To predict the number of used cars that Ben will sell if the interest rate is 17% per annum, we can use linear interpolation based on the given data.
First, we need to calculate the slope of the line that relates the interest rate and the number of cars sold:
Slope = (Change in number of cars sold) / (Change in interest rate)
= (300 - 250) / (20 - 18)
= 50 / 2
= 25.
Next, we can use the point-slope form of a linear equation to predict the number of cars sold at 17% interest rate:
Number of cars sold = Slope * (Interest rate - Interest rate at 20%) + Number of cars sold at 20%
Number of cars sold = 25 * (17 - 20) + 300
Number of cars sold = 25 * (-3) + 300
Number of cars sold = -75 + 300
Number of cars sold = 225.
Therefore, Ben is predicted to sell 225 used cars if the interest rate is 17% per annum at the beginning of a month.
Ben the owner of a used firm, is interested in the association between the interest rate (percent per annum) at the beginning of a month and the number of used cars that he sells that month. He has collected the following data over a period of six months.
When the interest rate is 20, then the number of cars sold is 300.
Interest rate 18, number of cars sold is 250.
Interest rate 15, number of cars sold is 190.
Interest rate 16, number of cars sold is 200.
Interest rate 14, number of cars sold is 180.
Interest rate 19, number of cars sold is 275.
Use the information above predict the number of used cars that Ben will sell if the interest rate is 17% per annum at the beginning of a month.
9 answers
Are you sure??
Let's try to make sure by using a more accurate method called linear regression. We will calculate the equation of the line of best fit for the data provided and use that equation to predict the number of cars sold at 17% interest rate.
1. Calculate the mean values for interest rate and number of cars sold:
Mean interest rate = (20 + 18 + 15 + 16 + 14 + 19) / 6
Mean interest rate = 16.333
Mean number of cars sold = (300 + 250 + 190 + 200 + 180 + 275) / 6
Mean number of cars sold = 231.67
2. Calculate the individual deviations from the mean:
Deviation for interest rate 20: 20 - 16.333 = 3.667
Deviation for interest rate 18: 18 - 16.333 = 1.667
Deviation for interest rate 15: 15 - 16.333 = -1.333
Deviation for interest rate 16: 16 - 16.333 = -0.333
Deviation for interest rate 14: 14 - 16.333 = -2.333
Deviation for interest rate 19: 19 - 16.333 = 2.667
Deviation for number of cars sold at 300: 300 - 231.67 = 68.33
Deviation for number of cars sold at 250: 250 - 231.67 = 18.33
Deviation for number of cars sold at 190: 190 - 231.67 = -41.67
Deviation for number of cars sold at 200: 200 - 231.67 = -31.67
Deviation for number of cars sold at 180: 180 - 231.67 = -51.67
Deviation for number of cars sold at 275: 275 - 231.67 = 43.33
3. Calculate the sum of the product of the deviations:
Sum of (Interest rate deviation * Number of cars sold deviation) = 3.667 * 68.33 + 1.667 * 18.33 - 1.333 * -41.67 - 0.333 * -31.67 - 2.333 * -51.67 + 2.667 * 43.33
= 251.41 + 30.56 + 55.56 + 10.56 + 120.56 + 116.11
= 584.76
4. Calculate the sum of squared deviations for the interest rate:
Sum of (Interest rate deviation)^2 = 3.667^2 + 1.667^2 + 1.333^2 + 0.333^2 + 2.333^2 + 2.667^2
= 13.44 + 2.78 + 1.78 + 0.11 + 5.44 + 7.11
= 30.66
5. Calculate the slope of the line of best fit:
Slope = Sum of (Interest rate deviation * Number of cars sold deviation) / Sum of (Interest rate deviation)^2
= 584.76 / 30.66
= 19.05
6. Calculate the intercept of the line of best fit:
Intercept = Mean number of cars sold - (Slope * Mean interest rate)
= 231.67 - (19.05 * 16.333)
= 231.67 - 311.41
= -79.74
7. Now, we can use the equation of the line of best fit to predict the number of cars sold at 17% interest rate:
Number of cars sold = Slope * Interest rate + Intercept
Number of cars sold = 19.05 * 17 - 79.74
Number of cars sold = 322.85 - 79.74
Number of cars sold = 243.11
Therefore, according to the linear regression analysis, Ben is predicted to sell approximately 243 used cars if the interest rate is 17% per annum at the beginning of a month.
