Question 7.6: Resin-based composites are used in restorative dentistry. Th...
Resin-based composites are used in restorative dentistry. The article “Reduction of Polymerization Shrinkage Stress and Marginal Leakage Using Soft-Start Polymerization”
(C. Ernst, N. Brand, et al. , Journal of Esthetic and Restorative Dentistry, 2003: 93-1 04) presents a comparison of the surface hardness of specimens cured for 40 seconds with constant power with that of specimens cured for 40 seconds with exponentially increasing power. Fifteen specimens were cured with each method. Those cured with constant power had an average surface hardness (in N/mm²) of 400.9 with a standard deviation of 1 0.6. Those cured with exponentially increasing power had an average sur-face hardness of 367.2 with a standard deviation of 6.1. Find a 98% confidence interval for the difference in mean hardness between specimens cured by the two methods.
We have \bar{X}=400.9, s_X=10.6, n_X=15, \bar{Y}=367.2, s_Y=6.1 \text{,} and n_y= 15. The number of degrees of freedom is given by Equation (7 .1 0) to be
v=\frac{\left(\frac{s_X^2}{n_X}+\frac{s_Y^2}{n_Y}\right)^2}{\frac{\left(s_X^2 / n_X\right)^2}{n_X-1}+\frac{\left(s_Y^2 / n_Y\right)^2}{n_Y-1}}\,\,\,\,\,\text{rounded down to the nearest integer.} (7.10)
\nu=\frac{\left(\frac{10.6^2}{15}+\frac{6.1^2}{15}\right)^2}{\frac{\left(10.6^2 / 15\right)^2}{15-1}+\frac{\left(6.1^2 / 15\right)^2}{15-1}}=22.36 \approx 22
From the t table (Table A.3 in Appendix A),we find that t_{22,01}=2.508. We use expression (7.11) to find that the 98% confidence interval is
\bar{X}-\bar{Y}\pm t_{v, \alpha / 2}\sqrt{\frac{s_X^2}{n_X}+\frac{s_Y^2}{n_Y}} (7.11)
400.9-367.2 \pm 2.508 \sqrt{10.6^2 / 15+6.1^2 / 15}\,\,\,\text{, or}\,\,\,33.7 \pm 7.9