Question

Data on corneal thickness in microns for 8 patients with glaucoma in one eye but not in the other is shown below. Patient 1 2 3 4 5 6 7 8 Normal 484 478 492 444 436 398 464 476 Galucoma 488 478 480 426 440 410 458 460 a) At alpha = 0.10, do the data suggest that mean corneal thickness is greater in normal eyes than in eyes with glaucoma? Assume that paired differences are normally distributed. (Use the critical value approach and the P-value approach). b) Find a 99% confidence interval for the difference between the mean corneal thickness of normal and glaucoma affected eyes. c) Use the Paired Wilcoxcon Signed-Rank Test at alpha = 0.10.

Answer 1

a)

Sample #1	Sample #2	difference , Di =sample1-sample2
484	488	-4
478	478	0
492	480	12
444	426	18
436	440	-4
398	410	-12
464	458	6
476	460	16

	sample 1	sample 2	Di
sum =	3672	3640	32
mean=	459	455	4

Ho :   µd=   0
Ha :   µd > 0
      
Level of Significance ,    α =    0.01
      
sample size ,    n =    8
      
mean of sample 1,    x̅1=   459.0000
      
mean of sample 2,    x̅2=   455.0000
      
mean of difference ,    D̅ =   4
      
std dev of difference , Sd =        10.743769

      
std error , SE =    Sd / √n =    3.7985
      
t-statistic =    (D̅ - µd)/SE =    1.0530

      
Degree of freedom, DF=   n - 1 =    7
t-critical value , t* = 1.415

      
p-value =        0.16365
Conclusion:     p-value>α =0.10, Do not reject null hypothesis  

so, there is no enough evidence to conclude that mean corneal thickness is greater in normal eyes than in eyes with glaucoma at α=0.10

b)

Degree of freedom, DF=   n - 1 =    7
t-critical value =    t α/2,df =    3.4995
      
std dev of difference , Sd =        10.7438
      
std error , SE =    Sd / √n =    3.7985
margin of error, E =    t*SE =    13.293
      
mean of difference ,    D̅ =   4.0000
confidence interval is       
Interval Lower Limit=   D̅ - E =   -9.2928
Interval Upper Limit=   D̅ + E =   17.2928

c)

Paired Wilcoxcon Signed-Rank

Ho: mean corneal thickness is equal in normal eyes than in eyes with glaucoma

H1: mean corneal thickness is greater in normal eyes than in eyes with glaucoma

sample 1	sample 2	difference=sample1-sample 2	absolue difference	rank	rank if positive	rank if negative
484	488	-4	4	1.5		1.5
478	478	0
492	480	12	12	4.5	4.5
444	426	18	18	7	7
436	440	-4	4	1.5		1.5
398	410	-12	12	4.5		4.5
464	458	6	6	3	3
476	460	16	16	6	6

number of non zero difference , n =    7
  
sum of positive ranks, W+ =    20.5
sum of negative ranks , W- =   7.5
  
T = W+ = 20.5
  
mean ,µ = n(n+1)/4 =    14
  
std dev ,σ = √(n(n+1)(2n+1)/24) =   5.916

Z-stat = (T - µ)/σ =    1.099
  
Z*, Z critical value = 1.28
  
P-value =    0.13594
Conclusion:     P-value>α=0.01 , Do not reject null hypothesis
so, there is no enough evidenc eto conclude that mean corneal thickness is greater in normal eyes than in eyes with glaucoma