Question

3) Listed below are prices of a certain model automobile and the age of the automobile in years. 1 1 Age Price 3 4 2 2 3 4 15,500 14,995 30,795 28,995 23,995 20,900 20,500 3 4 19,995 19,888 29,995 A scatterplot of the table data shows an approximately linear relationship between age and price. The linear correlation coefficient is r = -0.8632, and the regression equation is û = 34,052 – 4257.8x. (a) At the 0.05 significance level, is there sufficient evidence to support a claim of linear correlation between the automobile's age and price? (b) Find the predicted price of a 2-year-old automobile. (c) What percentage of total variation that can be explained by the linear relationship between price and age? Write your answer as a sentence.

Answer 1

a) r = -0.8632, n = 10, Degrees of freedom: df = n-2 = 8, Level of significance = 0.05

H₀: ρ = 0, There is no linear correlation between the automobile's age and price

H₁: ρ ≠ 0, There is a linear correlation between the automobile's age and price

Test statistic: t = r*((1-r*r)/(n-2))^0.5 = -0.8632*((1-(-0.8632^2))/(10-2))^0.5 = -0.154

Critical value (Using Excel function T.INV.2T(probability,df))) = -T.INV.2T(0.05,8) = -2.306

Since test statistic is more than critical value, we reject the null hypothesis.

So, there is a linear correlation between the automobile's age and price.

b) y = 34502-4257.8x

Automobile's age: x = 2

Price = 34502-4257.8*2 = 34502-8515.6 = 25986.4

c) R-square = r*r = -0.8632^2 = 0.7451

Percentage of total variation that can be exlained by the linear relationship between price and age = 74.51%