In the dynamic field of commercial production, choosing the right production company for your business requires a sophisticated understanding of the nuances inherent to the industry. As a brand, your ultimate goal is to ensure your commercial production mirrors your brand ethos, leverages cutting-edge technologies, and resonates with your target audience.

Here, we'll delve deeply into the critical questions you should pose to prospective commercial production companies. We'll interpret and explain these questions from multiple angles—historical, economic, statistical, and legal—and offer a comprehensive guide to aid you in your selection process.

The first question you should ask is, 'Who are the individuals that comprise your team?' It is essential to be cognizant of the team's collective skills, creativity, and experience. Research into Alan Page Fiske’s Relational Models Theory, a social science framework, can provide insights into how different individuals in the team may work together in an interplay of communal sharing, authority ranking, equality matching, and market pricing. Understanding these dynamics could help in predicting how effectively the team might work on your project.

Next, scrutinize 'What is your company’s creative approach?' A company's creative philosophy lays the foundation for its artistic decisions, which will inevitably influence your commercial's final output. Assess whether their approach aligns with the psychological profile of your target audience. Studies in cognitive psychology, such as the Elaboration Likelihood Model, can provide insights into how your audience might process and be persuaded by different types of advertising messages.

The third question is 'Where have your past commercials been aired or published?' The answer will reveal their contacts within the industry and potentially, the extent of exposure your commercial might receive. This is where an understanding of media economics is beneficial. Knowledge of the Pareto principle, a mathematical theory postulating that 80% of consequences come from 20% of the causes, might help you assess whether the company's past media placements align with your brand's marketing strategy.

When it comes to 'When can we expect the project to be completed?' it is essential to consider the time sensitivity of the project. Here, queuing theory, a statistical approach, could offer an insight into the company’s process efficiency and reliability.

The fifth question is 'How do you handle deadlines and budget constraints?' This question addresses potential legal and economic challenges that might arise during the course of the project. Knowledge of contract law could help you ensure your interests are protected, while understanding the concept of economies of scale could guide your discussions on budgeting.

Finally, ask ‘Why should we choose your company?’ This open-ended question allows the prospective commercial production company to highlight their unique selling proposition. Use your comprehensive understanding of your business, your competition, and the insights you've gleaned from their responses to the previous questions to evaluate their answer critically.

In conclusion, choosing the right commercial production company involves asking the right questions and interpreting their answers through the lens of varied academic disciplines. As a business leader, your paramount objective is to ensure your commercials project your brand's identity in an innovative and compelling manner. Through this deep-dive exploration of the critical questions to ask a prospective commercial production company, you can confidently navigate the selection process.