A basketball coach wants to know if the male athletes in his program are on average taller than the population. We know that in the general population, men have an average height of 173.9 cm with a standard deviation of 21.1 cm. Its basketball program claims that the average height of male athletes is 186cm. If the coach wants to achieve at least 85% power in a two-tailed test with an alpha of 1%, what sample size should he take from the athletes for his analyses?

1. Mathematically calculate the required sample size with the non-centrality parameter formula.

2. Graphically demonstrate your answer using the z-score technique.