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."
