$($f$)$ Among the above $4$ indexes, which one is the most closely watched? Explain your answer.

The most closely watched index is typically Cpk because it considers both process spread and centering. Cpk provides a comprehensive assessment of process capability by accounting for deviations from both specification limits and is therefore widely used for process monitoring and improvement.

$($g$)$ If you are the buyer of those ball bearings, what would you say to the manufacturer $($a subjective statement$)?$

As the buyer, I would express my concern about the process capability of the manufacturer's production process. While the ball bearings meet the specifications, the relatively low Cp and Cpk values indicate potential issues with process variability and centering. I would encourage the manufacturer to focus on improving process stability, reducing variation, and enhancing centering to ensure consistent quality and reliability of the ball bearings.