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 56 61 74 69 55 68 51 56

Construct a​ 95% confidence interval to judge whether the two indenters result in different​ measurements, where the differences are computed as​ 'diamond minus steel​ ball'.

The uppper bound is _______?

The lower bound is _________?

The test statistic is __________?

Answer 1

Given:

The manufacturer of hardness testing equipment uses​ steel-ball indenters to penetrate metal that is being tested.​

It is given that the differences are computed as​ 'diamond minus steel​ ball'

Table for calculating mean and standard deviation :

Specimen	steel ball, X	Diamond, Y	d = Y-X	d^2
1	51	53	-2	4
2	57	56	-1	1
3	61	61	0	0
4	70	74	4	16
5	68	69	1	1
6	54	55	1	1
7	65	68	3	9
8	51	51	0	0
9	53	56	3	9
			d = 9	d^2=41

Therefore

The uppper bound is -0.54

The lower bound is 2.54

The test statistic is 1.5