A manager wishes to determine the relationship between the number of miles (in hundreds of miles) the manager's sales representatives travel per month and the amount of sales (in thousands of dollars) per month. Find the equation of the regression line for the given data. Predict the value of sales when the sales representative travel 8 miles. Predict the value of sales when the sales representative traveled 11 miles.

Miles traveled x: 2,3,10,7,8,15,3,1,11
sales y: 31,33,78,62,65,61,48,55,120

3799

To find the equation of the regression line, we need to perform a linear regression analysis on the given data. Here are the step-by-step instructions:

Step 1: Calculate the means of the x-values (miles) and the y-values (sales). Let's denote the means as x̄ (x-bar) and ȳ (y-bar).

x̄ = (2 + 3 + 10 + 7 + 8 + 15 + 3 + 1 + 11) / 9 = 60 / 9 = 6.67 (rounded to two decimal places)
ȳ = (31 + 33 + 78 + 62 + 65 + 61 + 48 + 55 + 120) / 9 = 553 / 9 = 61.44 (rounded to two decimal places)

Step 2: Calculate the differences between each x-value and the mean of x (x - x̄), and the differences between each y-value and the mean of y (y - ȳ). Square these differences.

x - x̄: (-4.67, -3.67, 3.33, 0.33, 1.33, 8.33, -3.67, -5.67, 4.33)
(x - x̄)^2: (21.78, 13.47, 11.09, 0.11, 1.77, 69.51, 13.47, 32.19, 18.72)

y - ȳ: (-30.44, -28.44, 16.56, 0.56, 3.56, -0.44, -13.44, -6.44, 58.56)
(y - ȳ)^2: (926.19, 808.26, 274.18, 0.31, 12.67, 0.20, 180.62, 41.41, 3435.45)

Step 3: Calculate the sum of the products of the differences (x - x̄) and (y - ȳ).

(x - x̄) * (y - ȳ): (-140.00, 105.00, 203.00, 0.21, 4.74, -3.68, 49.33, 36.48, 252.64)

Step 4: Calculate the sum of the squared differences (x - x̄)^2.

Σ(x - x̄)^2 = 21.79 + 13.47 + 11.09 + 0.11 + 1.77 + 69.51 + 13.47 + 32.19 + 18.72 = 182.12 (rounded to two decimal places)

Step 5: Calculate the sum of the squared differences (y - ȳ)^2.

Σ(y - ȳ)^2 = 926.19 + 808.26 + 274.18 + 0.31 + 12.67 + 0.20 + 180.62 + 41.41 + 3435.45 = 5689.29 (rounded to two decimal places)

Step 6: Calculate the sum of the products of the differences (x - x̄) and (y - ȳ).

Σ(x - x̄) * (y - ȳ) = -140.00 + 105.00 + 203.00 + 0.21 + 4.74 - 3.68 + 49.33 + 36.48 + 252.64 = 506.72 (rounded to two decimal places)

Step 7: Calculate the slope (b) of the regression line:

b = Σ(x - x̄) * (y - ȳ) / Σ(x - x̄)^2 = 506.72 / 182.12 = 2.78 (rounded to two decimal places)

Step 8: Calculate the y-intercept (a) of the regression line:

a = ȳ - b * x̄ = 61.44 - 2.78 * 6.67 = 42.89 (rounded to two decimal places)

Therefore, the equation of the regression line is:

y = 2.78x + 42.89

Step 9: Predict the value of sales when the sales representative travels 8 miles.

y = 2.78 * 8 + 42.89 = 22.24 + 42.89 = 65.13 (rounded to two decimal places)

Therefore, when the sales representative travels 8 miles, the predicted value of sales is $65,130.

Step 10: Predict the value of sales when the sales representative travels 11 miles.

y = 2.78 * 11 + 42.89 = 30.58 + 42.89 = 73.47 (rounded to two decimal places)

Therefore, when the sales representative travels 11 miles, the predicted value of sales is $73,470.

To find the equation of the regression line, we can follow these steps:

Step 1: Calculate the mean values of x and y:
- Sum up all the x values and divide by the number of values to find the mean of x.
- Sum up all the y values and divide by the number of values to find the mean of y.

For the given data, the means are:
- Mean of x = (2+3+10+7+8+15+3+1+11) / 9 = 60 / 9 = 6.67
- Mean of y = (31+33+78+62+65+61+48+55+120) / 9 = 553 / 9 = 61.44

Step 2: Calculate the differences between each x value and the mean of x (x - mean of x). Similarly, calculate the differences between each y value and the mean of y (y - mean of y). Square each of these differences.

For each value:
- Difference between x and mean of x: (x - mean of x)
- Difference between y and mean of y: (y - mean of y)

For the given data, the differences between x and mean of x, and y and mean of y, squared are:
(2 - 6.67)^2 = 19.89
(3 - 6.67)^2 = 14.14
(10 - 6.67)^2 = 11.11
(7 - 6.67)^2 = 0.11
(8 - 6.67)^2 = 0.02
(15 - 6.67)^2 = 69.45
(3 - 6.67)^2 = 14.14
(1 - 6.67)^2 = 24.11
(11 - 6.67)^2 = 18.11

Step 3: Calculate the product of the differences between x and mean of x, and y and mean of y for each pair of values (x - mean of x)(y - mean of y).

For each pair of values:
- Product of differences: (x - mean of x)(y - mean of y)

For the given data, the products of the differences are:
(2 - 6.67)(31 - 61.44) = 180.89
(3 - 6.67)(33 - 61.44) = 232.44
(10 - 6.67)(78 - 61.44) = 138.78
(7 - 6.67)(62 - 61.44) = 5.78
(8 - 6.67)(65 - 61.44) = 11.78
(15 - 6.67)(61 - 61.44) = 3.00
(3 - 6.67)(48 - 61.44) = -31.78
(1 - 6.67)(55 - 61.44) = -28.44
(11 - 6.67)(120 - 61.44) = 333.67

Step 4: Sum up all the squared differences and products of differences.

- Sum of squared differences (Σ(x - mean of x)^2):
19.89 + 14.14 + 11.11 + 0.11 + 0.02 + 69.45 + 14.14 + 24.11 + 18.11 = 171.08

- Sum of products of differences (Σ(x - mean of x)(y - mean of y)):
180.89 + 232.44 + 138.78 + 5.78 + 11.78 + 3.00 - 31.78 - 28.44 + 333.67 = 846.32

Step 5: Calculate the slope (b) of the regression line:
- Slope (b) = Σ(x - mean of x)(y - mean of y) / Σ(x - mean of x)^2
- Plugging in the values, b = 846.32 / 171.08 ≈ 4.944286

Step 6: Calculate the y-intercept (a) of the regression line:
- Y-intercept (a) = mean of y - (slope * mean of x)
- Plugging in the values, a = 61.44 - (4.944286 * 6.67) ≈ 30.379706

Therefore, the equation of the regression line is:
- y = 4.944286x + 30.379706

To predict the value of sales when the sales representative traveled 8 miles:
- Plug in x = 8 into the equation: y = 4.944286(8) + 30.379706
- Approximating the calculation, y ≈ 66.75 thousand dollars

To predict the value of sales when the sales representative traveled 11 miles:
- Plug in x = 11 into the equation: y = 4.944286(11) + 30.379706
- Approximating the calculation, y ≈ 82.39 thousand dollars