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   Steel ball   Diamond
1 50 52
2 57 55
3 61 63
4 70 74
5 68 69
6 54 55
7 65 68
8 51 51
9 53 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 lower bound is

The upper bound is

​(Round to the nearest tenth as​ needed.)

Answer 1

Since, the indenters test on same sample, it is a paired sample test.
We are to test if there is significant difference in means. That is, we will be testing:

Or, in other words,

where,


The 95% confidence interval to judge whether the two indenters result in different​ measurements is given by:

We construct the following table for the ease of calculation:

Specimen	Stell Ball	Diamond	d
1	50	52	2
2	57	55	-2
3	61	63	2
4	70	74	4
5	68	69	1
6	54	55	1
7	65	68	3
8	51	51	0
9	53	56	3

Now,




Putting in the values, we get:

Thus,
The lower bound is 0.2.
The lower bound is 2.9.

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