1. Calculate the mean values for interest rate and number of cars sold:
Mean interest rate = (20 + 18 + 15 + 16 + 14 + 19) / 6
Mean interest rate = 16.333
Mean number of cars sold = (300 + 250 + 190 + 200 + 180 + 275) / 6
Mean number of cars sold = 231.67
2. Calculate the individual deviations from the mean:
Deviation for interest rate 20: 20 - 16.333 = 3.667
Deviation for interest rate 18: 18 - 16.333 = 1.667
Deviation for interest rate 15: 15 - 16.333 = -1.333
Deviation for interest rate 16: 16 - 16.333 = -0.333
Deviation for interest rate 14: 14 - 16.333 = -2.333
Deviation for interest rate 19: 19 - 16.333 = 2.667
Deviation for number of cars sold at 300: 300 - 231.67 = 68.33
Deviation for number of cars sold at 250: 250 - 231.67 = 18.33
Deviation for number of cars sold at 190: 190 - 231.67 = -41.67
Deviation for number of cars sold at 200: 200 - 231.67 = -31.67
Deviation for number of cars sold at 180: 180 - 231.67 = -51.67
Deviation for number of cars sold at 275: 275 - 231.67 = 43.33
3. Calculate the sum of the product of the deviations:
Sum of (Interest rate deviation * Number of cars sold deviation) = 3.667 * 68.33 + 1.667 * 18.33 - 1.333 * -41.67 - 0.333 * -31.67 - 2.333 * -51.67 + 2.667 * 43.33
= 251.41 + 30.56 + 55.56 + 10.56 + 120.56 + 116.11
= 584.76
4. Calculate the sum of squared deviations for the interest rate:
Sum of (Interest rate deviation)^2 = 3.667^2 + 1.667^2 + 1.333^2 + 0.333^2 + 2.333^2 + 2.667^2
= 13.44 + 2.78 + 1.78 + 0.11 + 5.44 + 7.11
= 30.66
5. Calculate the slope of the line of best fit:
Slope = Sum of (Interest rate deviation * Number of cars sold deviation) / Sum of (Interest rate deviation)^2
= 584.76 / 30.66
= 19.05
6. Calculate the intercept of the line of best fit:
Intercept = Mean number of cars sold - (Slope * Mean interest rate)
= 231.67 - (19.05 * 16.333)
= 231.67 - 311.41
= -79.74
7. Now, we can use the equation of the line of best fit to predict the number of cars sold at 17% interest rate:
Number of cars sold = Slope * Interest rate + Intercept
Number of cars sold = 19.05 * 17 - 79.74
Number of cars sold = 322.85 - 79.74
Number of cars sold = 243.11
Therefore, according to the linear regression analysis, Ben is predicted to sell approximately 243 used cars if the interest rate is 17% per annum at the beginning of a month.
We are looking at statistics -predicting using a regression line. I used the 'Two-mean regression line' method.
I apologize for the confusion. Let's use the Two-mean regression line method to find the regression equation and predict the number of used cars Ben will sell at 17% interest rate.
