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. LOADING... Click the icon to view the data table.

Specimen	1	2	3	4	5	6	7	8	9
Steel ball	5050	5757	6161	7070	6868	5454	6565	5151	5353
Diamond	5252	5656	6161	7474	6969	5555	6868	5151	56

Answer 1

Confidence interval for difference between two population means of paired samples is given as below:

Confidence interval = Dbar ± t*S_D/sqrt(n)

We take difference as diamond minus steel.

From given data, we have

Dbar = 1.444444444

S_d = 1.666666667

n = 9

df = n – 1 = 8

Confidence level = 95%

Critical t value = 2.3060

(by using t-table)

Confidence interval = Dbar ± t*S_D/sqrt(n)

Confidence interval = 1.444444444 ± 2.3060*1.666666667/sqrt(9)

Confidence interval = 1.444444444 ± 1.2811

Lower limit = 1.444444444 - 1.2811 = 0.1633

Upper limit = 1.444444444 + 1.2811 = 2.7256

0.1633 < µ_d < 2.7256