Do Counts of 120 vs 100 Events Show That One Process Has a Higher Poisson Rate?

Two independent processes generate events, and both are observed over the same period of time. By the end of the period, process A has recorded 100 events and process B has recorded 120. Is there evidence that process B is better, meaning that its underlying event rate is higher than process A's? Choose a model for the counts, carry out a hypothesis test, and state what you would conclude. The candidate recalls this question only in outline: two processes, counts of 100 and 120 after some time, a question about whether the one with 120 is better, and a sense that it called for a Poisson process and a hypothesis test. This version assumes both processes were observed for the same length of time and reads "better" as "has a higher event rate". ```hint Noise in a count Under a Poisson model the variance of a count equals its mean. How large is a gap of 20 compared with the random variation you would expect if the two rates were equal? ``` ```hint Condition on the total If the two rates were equal, how should a known total of 220 events be expected to split between the two processes? ``` ### Clarifying Questions - Were both processes observed for the same length of time? If not, how long was each observed? - Does "better" mean a higher event rate, and was the claim that B is better made before or after the counts were seen? - What significance level should the test use? - Do events arrive independently at a steady rate, or can they come in bursts or vary with the time of day or day of week? ### What a Strong Answer Covers - A model for the two counts, and null and alternative hypotheses stated in terms of rates rather than counts - A valid test, exact or approximate, with its statistic and p-value, and a justified one-sided or two-sided choice - A correctly worded conclusion that separates "not significant" from "no difference", supported by an interval or effect size - Checks of the Poisson assumptions (equal exposure, constant rate, overdispersion) and what changes when they fail ### Follow-up Questions - If process B had been observed for twice as long as process A, how would the test and the conclusion change? - Suppose B's true rate really is 20% higher than A's. Roughly how much longer would both processes need to be observed for a test to detect that difference reliably? - If counts per day vary far more than a Poisson model allows, how would you change the analysis?