Question 1 - A-Level Maths

OCR MEI S2 2005 June — Question 1 19 marks

Exam Board	OCR MEI
Module	S2 (Statistics 2)
Year	2005
Session	June
Marks	19
Paper	Download PDF ↗
Topic	Poisson distribution
Type	Frequency distribution and Poisson fit
Difficulty	Standard +0.3 This is a standard Poisson distribution question covering routine applications: stating conditions, calculating mean/variance from grouped data, comparing mean to variance, and using Poisson probabilities. Part (vi) requires normal approximation which is a standard technique. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average.
Spec	2.04d Normal approximation to binomial 5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.02l Poisson conditions: for modelling

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below.
\(x\) 0 1 2 3 4 5 \(> 5\)
Frequency, \(f\) 18 39 20 12 8 3 0
Show that the sample mean is 1.62 and calculate the sample variance.
Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of \(X\) ? Explain your answer. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 1.62 .
Find \(\mathrm { P } ( X = 2 )\). Comment on your answer in relation to the data in the table.
Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

Show LaTeX source

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable $X$ denotes the number of cars arriving in a randomly chosen period of ten seconds.\\
(i) State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of $X$. Comment briefly on whether these assumptions are likely to be valid.

The student counts the number of arrivals, $x$, in each of 100 ten-second periods. The data are shown in the table below.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & $> 5$ \\
\hline
Frequency, $f$ & 18 & 39 & 20 & 12 & 8 & 3 & 0 \\
\hline
\end{tabular}
\end{center}

(ii) Show that the sample mean is 1.62 and calculate the sample variance.\\
(iii) Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of $X$ ? Explain your answer.

For the remainder of this question you should assume that $X$ may be modelled by a Poisson distribution with mean 1.62 .\\
(iv) Find $\mathrm { P } ( X = 2 )$. Comment on your answer in relation to the data in the table.\\
(v) Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.\\
(vi) Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

\hfill \mbox{\textit{OCR MEI S2 2005 Q1 [19]}}

This paper (4 questions)

View full paper

Q1 19 Q2 18 Q3 17 Q9

\(x\)	0	1	2	3	4	5	\(> 5\)
Frequency, \(f\)	18	39	20	12	8	3	0