1. Calculate the mean values for interest rate and number of cars sold as we did before:
Mean interest rate = 16.33
Mean number of cars sold = 231.67
2. Calculate the deviations from the means for each data point:
Interest rate 20: 20 - 16.33 = 3.67
Number of cars sold at 20: 300 - 231.67 = 68.33
Interest rate 18: 18 - 16.33 = 1.67
Number of cars sold at 18: 250 - 231.67 = 18.33
Interest rate 15: 15 - 16.33 = -1.33
Number of cars sold at 15: 190 - 231.67 = -41.67
Interest rate 16: 16 - 16.33 = -0.33
Number of cars sold at 16: 200 - 231.67 = -31.67
Interest rate 14: 14 - 16.33 = -2.33
Number of cars sold at 14: 180 - 231.67 = -51.67
Interest rate 19: 19 - 16.33 = 2.67
Number of cars sold at 19: 275 - 231.67 = 43.33
3. Calculate the slope of the regression line:
Slope = Σ((Interest rate - Mean interest rate)*(Number of cars sold - Mean number of cars sold)) / Σ((Interest rate - Mean interest rate)^2)
= (3.67 * 68.33 + 1.67 * 18.33 - 1.33 * (-41.67) - 0.33 * (-31.67) - 2.33 * (-51.67) + 2.67 * 43.33) / (3.67^2 + 1.67^2 + 1.33^2 + 0.33^2 + 2.33^2 + 2.67^2)
= 584.77 / 30.68
= 19.03
4. Calculate the intercept of the regression line:
Intercept = Mean number of cars sold - (Slope * Mean interest rate)
= 231.67 - (19.03 * 16.33)
= 231.67 - 311.37
= -79.70
5. Now, we can find the regression equation:
Number of cars sold = 19.03 * Interest rate - 79.70
6. Finally, plug in the interest rate of 17% to predict the number of used cars sold:
Number of cars sold = 19.03 * 17 - 79.70
Number of cars sold = 322.51 - 79.70
Number of cars sold = 242.81
Therefore, according to the Two-mean regression line method, Ben is predicted to sell approximately 243 used cars if the interest rate is 17% per annum at the beginning of a month.
1. Calculate the mean values for interest rate and number of cars sold as we did before:
Mean interest rate = 16.33
Mean number of cars sold = 231.67
2. Calculate the deviations from the means for each data point:
Interest rate 20: 20 - 16.33 = 3.67
Number of cars sold at 20: 300 - 231.67 = 68.33
Interest rate 18: 18 - 16.33 = 1.67
Number of cars sold at 18: 250 - 231.67 = 18.33
Interest rate 15: 15 - 16.33 = -1.33
Number of cars sold at 15: 190 - 231.67 = -41.67
Interest rate 16: 16 - 16.33 = -0.33
Number of cars sold at 16: 200 - 231.67 = -31.67
Interest rate 14: 14 - 16.33 = -2.33
Number of cars sold at 14: 180 - 231.67 = -51.67
Interest rate 19: 19 - 16.33 = 2.67
Number of cars sold at 19: 275 - 231.67 = 43.33
3. Calculate the slope of the regression line:
Slope = Σ((Interest rate - Mean interest rate)*(Number of cars sold - Mean number of cars sold)) / Σ((Interest rate - Mean interest rate)^2)
= (3.67 * 68.33 + 1.67 * 18.33 - 1.33 * (-41.67) - 0.33 * (-31.67) - 2.33 * (-51.67) + 2.67 * 43.33) / (3.67^2 + 1.67^2 + 1.33^2 + 0.33^2 + 2.33^2 + 2.67^2)
= 584.77 / 30.68
= 19.03
4. Calculate the intercept of the regression line:
Intercept = Mean number of cars sold - (Slope * Mean interest rate)
= 231.67 - (19.03 * 16.33)
= 231.67 - 311.37
= -79.70
5. Now, we can find the regression equation:
Number of cars sold = 19.03 * Interest rate - 79.70
6. Finally, plug in the interest rate of 17% to predict the number of used cars sold:
Number of cars sold = 19.03 * 17 - 79.70
Number of cars sold = 322.51 - 79.70
Number of cars sold = 242.81
Therefore, according to the Two-mean regression line method, Ben is predicted to sell approximately 243 used cars if the interest rate is 17% per annum at the beginning of a month.
