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 indentations can be made. Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings.

Construct a [tex]$95\%$[/tex] confidence interval to judge whether the two indenters result in different measurements. The differences are computed as 'diamond minus steel ball.'

Data Table
\begin{tabular}{|lccccccccc|}
\hline Specimen & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline Steel ball & 50 & 57 & 61 & 71 & 68 & 54 & 65 & 51 & 53 \\
Diamond & 52 & 55 & 63 & 74 & 69 & 56 & 68 & 51 & 56 \\
\hline
\end{tabular}

Steps:

1. Construct a [tex]$95\%$[/tex] confidence interval to determine if the two indenters produce different measurements.
2. The lower bound is [tex]$\square$[/tex]
3. The upper bound is [tex]$\square$[/tex]
(Round to the nearest tenth as needed.)

Conclusion:
Choose the correct statement:

A. There is sufficient evidence to conclude that the two indenters result in different measurements.
B. There is insufficient evidence to conclude that 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.

To determine whether the steel-ball and diamond indenters yield different hardness readings, we can construct a 95% confidence interval for the mean difference in hardness measurements (diamond minus steel-ball).

### Step-by-Step Solution

1. Sample Data:
- Steel Ball Hardness Readings: [tex]$ [50, 57, 61, 71, 68, 54, 65, 51, 53] $[/tex]
- Diamond Hardness Readings: [tex]$ [52, 55, 63, 74, 69, 56, 68, 51, 56] $[/tex]

2. Differences Between Diamond and Steel Ball Readings:
Let's compute the differences between each corresponding pair of values (Diamond - Steel Ball):

[tex]\[ \text{differences} = [2, -2, 2, 3, 1, 2, 3, 0, 3] \][/tex]

3. Calculating the Mean and Standard Error of the Differences:
- Mean of the differences ([tex]$\bar{d}$[/tex]):
[tex]\[ \bar{d} = 1.55 \][/tex]
- Standard Error of the Mean ([tex]$SEM$[/tex]) of the differences:
[tex]\[ SEM = 0.56 \][/tex]

4. Degrees of Freedom (df):
[tex]\[ df = n - 1 = 9 - 1 = 8 \][/tex]

5. Constructing the 95% Confidence Interval:
Using the t-distribution for a 95% confidence level and with 8 degrees of freedom, the confidence interval is:

[tex]\[ \text{Confidence Interval} = (\bar{d} - t^ \cdot SEM, \bar{d} + t^ \cdot SEM) \][/tex]

Where [tex]$t^*$[/tex] is the critical value for t-distribution at 95% confidence interval with 8 degrees of freedoms.

The resulting confidence interval is given by:

[tex]\[ \text{Lower Bound} = 0.27 \][/tex]
[tex]\[ \text{Upper Bound} = 2.84 \][/tex]

### Conclusion

Since the 95% confidence interval for the mean difference ([tex]$0.27, 2.84$[/tex]) does not include 0, we can conclude that there is sufficient evidence to suggest that there is a statistically significant difference in the hardness readings between the steel-ball and diamond indenters.

Therefore, we would state the conclusion as:

"There is sufficient evidence to conclude that the steel-ball and diamond indenters produce different hardness readings."