Question

An engineer wished to test two recyclable materials to compare their suitability for making tyres. She tested two materials, A and B, on the left and right rear positions of ten tractors. The wear on the tread was measured in millimetres of wear, with a lower number indicating less wear. The results are shown in the following table:

Tractor 1 2 3 4 5 6 7 8 9 10

Material A 13.2 8.2 10.9 14.3 10.7 6.6 9.5 10.8 8.8 13.3

Material B 14.0 8.8 11.2 14.2 11.8 6.4 9.8 11.3 9.3 13.6

Difference -0.8 -0.6 -0.3 0.1 -1.1 0.2 -0.3 -0.5 -0.5 -0.3

R was used to obtain the following summary statistics for the differences:

summary(tyres$Difference)

Min. 1st Qu. Median Mean 3rd Qu. Max.

-1.100 -0.575 -0.400 -0.410 -0.300 0.200

sd(tyres$Difference)

[1] 0.3871549

(a) In this experiment, how should the two materials, A and B, be assigned to the two rear positions, left and right, on the tractors? [2 marks] (b) Using the summary statistics, sketch the boxplot of the differences in tyre wear. [3 marks]

(c) Describe the distribution of the differences from (b). [2 marks] (d) We have two sets of tyre wear results, one for Material A and one for Material B. Briefly explain why we work with the differences rather than carrying out a twosample t test to compare the materials. [2 marks] (e) State the null and alternative hypotheses to address the research question. [2 marks]

Answer 1

a) The make the experiment as a random experiment, the two materials A and B should be assigned randomly to the two rear positions. And also, to be sure that there is no tractor effect, one should also try to select ten tractors which are more or less same with respect to other consitions, i.e. there should be homogeneity between them.

b) The boxplot using min, max, 1st and 3rd quartile and median, we get as

c) The distribution of the differences is not symmetric. There are two outlier observations. More over, the mean is lesser than median.

d) In two sample t-test there has to be two independent samples, but here, the two samples arise from a set of tractors, as the two left and right position outcomes come from the same tractor. So, the sample is not different. So, it is a paired sample.

e) Let be population mean of the difference. We want to test