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. Click the icon to view the data table. 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.) 0 Data Table Specimen 1 2 3 4 Steel ball 50 57 61 70 Diamond 52 55 63 74 5 68 69 6 7 8 9 54 65 51 53 56 68 51 56 Print Done Enter your answer in the edit fields and then click Check Answer. 1 part remaining Clear All Check Answer

Answer 1

From the given data, values of d = Diamond - Steel are got as follows:

d = 2, - 2, 2, 4, 1, 2, 3, 0, 3

From the d values, the following Table is calculated:

n = 9

$\bar{d}$ = 15/9 = 1.6667

d	d - $\bar{d}$	(d - $\bar{d}$ )2
2	0.3333	0.1111
-2	-3.6666	13.4444
2	0.3333	0.1111
4	2.3333	5.4444
1	-0.6667	0.4444
2	0.3333	0.1111
3	1.3333	1.7778
0	-1.6667	2.7778
3	1.3333	1.7778
Total =		26.0

Sample Standard Deviation is given by:

$s=\sqrt{\frac{26.0}{8}}=\sqrt{3.25}=1.8028$

SE = s/$\sqrt{n}$

= 1.8028$\sqrt{9}$

= 0.6009

$\alpha$ = 0.05

ndf = 9 - 1 = 8

From Table, critical values of t = $\pm$ 2.3060

The lower bound is 1.6667 - (2.3060 X 0.6009) = 1 6667 - 1.3858 = 0.2809

The upper bound is 1.6667 + (2.3060 X 0.6009) = 1 6667 + 1.3858 = 3.0525

So,

Answers are:

The lower bound is = 0.3

The upper bound is = 3.1