$7\mathrm{.}1$ The design engineer specifies a concrete strength of $5500$ psi. Determine the required average compressive strength for:

a$.$ a new plant where $s$ is unknown

b$.$ a plant where $s=500$ for $22$ test results

c$.$ a plant with extensive history of producing concrete with $s=400$

d$.$ a plant with extensive history of producing concrete with $s=600$