- Null hypothesis: $H_0$: There is no significant difference in the accident rates between red and yellow fire trucks.
- Alternative hypothesis: $H_1$: The accident rate of red fire trucks is significantly lower than that of yellow fire trucks.
We will use the chi-square test of independence to test the hypothesis. The expected frequencies for each category can be calculated as follows:
| | Red Trucks | Yellow Trucks | Total |
|---|---|---|---|
| Accidents | 20 | 80 | 100 |
| No Accidents | 153328 | 134955 | 134983 |
| Total | 153348 | 135035 | 135083 |
The chi-square statistic is calculated as:
$$\chi^2 = \sum (O_i - E_i)^2 / E_i$$
where $O_i$ is the observed frequency and $E_i$ is the expected frequency.
The degrees of freedom for the chi-square test is calculated as:
$$df = (r-1)(c-1)$$
where $r$ is the number of rows and $c$ is the number of columns.
In this case, we have $r=2$ rows and $c=2$ columns, so the degrees of freedom is:
$$df = (2-1)(2-1) = 1$$
Using a chi-square table or calculator, we find that the critical value for a chi-square test with 1 degree of freedom and a significance level of 0.01 is 6.635.
The calculated chi-square statistic is:
$$\chi^2 = (20-25)^2/25 + (80-75)^2/75 + (153328-153323)^2/153323 + (134955-134960)^2/134960 \\= 5.16$$
Since the calculated chi-square statistic (5.16) is less than the critical value for the chi-square test (6.635), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the red fire trucks have a significantly lower accident rate than the yellow fire trucks at a significance level of 0.01.
