Compute averages and binomial/Poisson probabilities

Part A (streaming mean): Consider a dataset of 1,000 values whose average is 12.4. You delete 50 entries with an average of 20.1, then add 30 fresh entries with an average of 9.8. What is the precise new mean, without going back to the original individual data points? Present the formula you apply and the result rounded to three decimal places.

Part B (binomial versus Poisson): Suppose $X \sim \text{Binomial}(n=40, p=0.07)$. 1) Work out $P(X=0)$, $P(X \le 2)$, and $P(X \ge 3)$ to four decimal places using the exact binomial distribution. 2) Compute approximations of those same three probabilities using a Poisson model with $\lambda = n \cdot p$, applying a continuity correction where it is warranted; give the absolute error for each one, meaning $| \text{exact} - \text{approximate} |$. 3) Explain in a few sentences why the Poisson approximation is suitable for this case, and describe how you could improve the approximation if $p$ were higher.

Overview: The item assesses skill in updating summary statistics in a streaming context and in probability computation, in particular deriving a revised mean from summary aggregates and contrasting exact binomial probabilities against Poisson approximations, all within the Statistics & Math area.