| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Binomial |
| Difficulty | Standard +0.3 This is a standard chi-squared goodness of fit question with routine procedures: (a) fitting binomial distribution with given p, calculating expected frequencies, and performing test; (b) finding missing expected frequencies using Poisson properties; (c) performing another chi-squared test. All steps are textbook applications with no novel insight required, making it slightly easier than average for S3 level. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous |
| Number of suitcases | 0 | 1 | 2 | 3 | 4 |
| Frequency | 6 | 25 | 12 | 6 | 1 |
| Number of passengers | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Observed frequency | 5 | 40 | 31 | 18 | 6 | 0 |
| Expected frequency | 16.53 | 29.75 | \(r\) | \(s\) | 7.23 | 3.64 |
An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Number of suitcases & 0 & 1 & 2 & 3 & 4 \\
\hline
Frequency & 6 & 25 & 12 & 6 & 1 \\
\hline
\end{tabular}
\end{center}
The manager claims that the data can be modelled by a binomial distribution with $p = 0.3$
\begin{enumerate}[label=(\alph*)]
\item Test the manager's claim at the 5\% level of significance. State your hypotheses clearly.
Show your working clearly and give your expected frequencies to 2 decimal places.
(8)
The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals.
The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Number of passengers & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline
Observed frequency & 5 & 40 & 31 & 18 & 6 & 0 \\
\hline
Expected frequency & 16.53 & 29.75 & $r$ & $s$ & 7.23 & 3.64 \\
\hline
\end{tabular}
\end{center}
\item Find the value of $r$ and the value of $s$ giving your answers to 2 decimal places.
(3)
\item Stating your hypotheses clearly, use a 1\% level of significance to test the manager's new claim.
(6)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2016 Q6}}