Question

A company has just built a new factory to manufacture their widgets with a method they believe should be more efficient than their old method on a daily basis. To compare they recorded the number of widgets produced by the old factory for the last 8 days before it was shutdown and the first 9 days of the new factory's production. The company has reason to believe that daily production of widgets does not follow the normal distribution and have requested the use of the Wilcoxon rank sum test.

New Factory:	Old Factory:
275	280
228	219
238	267
314	255
238	240
295	297
246	293
251	265
297

What would be the correct hypothesis test assuming that μ₁ is the mean number produced by the old factory and μ₂ is the mean number produced by the new factory?

H₀: μ₁ = μ₂ vs. H_a: μ₁ ≠ μ₂
H₀: μ₁ = μ₂ vs. H_a: μ₁ > μ₂
H₀: μ₁ > μ₂ vs. H_a: μ₁ = μ₂
H₀: μ₁ = μ₂ vs. H_a: μ₁ < μ₂

What rank should the value 267 from the old factory data have?

Calculate μ_W.   and Calculate the test statistic W

What is the approximate p value? (Hint: treat W as normally distributed)

Answer 1

H₀: μ₁ = μ₂ vs. H_a: μ₁ < μ₂

_........

Trt A	Trt B	rank for sample 1	rank for sample 2
280	275	12	11
219	228	1	2
267	238	10	3.5
255	314	8	17
240	238	5	3.5
297	295	15.5	14
293	246	13	6
265	251	9	7
	297		15.5

10th rank should the value 267 from the old factory data have

..........

Trt A      
sample size ,    n1 =    8
sum of ranks ,    R1 =    73.5
      
Trt B      
sample size ,    n2 =    9
sum of ranks ,   R2 =    79.5
      
W=sum of ranks for smaller sample size =       73.5
      
mean ,µ =    n1(n1+n2+1)/2 =    72
      
std dev,σ =    √(n1*n2*(n1+n2+1)/12) =   10.3923
      
Z-stat =    (W - µ)/σ =    0.1443
      
      
P-value =      0.4426
.............

Please let me know in case of any doubt.

Thanks in advance!

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