SPS SPS FM Statistics 2024 January — Question 4 9 marks

Exam BoardSPS
ModuleSPS FM Statistics (SPS FM Statistics)
Year2024
SessionJanuary
Marks9
TopicPoisson distribution
TypeTwo independent Poisson sums
DifficultyModerate -0.3 This is a straightforward Poisson distribution question testing standard knowledge: stating the constant rate assumption, recalling that SD = √λ, scaling the parameter for different time periods, and adding independent Poisson variables. Part (e) requires basic interpretation but no formal hypothesis testing. All parts are routine applications of textbook Poisson properties with no novel problem-solving required.
Spec5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02n Sum of Poisson variables: is Poisson
4. The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.
  1. State a further assumption needed for the Poisson model to be valid.
  2. State the value of the standard deviation of \(R\).
  3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\mathrm { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
  4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
  5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.
4.

The manager of a car breakdown service uses the distribution $\operatorname { Po } ( 2.7 )$ to model the number of punctures, $R$, in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.
\begin{enumerate}[label=(\alph*)]
\item State a further assumption needed for the Poisson model to be valid.
\item State the value of the standard deviation of $R$.
\item Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur.

The manager uses the distribution $\mathrm { Po } ( 0.8 )$ to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
\item Assume first that both the manager's models are correct.

Calculate the probability that a randomly chosen day is a "bad" day.
\item It is found that 12 of the next 100 days are "bad" days.

Comment on whether this casts doubt on the validity of the manager's models.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q4 [9]}}
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