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Surf lifesaving rescue call-outs

Poisson and normal models: calculation, conditional probability, model fit and assumptions

water safety

A surf lifesaving club patrols a beach on summer weekend days. Club records suggest a mean of 2.4 rescue call-outs per patrol day. Table 1 and Figure 1 show the number of call-outs on each of 60 randomly selected patrol days from the last three summers (sample mean 2.43, sample variance 6.35). The club also records response time — the time from a call-out to a lifeguard reaching the swimmer. Response times are approximately normally distributed with mean 9.5 minutes and standard deviation 2.8 minutes (Figure 2). Table 1: Call-outs per day (60 days) | Call-outs | Number of days | |---|---| | 0 | 17 | | 1 | 8 | | 2 | 13 | | 3 | 6 | | 4 | 6 | | 5 | 2 | | 6 | 3 | | 7 | 2 | | 8 | 1 | | 9 | 0 | | 10 | 2 |

Bar graph of the number of patrol days with each number of rescue call-outs, from 0 to 10, for 60 randomly selected patrol days; 17 days had no call-outs and a few days had 7 to 10.

Sketch of a normal distribution for response time with mean 9.5 minutes and standard deviation 2.8 minutes; the region between 12 and 15 minutes is shaded.

Facts you may need

(a)

The club plans for a second rescue boat to be on standby when there are 5 or more call-outs in a day. Using a Poisson model with

\lambda = 2.4

, find the probability that the standby boat is needed on a given patrol day, and the probability it is needed on at least one day of a two-day weekend. Comment on the independence assumption.

(b)

Use Table 1 and Figure 1 to evaluate whether a Poisson model is appropriate for the number of call-outs per day. Support your answer with calculations, and explain in context why the model may or may not be appropriate.

(c)

Find the probability that a response takes more than 12 minutes, given that it takes less than 15 minutes.

(d)

Discuss TWO reasons why a normal model may not be appropriate for response times, and how the club could check one of them.