An engineering laboratory is conducting research on the tensile strength of two newly developed carbon-fiber composites, Apex and Zenith. A quality control engineer randomly selects 250 samples of composite Apex and 250 samples of composite Zenith. The tensile strength measurements (in MPa) are summarized in the following table:
| Composite | Mean Tensile Strength (xˉ\bar{x}xˉ) | Variance of Strength (s2s^2s2) |
|---|---|---|
| Apex | 455.5 | 1800 |
| Zenith | 447.1 | 2200 |
The engineer asserts that there is no significant difference between the mean tensile strengths of these two composites.
Evaluate the engineer's assertion at the 5% significance level. You must clearly state your hypotheses, the calculated test statistic, and the critical value(s) used.
Discuss the importance of the Central Limit Theorem in the context of the test performed in part (a).