The selling of prices Of six second hand bicycles in relation to their ages were noted for advertisements appearing in a local newspaper . The details are given in the table bellow.
Age x ( years) 4 10 2 1 3 4
Price, y. 9 1 14 18 12 7
( a) calculate the value of the product - moment correlation coefficient and state wag you infer from its value.
( b) fit a regression line to a model y equals a ➕ bx for the above data
http://www.alcula.com/calculators/statistics/correlation-coefficient/
http://www.alcula.com/calculators/statistics/linear-regression/
To calculate the product-moment correlation coefficient, also known as Pearson's correlation coefficient, we need to follow these steps:
1. Calculate the mean of both the age (x) and price (y) columns.
Mean of x (age): (4 + 10 + 2 + 1 + 3 + 4) / 6 = 4.67
Mean of y (price): (9 + 1 + 14 + 18 + 12 + 7) / 6 = 10.17
2. Calculate the deviations for both x and y by subtracting the mean from each value.
Deviations for x: (4 - 4.67) = -0.67, (10 - 4.67) = 5.33, (2 - 4.67) = -2.67,
(1 -4.67) = -3.67, (3 - 4.67) = -1.67, (4 - 4.67) = -0.67
Deviations for y: (9 - 10.17) = -1.17, (1 - 10.17) = -9.17, (14 - 10.17) = 3.83,
(18 - 10.17) = 7.83, (12 - 10.17) = 1.83, (7 - 10.17) = -3.17
3. Calculate the product of each pair of deviations.
Products of deviations: (-0.67 * -1.17), (5.33 * -9.17), (-2.67 * 3.83),
(-3.67 * 7.83), (-1.67 * 1.83), (-0.67 * -3.17)
4. Sum up all the products of deviations.
Sum of products of deviations = -2.67 + (-48.81) + (-10.22) + (-28.68) + (-3.06) + 2.12 = -91.32
5. Calculate the sum of the squares of the deviations for both x and y.
Sum of squares of deviations for x: (-0.67)^2 + (5.33)^2 + (-2.67)^2 + (-3.67)^2 + (-1.67)^2 + (-0.67)^2
= 0.4489 + 28.4089 + 7.1289 + 13.4689 + 2.7889 + 0.4489 = 52.6844
Sum of squares of deviations for y: (-1.17)^2 + (-9.17)^2 + (3.83)^2 + (7.83)^2 + (1.83)^2 + (-3.17)^2
= 1.3689 + 84.0289 + 14.6689 + 61.1889 + 3.3489 + 10.0489 = 175.65
6. Calculate the square root of the product of the sums of squares of deviations for both x and y.
Square root of (sum of squares of deviations for x) * (sum of squares of deviations for y) =
sqrt(52.6844 * 175.65) = sqrt(9240.75946) = 96.03
7. Calculate the product-moment correlation coefficient (r) using the formula:
r = (sum of products of deviations) / (square root of (sum of squares of deviations for x) * (sum of squares of deviations for y))
r = -91.32 / 96.03 = -0.95 (approximately)
Now, let's interpret the result:
(a) The value of the product-moment correlation coefficient (r) is -0.95. The range of possible values for r is between -1 and 1. In this case, -0.95 indicates a strong negative correlation between the age and price of the second-hand bicycles. This means that as the age of the bicycles increases, the price tends to decrease.
To fit a regression line to the model y = a + bx for the given data, we need to calculate the coefficient estimates (a and b) using the following formulas:
b = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)^2)
a = ȳ - b * x̄
where Σ denotes the sum, x̄ represents the mean of x, and ȳ represents the mean of y.
Let's calculate:
Σ((x - x̄)(y - ȳ)) = (-0.67 * -1.17) + (5.33 * -9.17) + (-2.67 * 3.83) + (-3.67 * 7.83) + (-1.67 * 1.83) + (-0.67 * -3.17) = -91.32 (same as before)
Σ((x - x̄)^2) = (0.4489) + (28.4089) + (7.1289) + (13.4689) + (2.7889) + (0.4489) = 52.6844 (same as before)
Using the formulas, we can calculate:
b = -91.32 / 52.6844 = -1.735 (approximately)
a = 10.17 - (-1.735 * 4.67) = 17.68 (approximately)
Therefore, the regression line model for the given data is y = 17.68 - 1.735x.