Question

Measurement of inspection time, from a large sample of outsourced components, gave the following distribution:

Time	20	22	24	25	27	28	29	31
Number	1	2	3	3	2	3	2	2

   Your manager thinks that the inspection time should be the same for all outsourced components. Using the data provided, test (at the 5% significance level) this hypothesis and indicate whether there is a correlation or not.

Answer 1

X	Y	XY	X²	Y²
20	1	20	400	1
22	2	44	484	4
24	3	72	576	9
25	3	75	625	9
27	2	54	729	4
28	3	84	784	9
29	2	58	841	4
31	2	62	961	4
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
206	18	469	5400	44

Correlation coefficient, r = [n*∑xy - ∑x*∑y] / √[(n∑x²-(∑x)²)*(n∑y²-(∑y)²)]

= [8*469 - 206*18] / √[(8*5400 - 206²)*(8*44 - 18²)] =   0.3008

Null and alternative hypothesis:  

Ho: ρ = 0 ; Ha: ρ ≠ 0  

Correlation coefficient, r = 0.3008

Test statistic :    

t = r*√(n-2)/√(1-r²) = 0.3008 *√(8 - 2)/√(1 - 0.3008²) = 0.7727

df = n-2 =    6

p-value = T.DIST.2T(ABS(0.7727), 6) =    0.4690

Conclusion:  

p-value > α , Fail to reject the null hypothesis. There is no correlation between x and y.