A case control study was conducted to investigate a relationship between the colors of helmets born by motorcycle drivers and whether they are entertaining crash. Results are given in the table. Use 80.05 significance level to test the claim. The injuries are independent of helmet color. Find the test, statistics and critical value. State the conclusion, color of helmets, black, white, yellow, orange, red, blue. Not injured 491, 377, 31, 170, 55. Injured, 213, 112, eight, 70, 26.
To test the claim that the color of helmets worn by motorcycle drivers is related to whether they are injured in a crash, we can use a chi-square test of independence.
First, we need to set up a contingency table:
| | Not Injured | Injured |
|-------------|-------------|---------|
| Black | 491 | 213 |
| White | 377 | 112 |
| Yellow | 31 | 8 |
| Orange | 170 | 70 |
| Red | 55 | 26 |
Next, we calculate the expected values for each cell in the contingency table assuming independence:
Expected value = (row total * column total) / grand total
For example, the expected value for the first cell (black helmet, not injured) would be:
(704 * 1060) / 2068 = 359.26
We calculate the expected values for each cell in the table and then calculate the chi-square test statistic:
chi-square = Σ ((observed - expected)^2 / expected)
The degrees of freedom for a chi-square test of independence with a 5x2 contingency table are given by df = (number of rows - 1) * (number of columns - 1) = (5-1)*(2-1) = 4
We then compare the calculated chi-square test statistic with the critical value from the chi-square distribution with 4 degrees of freedom at a significance level of 80.05%.
Finally, we state the conclusion based on the comparison of the test statistic with the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis that injuries are independent of helmet color. If the test statistic is less than the critical value, we fail to reject the null hypothesis.