Z-test, power and sample size for chef output

A restaurant manager suspects his chefs average at most $13$ dishes per day. Throughout, the number of dishes per chef has variance $4$ and he tests $H_0:\mu=13$ against an upper-tail alternative.

(a) He samples $49$ chefs on one day and finds a mean of $15$ dishes. Compute the test statistic for $H_a:\mu>13$.

(b) Using $49$ chefs and $\alpha=0.05$, he wants to detect a true mean of $14$ (one dish above the null). How likely is he to commit a Type II error?

(c) For $H_0:\mu=13$ versus $H_a:\mu=14$, what sample size makes $\alpha=\beta=0.05$?

(a) $z=7$; (b) $\beta\approx0.03$; (c) $n=44$. (a) $z=(15-13)/(2/7)=7$. (b) Reject when $\bar x>13+1.645\cdot\tfrac27\approx13.47$; under $\mu=14$ the chance of falling below that is $\Phi(-1.855)\approx0.032$. (c) $n=(z_\alpha+z_\beta)^2\sigma^2/\Delta^2=(3.29)^2\cdot4\approx43.3$, rounded up to $44$.