a line that minimizes the sum of the squared differences between the predicted values (y hat) and the actual values (y).

Step 1: Calculate the mean of x (x-bar) and y (y-bar).
x-bar = (1 + 1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 + 9) / 10 = 5
y-bar = (3 + 2 + 6 + 1 + 3 + 3 + 3 + 2 + 5 + 3) / 10 = 3.1

Step 2: Calculate the deviations (d) for both x and y.
d(x) = x - x-bar
d(y) = y - y-bar
For example, d(x) for the first point (x=1, y=3) would be 1 - 5 = -4.

Step 3: Calculate the product of the deviations (d(x) * d(y)) for each point.
For example, for the first point, (1-5) * (3-3.1) = -0.4.

Step 4: Calculate the square of the deviations (d(x)^2) for each point.
For example, for the first point, (-4)^2 = 16.

Step 5: Calculate the sum of the products of deviations (d(x) * d(y)) and the sum of the squares of deviations (d(x)^2).
∑(d(x) * d(y)) = -0.4 + (-0.4) + 2.6 + (-0.6) + (-0.4) + (-0.4) + (-0.4) + (-0.8) + 1.6 + (-0.4) = -1.6
∑(d(x)^2) = 16 + 16 + 4 + 4 + 0 + 0 + 4 + 4 + 16 + 16 = 80

Step 6: Calculate the slope of the line (b).
b = ∑(d(x) * d(y)) / ∑(d(x)^2) = -1.6 / 80 = -0.02

Step 7: Calculate the y-intercept of the line (a).
a = y-bar - b * x-bar
a = 3.1 - (-0.02) * 5 = 3.1 + 0.1 = 3.2

Therefore, the equation of the line of best fit is:
y hat = 3.2 - 0.02x

To find the ordinates (y hat) for the given x-values of 1, 3, 5, 7, and 9, substitute those x-values into the equation of the line of best fit.

For x = 1:
y hat = 3.2 - 0.02 * 1 = 3.2 - 0.02 = 3.18 (rounded to two decimal places)

For x = 3:
y hat = 3.2 - 0.02 * 3 = 3.2 - 0.06 = 3.14 (rounded to two decimal places)

For x = 5:
y hat = 3.2 - 0.02 * 5 = 3.2 - 0.10 = 3.10 (rounded to two decimal places)

For x = 7:
y hat = 3.2 - 0.02 * 7 = 3.2 - 0.14 = 3.06 (rounded to two decimal places)

For x = 9:
y hat = 3.2 - 0.02 * 9 = 3.2 - 0.18 = 3.02 (rounded to two decimal places)