Question

The manufacturer of hardness testing equipment uses​ steel-ball indenters to penetrate metal that is being tested.​ However, the manufacturer thinks it would be better to use a diamond indenter so that all types of metal can be tested. Because of differences between the two types of​ indenters, it is suspected that the two methods will produce different hardness readings. The metal specimens to be tested are large enough so that two indentions can be made.​ Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings. Construct a​ 95% confidence interval to judge whether the two indenters result in different measurements. ​Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Specimen	1	2	3	4	5	6	7	8	9
Steel ball	51	57	61	70	68	54	65	51	53
Diamond	53	55	63	74	69	56	68	51	56

Answer 1

S. No	Diamond	Steel ball	diff:(d)=x1-x2	d2
1	53	51	2	4.00
2	55	57	-2	4.00
3	63	61	2	4.00
4	74	70	4	16.00
5	69	68	1	1.00
6	56	54	2	4.00
7	68	65	3	9.00
8	51	51	0	0.00
9	56	53	3	9.00
	total	=	Σd=15	Σd2=51
	mean dbar=	d̅                           =	1.667
	degree of freedom =n-1                            =		8.000
	Std deviaiton SD=√(Σd2-(Σd)2/n)/(n-1) =		1.803
	std error=Se=SD/√n=		0.6009

for 95% CI; and 8 degree of freedom, value of t=			2.306
therefore confidence interval=sample mean -/+ t*std error
margin of errror          =t*std error=		1.386
lower confidence limit                     =		0.2809
upper confidence limit                    =		3.0524
from above 95% confidence interval for population mean =(0.3 ,3.1)

since interval values are above 0 , there is a significant difference between them