This is a standard question that is likely to be asked in your interview. Before I get down to answering this question, lets walk through the approach and the framework
The most important (and obvious) point here is answering the question. Make sure that any example you give has the two elements mentioned here namely (a) It is a difficult challenge and (b) it clearly shows you leading the team through the challenge mentioned
Make sure you communicate your answer to the question in a structured way - I prefer the situation, complication and resolution framework because as a listener it is closest to a story format (beginning, middle and an end)
Be enthusiastic and interested in your own story - you want the person listening to be engaged and interested in what you have to say. People react best to others who are interested in what they have to say
Use a style of speaking that is clear and easy to follow - for example, I like using numbered points so people can tell where you are at any point
And lastly be prepared for detailed follow-ups
Below is my personal answer to the question
We were working on the launch of an infant formula brand in the Indian market. The country had not seen a new launch from a global infant nutrition company in over a decade due to various factors. The most important of these factors was that 90% market share of the market is with Nestle - who forced the government to shut off competition to other brands using manipulative tactics. As part of the project we had to estimate the total market size for infant nutrition at that time. With no publicly available data and no breakdown from Nestle of their infant nutrition sales, we were at a loss as to how to come up with the estimate.
The current consensus in the team was that in order to understand the market size, we launch a survey to few hundred mothers in the target market to assess whether they would use infant nutrition and furthermore what % of those would likely use the brand we were going to launch. However, as a statistician I knew that surveys can be easily manipulated when it comes to data gathering - people are paid to do them, there is a strong favor-ability bias and for a country like India, getting a representative sample is not realistic in a quick survey. I had thought about this problem as well and had suggested an alternative - there were other markets that were similar to India in target demographics, at one point highly regulated like India and where this brand had launched such as Vietnam. We could look into data from those markets to get ratios to apply to our market to estimate current market size, additionally given that the brand had already launched there, we could use this data to also see what would happen when we launched the product in India. In our problem solving sessions, my manager and the partner were not in favor of this approach as they felt it would be 'too theoretical' and the client would not accept it.
Upon submitting my suggestion a couple of times, I spent time to think about what it was that my team was not convinced about. I was able to narrow down the concerns as follows.
They were not sold on the methodology and its accuracy
They were not sure the client would be bought in on it given the different nature of Indian consumers
It was one of my first projects and there was some hesitation to listen to my opinion from my Manager
I decided that the best approach would be to tackle each of these things independently and then try the approach again
Firstly, I used the method I was proposing to create initial back of the envelope calculations to show that if we applied it to other markets that we had data for, we would get ~70% accuracy of the data which was quite high given how little we knew about the market in total
Secondly, I had built up a relationship with the finance head of the team and I asked them for some time to discuss my approach. I walked them through my methodology to see if this made sense and if the company would be open to it. He seemed happy enough with the approach and even gave me some pointers to make it better
Lastly, I could not change my being new but I then booked time with my manager 1:1. I requested his time and before the meeting began I said I know I am new but reminded him that he was also new at some point and that I am open to feedback but I cannot change that fact. During the meeting I walked him through the analysis and the fact that I had discussed this informally with the finance team. He was very pleased with my initiative and agreed to give it genuine time in the next team meeting.
He followed through on his promise, and in the next team meeting I presented my idea with the backups and the suggestions of the finance team. The team agreed that my approach had merit and would also be quicker to implement, and a modified version of my suggested approach was what we ended up using to solve this problem
Above is one example of how to use my outlined approach to tackle PEI questions. For more such examples here is a link to my book