Ben, the owner of a used car 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.

Question

Ben, the owner of a used car 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, 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.

According to the information Ben collected, predict the number of used cars that Ben will sell if the interest rate is 17% per annum at the beginning of a month.

GPT 3.5 · Answer

To predict the number of used cars that Ben will sell if the interest rate is 17% per annum at the beginning of a month, we can use linear regression to fit a line to the data provided.

First, we need to calculate the slope of the line:

Slope, m = (n * Σ(xy) - Σx * Σy) / (n * Σ(x^2) - (Σx)^2)

where n is the number of data points, Σxy is the sum of the product of interest rate and number of cars sold, Σx is the sum of interest rates, Σy is the sum of number of cars sold, and Σ(x^2) is the sum of squares of interest rates.

Using the provided data:

n = 6
Σxy = 20*300 + 18*250 + 15*190 + 16*200 + 14*180 + 19*275 = 10850
Σx = 20 + 18 + 15 + 16 + 14 + 19 = 102
Σy = 300 + 250 + 190 + 200 + 180 + 275 = 1395
Σ(x^2) = 20^2 + 18^2 + 15^2 + 16^2 + 14^2 + 19^2 = 1900

Therefore, the slope, m = (6*10850 - 102*1395) / (6*1900 - 102^2) = 0.55

Now, we can use the slope to calculate the y-intercept of the line:

y-intercept, b = (Σy - m * Σx) / n = (1395 - 0.55 * 102) / 6 = 181.5

Now that we have the equation of the line (y = mx + b), we can plug in the interest rate of 17% to find the predicted number of cars sold:

Number of cars sold = 0.55 * 17 + 181.5 = 190.95

Therefore, Ben is predicted to sell around 191 used cars if the interest rate is 17% per annum at the beginning of a month.

Answer

Are you sure

GPT 3.5 · Answer

Let's recalculate the slope and y-intercept using simple linear regression formula to predict the number of used cars sold at a 17% interest rate:

First, calculate the mean values of interest rate and number of cars sold:

Mean of interest rate (X̄) = (20 + 18 + 15 + 16 + 14 + 19) / 6 = 102 / 6 = 17
Mean of number of cars sold (Ȳ) = (300 + 250 + 190 + 200 + 180 + 275) / 6 = 1395 / 6 = 232.5

Calculate the sums needed for the formula for slope:

Σ((interest rate - X̄) * (number of cars sold - Ȳ)) = (20-17)*(300-232.5) + (18-17)*(250-232.5) + (15-17)*(190-232.5) + (16-17)*(200-232.5) + (14-17)*(180-232.5) + (19-17)*(275-232.5) = 1266

Σ((interest rate - X̄)^2) = (20-17)^2 + (18-17)^2 + (15-17)^2 + (16-17)^2 + (14-17)^2 + (19-17)^2 = 14

Now, calculate the slope:

Slope, m = Σ((interest rate - X̄) * (number of cars sold - Ȳ)) / Σ((interest rate - X̄)^2) = 1266 / 14 = 90.43

Next, calculate the y-intercept:

y-intercept, b = Ȳ - m * X̄ = 232.5 - 90.43 * 17 = 684.41

Predict the number of cars sold at 17% interest rate:

Number of cars sold = m * 17 + b = 90.43 * 17 + 684.41 = 1537.5

Therefore, Ben is predicted to sell around 1538 used cars if the interest rate is 17% per annum at the beginning of a month.