In the dynamic landscape of a multinational corporation operating across three distinct regions, each characterized by intricate economic factors, the company is meticulously strategizing its production cost estimations. Seeking precision, the company aims to estimate the average cost of manufacturing a vital component over a specified period. The anticipated production quantities for each region during this period are Region $1($N$1=9,500$ parts$),$ Region $2($N$)11,200$ parts$),$ anc Region $3($N$3=14,800$ parts$),$ culminating in a total expected production of $N=35,500$ parts.

Undertaking a nuanced approach, the company meticulously collected cost data for $12$ randomly selected parts from Region $1,12$ parts from Region $2,$ and $16$ parts from Region $3.$ The intricately measured cost data, expressed in dollars, are as follows:

Sample from Region $1$: $$720,$$$695,$$$810,$$$635,$$$740,$$$725,$$$770,$$$690,$$$710,$$$705,$$$760,$$$695$

Sample from Region $3$:$$2535,$$$2478,$$$2621,$$$2619,$$$2590,$$$2525,$$$2570,$$$2485,$$$2517,$$$2590,$$$2631,$$$2600,$$$2421,$ $$2591,52458,$$$2623$

Embark on this analysis to provide comprehensive insights into:

a$)$ A sophisticated understanding of the intricacies surrounding the estimated average cost and the total cost of producing parts in each region, considering potential outliers and variations.

b$)$ An intricate assessment of the margin of error in estimating the average cost and total cost, factoring in the complex interplay of statistical distributions and sample characteristics, with a stringent $95\%$ probability requirement

c$)$ The formulation of confcence intenrais for both the average cost and total cost, incorporating a confidence ceefficient of $95\%,$ while accounting for potential non$-$normality in the data distribution.