91Ó°ÊÓ

The data is recorded for the two variables, old and new ratings.

A plot described using a paired set of observations is known as a scatterplot.

It givesa tentative relationship between two variables.

Steps to sketch a scatterplot:

1. Mark the horizontal for old ratings and the vertical for new ratings.
2. Mark each point in a pair onto the curve.

The resultant scatterplot is shown below.

The correlation coefficient formula is

\(r = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}} \sqrt {n\left( {\sum {{y^2}} } \right) - {{\left( {\sum y } \right)}^2}} }}\).

Define variable xas old ratings and variable y as new ratings.

The valuesare listedin the table below:

Substitute the values in the formula:

\(\begin{aligned} r &= \frac{{11\left( {5569} \right) - \left( {225} \right)\left( {229} \right)}}{{\sqrt {11\left( {6213} \right) - {{\left( {255} \right)}^2}} \sqrt {11\left( {4993} \right) - {{\left( {229} \right)}^2}} }}\\ &= 0.998\end{aligned}\)

Thus, the correlation coefficient is 0.998.

Definethe actual measure of the correlation coefficient as\(\rho \).

For testing the claim, form the hypotheses:

\(\begin{array}{l}{H_o}:\rho = 0\\{H_a}:\rho \ne 0\end{array}\)

The samplesize is 11 (n).

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\\ &= \frac{{0.998}}{{\sqrt {\frac{{1 - {{\left( {0.998} \right)}^2}}}{{11 - 2}}} }}\\ &= 47.363\end{aligned}\)

Thus, the test statistic is 47.363.

The degree of freedom is

\(\begin{aligned} df &= n - 2\\ &= 11 - 2\\ &= 9.\end{aligned}\)

The p-value is computed from the t-distribution table.

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > 47.363} \right)\\ &= 0.000\end{aligned}\)

Thus, the p-value is 0.000.

Since thep-value is less than 0.05, the null hypothesis is rejected.

Therefore, there is enough evidence to conclude that old ratings are linearly correlated with new ratings.

The data suggests that throughout the period, the old ratings were higher than the new rating for each car.