Question

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

Short answer

Injuries are dependent on helmet color.

The proportion of the subjects that were not injured was least when the subjects wore blue color.

Step-by-step solution

Given information

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

Check the requirements of the test

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,

	Black	White	Yellow/Orange	Red	Blue	Row total
Controls(not injured)	491	377	31	170	55	1124
Cases(injured or killed)	213	112	8	70	26	429
Column Total	704	489	39	240	81	1553

Theexpected frequency tableis represented as,

	Black	White	Yellow/ Orange	Red	Blue
Controls (not injured)	509.5274	353.9189	28.2267	173.7025	58.6246
Cases (injured or killed)	194.4726	135.0811	10.7733	66.2975	22.3754

Assume the experimental units are randomly selected.

The expected frequencies are greater than 5.

Thus, the requirements of the test are satisfied.

Formulate the hypotheses

The hypotheses are formulated as follows:

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

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

Compute the test statistic

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.

Compute the degrees of freedom

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.

Compute the critical value

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.

State the decision

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.

State the conclusion

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:

	Black	White	Yellow/Orange	Red	Blue
Controls(not injured)	0.6974\(\left( {\frac{{491}}{{704}}} \right)\)	0.7710 \(\left( {\frac{{377}}{{489}}} \right)\)	0.7949 \(\left( {\frac{{31}}{{39}}} \right)\)	0.7083 \(\left( {\frac{{170}}{{240}}} \right)\)	0.6790 \(\left( {\frac{{55}}{{81}}} \right)\)

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