For a project, the structural design engineer specifies a concrete strength of $5,200$ psi. Determine the required average compressive strength of concrete to be prepared for the following conditions:

a$)$ The standard deviation of the concrete strength is unknown.

b$)$ The standard deviation of the concrete strength is $250$ psi for $17$ test results available from the past.

c$)$ The standard deviation of the concrete strength is $210$ psi for the intensive history of the plant.