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The repair costs are tabulated for cars crashed at 6 mi/h in full-front crashes and for cars crashed at 6mi/h in full rear crashes.

The null hypothesis is as follows:

There is no linear correlation between the repair costs of two different crashes.

The alternative hypothesis is as follows:

There is a linear correlation between the repair costs of two different crashes.

Let x denote the repair costs of cars crashed at six mi/h in full-front crashes.

Let y denote the repair costs of cars crashed at six mi/h in full rear crashes.

Here, n is equal to 7.

The formula for computing the correlation coefficient (r) is as follows:

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

The following calculations are done to compute the value of r:

Substituting the above values, the following value of r is obtained:

\[\begin{aligned}{c}r = \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\\ = \frac{{7\left( {24833485} \right) - \left( {16890} \right)\left( {11303} \right)}}{{\sqrt {7\left( {53892334} \right) - {{\left( {16890} \right)}^2}} \sqrt {7\left( {23922183} \right) - {{\left( {11303} \right)}^2}} }}\\ = - 0.283\end{aligned}\]

Therefore, the value of r is equal to -0.283.

The test statistic value is computed as follows:

\(\begin{aligned}{c}t = \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\;\;\; \sim {t_{\left( {n - 2} \right)}}\\ = \frac{{ - 0.283}}{{\sqrt {\frac{{1 - {{\left( { - 0.283} \right)}^2}}}{{7 - 2}}} }}\\ = - 0.659\end{aligned}\)

Here, n=7.

The degrees of freedom are computed as follows:

\(\begin{aligned}{c}df = n - 2\\ = 7 - 2\\ = 5\end{aligned}\)

The critical correlation value of r for\(\alpha = 0.05\)and degrees of freedom equal to 5 is equal to 0.754.

The p-value obtained dusing the test statistic is equal to 0.539.

Since the absolute value of the test statistic is less than the critical value and the p-value is greater than 0.05, so the null hypothesis is rejected.

Therefore, there is no linear correlation between the repair costs of the two types of car crashes.