Question: HSBC Bank is a large, London-based international banking company. One of its most important sources of income is home loans. A component of its effort to maintain and increase its customer base is excellent service. The loan manager at one of its branches in New York keeps track of the number of loan applicants who visit his branch's loan department per week. Having enough loan officers available is one of the ways of providing excellent service. Over the last year, the loan manager accumulated the following data:
Number of Customers 0 1 2 3 4 5 ≥6
Frequencies 1 2 9 11 14 6 9
From previous years, the manager believes that the distribution of the number of customer arrivals has a Poisson distribution with an average of 3.5 loan applicants per week. Determine if the loan officer's belief is correct using a significance level of 0.025.