Part A. Find the equation of the trend line.
Years:1, 2, 3, 4, 5
Cars sold: 15, 20, 23, 30, 34
Part B. The correlation coefficient r is:
Part C. The predicted number of cars sold in year 10 is_____
To find the equation of the trend line, we need to calculate the slope and the y-intercept.
Step 1: Calculate the mean of the x-values (years) and the y-values (cars sold).
Mean of years (x-values): (1 + 2 + 3 + 4 + 5)/5 = 3
Mean of cars sold (y-values): (15 + 20 + 23 + 30 + 34)/5 = 24.4
Step 2: Calculate the differences between each year and the mean of years (x-values) and between each number of cars sold and the mean of cars sold (y-values).
Year 1 - mean of years: 1 - 3 = -2
Year 2 - mean of years: 2 - 3 = -1
Year 3 - mean of years: 3 - 3 = 0
Year 4 - mean of years: 4 - 3 = 1
Year 5 - mean of years: 5 - 3 = 2
Cars sold in year 1 - mean of cars sold: 15 - 24.4 = -9.4
Cars sold in year 2 - mean of cars sold: 20 - 24.4 = -4.4
Cars sold in year 3 - mean of cars sold: 23 - 24.4 = -1.4
Cars sold in year 4 - mean of cars sold: 30 - 24.4 = 5.6
Cars sold in year 5 - mean of cars sold: 34 - 24.4 = 9.6
Step 3: Calculate the product of the differences for each x-value and y-value.
Product of (-2) * (-9.4) = 18.8
Product of (-1) * (-4.4) = 4.4
Product of (0) * (-1.4) = 0
Product of (1) * (5.6) = 5.6
Product of (2) * (9.6) = 19.2
Step 4: Calculate the sum of the product of the differences.
18.8 + 4.4 + 0 + 5.6 + 19.2 = 48
Step 5: Calculate the sum of the squared differences for each x-value.
(-2)^2 = 4
(-1)^2 = 1
(0)^2 = 0
(1)^2 = 1
(2)^2 = 4
4 + 1 + 0 + 1 + 4 = 10
Step 6: Calculate the slope of the trend line.
Slope (m) = Sum of product of differences / Sum of squared differences
Slope (m) = 48/10 = 4.8
Step 7: Calculate the y-intercept of the trend line.
y-intercept (b) = mean of cars sold - (slope * mean of years)
y-intercept (b) = 24.4 - (4.8 * 3) = 24.4 - 14.4 = 10
Therefore, the equation of the trend line is y = 4.8x + 10.
To find the correlation coefficient r, we can use the formula:
r = (nΣxy - ΣxΣy) / sqrt((nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2))
Using the values from the calculations above:
n = 5 (number of data points)
Σxy = 48 (sum of the product of differences)
Σx = 0 (sum of the differences between x-values and the mean of x-values)
Σy = 0 (sum of the differences between y-values and the mean of y-values)
Σx^2 = 10 (sum of the squared differences for x-values)
Σy^2 = 221.2 (sum of the squared differences for y-values)
r = (5*48 - 0) / sqrt((5*10 - 0)(5*221.2 - 0))
r = 240 / sqrt(50*1106)
r = 240 / sqrt(55300)
Using a calculator, we find that r ≈ 0.954.
Therefore, the correlation coefficient r is approximately 0.954.
To find the predicted number of cars sold in year 10, substitute x = 10 into the equation of the trend line:
y = 4.8x + 10
y = 4.8 * 10 + 10
y = 48 + 10
y = 58
Therefore, the predicted number of cars sold in year 10 is 58.