Question

The data below are the head injury readings for driver-side crash dummies for various classes of vehicles. The vehicle types are coded as 1=Compact, 2=Heavy, 3=Light, 4=Medium, 5=MPV, 6=Pickup, and 7=Van. Larger numbers indicate more serious injuries.

Injury  Type
791     1
846     1
1024    1
1007    1
399     1
1279    1
407     1
1656    1
1036    1
1226    1
423     2
541     2
517     2
1328    2
471     2
533     2
863     2
786     2
551     2
1068    2
551     3
1068    3
757     3
1114    3
920     3
1809    3
1238    3
918     3
1339    3
603     3
712     4
435     4
298     4
733     4
1200    4
1701    4
707     4
790     4
800     4
480     4
1345    5
1269    5
1477    5
758     5
996     5
1306    5
968     5
943     5
1026    5
1564    5
985     6
1074    6
742     6
985     6
1342    6
1184    6
977     6
1465    6
892     6
1074    6
805     7
2613    7
903     7
949     7
1183    7
1051    7
1387    7
1320    7
1434    7
1603    7

(a) Apply the Kruskal-Wallis test to this data. State your test statistic and p-value using the approximate chi-square distribution. What is your conclusion if α = 0.05?

Answer 1

To test: No. of head injuries differs with the type of vehicle.

To run a Kruskal Wallis test:

First, sort the data (Injury), and rank them:

Write the corresponding ranks accross each observation in the data:

Find the sum of ranks for each group.Here, for group 1, $R_{1}=22+25+.....+41+51=342$ . Similarly, find the sum of ranks for each groups.

Let N be the total no. of observations.Here, N = 70.

The test statistic is given by:

$H=\frac{12}{N(N+1) }\sum\frac{R_{i}^{2}}{N_{i}}$

We have more than 5 datapoints per group.Hence, we may treat H as chi square.

Substituting the values, we get the test statistic value H = 18.3. Next, we compare the test statistic value obtained with the critical chi square value for n-1 = 8 - 1 = 7 df. ( $\chi ^{2}_{0.05,7}=14.07$ )

(Since the sample size is large, we interpret the result using the p value using the approximate chi-square distribution.)

As the calculated test statistic 18.3 > 14.07, we do not have sufficient evidence to support the null hypothesis that the no. of injuries does nor differ with the type of vehicles.Hence, reject H₀ at 5% level.

Running, kruskal wallis test in SPSS,

The output is obtained as:

Since, the p value is less than 0.05, we reject H_0. We may conclude that no. of injuries differs significantly, accross different class of vehicles.