Answer :
To solve the problems about the relationship between the speed of a car (X) and the fuel consumption rates (Y), we need to tackle each part step by step.
a) Find the standard deviation of X and Y.
The standard deviation [tex]s[/tex] is the square root of the variance. Given:
Variance of X [tex]s^2_x = 16.17[/tex]
Variance of Y [tex]s^2_y = 198[/tex]
The standard deviations are:
[tex]s_x = \sqrt{16.17} = 4.02[/tex]
[tex]s_y = \sqrt{198} = 14.07[/tex]
b) Compute the correlation coefficient [tex]r[/tex], and comment on the value obtained.
The correlation coefficient [tex]r[/tex] is calculated using the formula:
[tex]r = \frac{n \Sigma XY - \Sigma X \Sigma Y}{\sqrt{(n \Sigma X^2 - (\Sigma X)^2)(n \Sigma Y^2 - (\Sigma Y)^2)}}[/tex]
Substituting the given values:
[tex]r = \frac{10 \times 3751.7 - 419 \times 97.4}{\sqrt{(10 \times 19909 - 419^2)(10 \times 1006.8 - 97.4^2)}}[/tex]
Calculating, we find:
[tex]r = -0.31[/tex]
This negative correlation indicates that as the speed increases, the fuel consumption rate decreases slightly.
c) Which variable (X or Y) has more variation? Determine by the coefficient of variation (c.v.).
The coefficient of variation [tex]c.v.[/tex] is calculated as follows:
[tex]c.v._x = \left(\frac{s_x}{\bar{X}}\right) \times 100[/tex]
[tex]c.v._y = \left(\frac{s_y}{\bar{Y}}\right) \times 100[/tex]
Where [tex]\bar{X} = \frac{\Sigma X}{n} = 41.9[/tex] and [tex]\bar{Y} = \frac{\Sigma Y}{n} = 9.74[/tex].
[tex]c.v._x = \left(\frac{4.02}{41.9}\right) \times 100 \approx 9.6\%[/tex]
[tex]c.v._y = \left(\frac{14.07}{9.74}\right) \times 100 \approx 144.5\%[/tex]
Hence, Y has more variation.
d) Find the equation of the relationship between X and Y.
The equation of the linear relationship is expressed as:
[tex]Y = a + bX[/tex]
We use the given slope [tex]b = -0.14[/tex]. To find the intercept [tex]a[/tex]:
[tex]a = \bar{Y} - b\bar{X}[/tex]
[tex]= 9.74 - (-0.14 \times 41.9)[/tex]
[tex]= 9.74 + 5.866[/tex]
[tex]= 15.606[/tex]
Thus, the equation is:
[tex]Y = 15.606 - 0.14X[/tex]
e) Find the coefficient of determination [tex]r^2[/tex] and comment on its value.
The coefficient of determination [tex]r^2[/tex] is the square of the correlation coefficient:
[tex]r^2 = (-0.31)^2 = 0.0961[/tex]
This means that only about 9.61% of the variation in fuel consumption rates can be explained by car speed, indicating a weak relationship.
