A case-control (or retrospective) study was conductedto investigate a relationship between the colors of helmets worn by motorcycle drivers andwhether they are injured or killed in a crash. Results are given in the table below (based on datafrom “Motorcycle Rider Conspicuity and Crash Related Injury: Case-Control Study,” by Wellset al., BMJ USA,Vol. 4). Test the claim that injuries are independent of helmet color. Shouldmotorcycle drivers choose helmets with a particular color? If so, which color appears best?

	Color of helmet
	Black	White	Yellow/Orange	Red	Blue
Controls (not injured)	491	377	31	170	55
Cases (injured or killed)	213	112	8	70	26

Data for relationship between the colors of helmets worn by motorcycle drivers and their injuries or deaths in crashes.

Theexpected frequency formulais,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The observation table with row and column total is,

Theexpected frequency tableis represented as,

Assume the experimental units are randomly selected.

The expected frequencies are greater than 5.

Thus, the requirements of the test are satisfied.

The hypotheses are formulated as follows:

\({H_0}:\)Injuries are independent of helmet color.

\({H_1}:\)Injuries are dependent on helmet color.

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {491 - 509.5274} \right)}^2}}}{{509.5274}} + \frac{{{{\left( {377 - 353.9189} \right)}^2}}}{{353.9189}} + ... + \frac{{{{\left( {26 - 22.3754} \right)}^2}}}{{22.3754}}\\ = 9.971\end{aligned}\]

Therefore, the value of the test statistic is 9.971.

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {5 - 1} \right)\\ = 4\end{aligned}\)

Therefore, the degrees of freedom are 4.

From the chi-square table, the critical value corresponding to 4 degrees of freedom and at 0.05 level of significance 9.488.

Therefore, the critical value is 9.488.

The p-value is obtained as 0.041.

Since the critical value (9.488) is less than the value of the test statistic (9.971). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

There isinsufficient evidence to support the claim that Injuries are independent of helmet color.

Thus, the helmet color is dependent on the injuries.

Therefore, the motocycle drivers must choose a particular color to avoid injuries. From the sample data, the proportion of least controls (not injured) under each of the color categories is:

Thus, lowest proportion of no injuries occurred when subjects wore blue helmets.