Question

Underinflated or overinflated tires can increase tire wear and decrease gas mileage. A manufacturer of a new tire tested the tire for wear at different pressures. Construct scatterplot and calculate Pe, arson. Corr. Coeff. The reasearcher for this study calcualted a pearson correlation coefficient and concluded no relationships existed between the two variables. Was he correct in his conclusion?
col1 Pressure - 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36
col2 Milege (thous) - 29.5, 30.2, 32.1, 34.5, 36.3, 35, 38.2, 37.6, 37.7, 36.1, 33.6, 34.2, 26.8, 27.4

Answer 1

Answer:

The correlation coefficient between pressure and mileage is -0.1137.

The researcher was not correct in his conclusion.

Step-by-step explanation:

The correlation coefficient is a statistical degree that computes the strength of the linear relationship amid the relative movements of the two variables (i.e. dependent and independent).

It is a mathematical measure of certain kind of correlation, in sense a statistical relationship amid two variables. It ranges from -1 to +1.

A correlation coefficient of 0 implies that there is no relationship between the two variables.

Let X = pressure and Y = mileage.

The formula to compute the Pearson's correlation coefficient is:

$\begin{aligned}r~&=~\frac{n\cdot\sum{XY} - \sum{X}\cdot\sum{Y}} {\sqrt{\left[n \sum{X^2}-\left(\sum{X}\right)^2\right] \cdot \left[n \sum{Y^2}-\left(\sum{Y}\right)^2\right]}} \\\end{aligned}$

The values of $\sum X,\ \sum,\ \sum Y,\ \sum X^{2},\ \sum Y^{2}\ and\ \sum XY$ are computed in the table below.

Compute the correlation coefficient between pressure and mileage as follows:

$\begin{aligned}r~&=~\frac{n\cdot\sum{XY} - \sum{X}\cdot\sum{Y}} {\sqrt{\left[n \sum{X^2}-\left(\sum{X}\right)^2\right] \cdot \left[n \sum{Y^2}-\left(\sum{Y}\right)^2\right]}} \\r~&=~\frac{ 14 \cdot 15472 - 462 \cdot 469.2 } {\sqrt{\left[ 14 \cdot 15302 - 462^2 \right] \cdot \left[ 14 \cdot 15910.74 - 469.2^2 \right] }} \approx -0.1137\end{aligned}$

The correlation coefficient between pressure and mileage is -0.1137.

Since the correlation coefficient is not equal to 0, there exist a relationship between the two variables.

The correlation coefficient value is negative. This implies that there is an inverse relationship between the two variables.

That is, as the pressure increases the mileage decreases.

Thus, the researcher was not correct in his conclusion.