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.

Data Table

Specimen	1	2	3	4	5	6	7	8	9
Steel ball	51	57	61	70	68	54	65	51	53
Diamond	53	55	63	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 lower bound is _________________(Round to the nearest tenth as needed.)

The upper bound is _________________(Round to the nearest tenth as needed.)

State the appropriate conclusion. Choose the correct answer below:

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

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

Answer 1

For calculating confidence interval we need to first determine standard deviation and mean of hardness reading for both Steel ball and Diamond

Diamond

x̅1 = (53+55+63+74+69+55+68+51+56)/9 = 60.4444

s1 = 8.2479

Steel Ball

x̅2 = (51+57+61+70+68+54+65+51+53)/9 = 58.8889

s2 = 7.373

Confidence Interval Calculation

We have calculated confidence interval assuming equal variances:

The lower bound is -6.3

The upper bound is 9.4

Appropriate Conclusion: Option B is correct

Null Hypothesis H0: μ1 - μ2 = 0

Alternate Hypothesis Ha: μ1- μ2 ≠ 0

In this case, confidence interval (-6.3, 9.4) contains value of null hypothesis that is zero, so we cannot reject null hypothesis. In other words, there is insufficient evidence to conclude that the two indenters produce different hardness readings.