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. 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 nothing. The upper bound is nothing. ​(Round to the nearest tenth as​ needed.) State the appropriate conclusion. Choose the correct answer below. There is insufficient evidence to conclude that the two indenters produce different hardness readings. There is sufficient evidence to conclude that the two indenters produce different hardness readings.

Specimen	1	2	3	4	5	6	7	8	9
Steel ball	50	57	61	70	68	54	65	51	53
Diamond	52	55	63	74	69	55	68	51	56

Answer 1

From the given data, values of the difference:

d = Diamond - Steel ball

are got as follows:

2, - 2, 2, 4, 1,1, 3, 0, 3

From values of d, the following statistics are calculated:

n = 9

= 1.5556

s_d = 1.8105

SE = s_d/

= 1.8105/ =0.6035

= 0.05

ndf = n - 1 = 9 - 1 = 8

From Table, critical values of t = 2.3060

Confidence Interval:

Lower bound is:

1.5556 - (2.3060 X 0.6035)

= 1.5556 - 1.3917

= 0.16

Upper bound is:

1.5556 + (2.3060 X 0.6035)

= 1.5556 + 1.3917

= 2.95

Since 0 is not included in the confidence interval, :

Correct option:

There is sufficient evidence to conclude that the two indenters produce different hardness readings.

Answers to Questions asked:

Lower bound is:

0.16

Upper bound is:

2.95

Correct option:

There is sufficient evidence to conclude that the two indenters produce different hardness readings.