Question

Are the proportions of road accidents similar in various highways of Bangladesh?                    

Highways

1

2

3

4

Total

No. of road accidents (O_i)

50

42

32

82

206

Answer 1

The hypothesis tested in this question are,

$H_0:$  Proportions of road accidents are similar.

$Against$  

$H_1:$  Proportions of road accidents are not similar.

To perform this test we shall the chi-square goodness of fit test.

If the null hypothesis is true, then the proportion for each highway should be the same. Since there are 4 highways, the proportion for each highway is $p=1/4$ .

We proceed to find the expected count for each highway as follows,

$E_k=n*p$ for $k=1,2,3,4$ and $n=206$

$E_1=206*1/4=51.5$

$E_2=206*1/4=51.5$

$E_3=206*1/4=51.5$

$E_5=206*1/4=51.5$

The test statistic is given as,

$\chi^2_c=\displaystyle \sum^4_{k=1}(O_k-E_k)^2/E_k$

$\chi^2_c=(50-51.5)^2/51.5+(42-51.5)^2/51.5+(32-51.5)^2/51.5+(82-51.5)^2/51.5= 27.24272$

$\chi^2_c$ is compared with the chi square table value at $\alpha=5\%$ with $(k-1)=(4-1)=3$, where $k$ is the number of highways.

The table value is given as, $\chi^2_{\alpha, 3}=\chi^2_{0.05,3}=7.81473$ and the null hypothesis is rejected if $\chi^2_{c}\gt \chi^2_{0.05,3}$.

Since $\chi^2_{c}=27.24272\gt \chi^2_{0.05.3}=7.81473$ the null hypothesis is rejected and we conclude that there is not enough evidence to show that the proportions of road accidents are similar in various highways of Bangladesh at 5% level of significance.