Question

A construction company is suspicious that the alloy beams they are receiving at a construction site have masses that are on average lighter than what the manufacturer claims to be sending out. They asked for an audit of beam weight measurements from the manufacturer to compare with their on- site weights. The results from their comparisons are given in the table below: Alloy sample 1 34 Manufacturer's mass (kg) 165 201 192 198 155 141 150 187 161 195 193 193 150 143 146 183 On-site mass (kg) i. Is there any evidence that the on-site weights are lighter on average than the manufacturer's reported weights? Answer this question by carrying out an appropriate hypothesis test, clearly stating the null and alternative hypotheses, computing and interpreting a P-value, and formulating a conclusion

Answer 1

wilcoxon signed rank test

median difference = manufacture'mass - on site mass

Ho: median difference=0

H1: median difference>0

sample 1	sample 2	difference=sample1-sample 2	absolue difference	rank	rank if positive	rank if negative
165	161	4	4	4	4
201	195	6	6	8	8
192	193	-1	1	1		1
198	193	5	5	6.5	6.5
155	150	5	5	6.5	6.5
141	143	-2	2	2		2
150	146	4	4	4	4
187	183	4	4	4	4

number of non zero difference , n =    8
  
sum of positive ranks, W+ =    33
sum of negative ranks , W- =   3
  
T = min(W+, W-) =  3

The critical value for the signficance level \alpha = 0.05α=0.05 provided, and the type of tail specified is T^* = 5, and the null hypothesis is rejected if T≤5.

Since in this case T = 3 ≤ 5 , reject Ho

conclusion: there is enough evidence that on site weights are lighter on average than manufacture's reported weights at α=0.05