https://file.aiquickdraw.com/mathbot/file/1726589623467zjl9peyk.pdf

The case problem provided is from *Statistics for Business and Economics* and centers around Harriet McNeil's Auto Mall. Here is a breakdown of the problem:

### Key Problem:
Ms. McNeil wants to ensure she has the right number of salespeople on her lot on busy Saturday mornings, where an average of 6.8 customers arrive per hour. Her goal is to always have two more customers than salespeople available without letting the number of unserved customers (those who exceed the capacity of the salespeople by more than two) rise above 10% of the time. 

### Questions to Address:
1. **Distribution of customer arrivals**:
   The number of customers arriving per hour follows a **Poisson distribution** since customers arrive randomly at a constant rate (6.8 customers/hour).

2. **Current Strategy (5 salespeople)**:
   Using 5 salespeople, the task is to calculate the probability that more than 7 customers (5 salespeople + 2 extra) are on the lot at any given time. This can be calculated using the Poisson distribution.

3. **Optimal number of salespeople**:
   The goal is to find the minimum number of salespeople such that the probability of exceeding the two extra customers more than 10% of the time is acceptable.

Would you like to dive into solving each of these questions with the detailed steps, starting with calculating the probabilities? 

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Here are five questions to explore further:
1. How do we compute the Poisson distribution for a given rate?
2. What are the assumptions behind the Poisson process in this scenario?
3. How do you calculate cumulative probabilities using the Poisson distribution?
4. What impact would increasing the average service time per customer have?
5. How do you determine the trade-off between customer satisfaction and staffing?

**Tip**: Poisson distributions are commonly used in scenarios where events occur at a constant average rate, but each event happens independently of the others.

How many salespeople should Ms. McNeil have on her lot on Saturday mornings to ensure an optimal customer service balance, given the average customer arrival rate is 6.8 per hour?

Determine the optimal number of salespeople needed on Saturday mornings at McNeil's Auto Mall using Poisson distribution. With an average customer arrival rate of 6.8 per hour and a goal to keep unserved customers under 10% of the time, learn how to balance staffing with customer demand.

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