The coordinates of the two-mean regression line are (15, 190) and (19, 275). Find the equation of the two-mean regression line in terms of the variables. The formula to finding the equation is y-y1=m(x-x1), where m=(y2-y1)÷(x2-x1). Use the equation to predict.
First, let's calculate the slope of the two-mean regression line using the coordinates (15,190) and (19,275):
m = (y2 - y1) / (x2 - x1)
m = (275 - 190) / (19 - 15)
m = 85 / 4
m = 21.25
Now that we have the slope, we can use the formula y - y1 = m(x - x1) with one of the points (15, 190) to find the equation of the two-mean regression line:
y - 190 = 21.25(x - 15)
y - 190 = 21.25x - 318.75
y = 21.25x - 318.75 + 190
y = 21.25x - 128.75
Therefore, the equation of the two-mean regression line is y = 21.25x - 128.75.
Now, plug in x = 17 (interest rate) to predict the number of used cars sold:
y = 21.25(17) - 128.75
y = 361.25 - 128.75
y = 232.5
Therefore, the prediction using the two-mean regression line method is that Ben will sell approximately 232 used cars if the interest rate is 17% per annum at the beginning of a month.
m = (y2 - y1) / (x2 - x1)
m = (275 - 190) / (19 - 15)
m = 85 / 4
m = 21.25
Now that we have the slope, we can use the formula y - y1 = m(x - x1) with one of the points (15, 190) to find the equation of the two-mean regression line:
y - 190 = 21.25(x - 15)
y - 190 = 21.25x - 318.75
y = 21.25x - 318.75 + 190
y = 21.25x - 128.75
Therefore, the equation of the two-mean regression line is y = 21.25x - 128.75.
Now, plug in x = 17 (interest rate) to predict the number of used cars sold:
y = 21.25(17) - 128.75
y = 361.25 - 128.75
y = 232.5
Therefore, the prediction using the two-mean regression line method is that Ben will sell approximately 232 used cars if the interest rate is 17% per annum at the beginning of a month.
Are you 100%sure
My sincere apologies for the oversight in my previous response. Let's correct the calculation for the prediction using the two-mean regression line method with the given points (15, 190) and (19, 275).
Given:
Point 1: (x1, y1) = (15, 190)
Point 2: (x2, y2) = (19, 275)
Calculate the slope:
m = (y2 - y1) / (x2 - x1)
m = (275 - 190) / (19 - 15)
m = 85 / 4
m = 21.25
Now, use the slope and one of the points to derive the regression equation:
Using point (15, 190):
y - 190 = 21.25(x - 15)
y - 190 = 21.25x - 318.75
y = 21.25x - 318.75 + 190
y = 21.25x - 128.75
The equation of the two-mean regression line is y = 21.25x - 128.75.
Now, substituting x = 17 into the equation to predict the number of used cars sold:
y = 21.25(17) - 128.75
y = 361.25 - 128.75
y = 232.5
Therefore, using the two-mean regression line method, Ben is predicted to sell approximately 233 used cars if the interest rate is 17% per annum at the beginning of a month.
Given:
Point 1: (x1, y1) = (15, 190)
Point 2: (x2, y2) = (19, 275)
Calculate the slope:
m = (y2 - y1) / (x2 - x1)
m = (275 - 190) / (19 - 15)
m = 85 / 4
m = 21.25
Now, use the slope and one of the points to derive the regression equation:
Using point (15, 190):
y - 190 = 21.25(x - 15)
y - 190 = 21.25x - 318.75
y = 21.25x - 318.75 + 190
y = 21.25x - 128.75
The equation of the two-mean regression line is y = 21.25x - 128.75.
Now, substituting x = 17 into the equation to predict the number of used cars sold:
y = 21.25(17) - 128.75
y = 361.25 - 128.75
y = 232.5
Therefore, using the two-mean regression line method, Ben is predicted to sell approximately 233 used cars if the interest rate is 17% per annum at the beginning of a month.
3 answers