Challenging +1.2 This is a comprehensive chi-squared goodness of fit question with standard calculations (mean, variance, expected frequencies) and routine hypothesis testing. Parts (a)-(f) are textbook applications requiring no novel insight. Part (g) adds mild complexity by asking students to fit a custom discrete distribution, but this is still a straightforward parameter estimation exercise using observed frequencies. The multi-part structure and length push it slightly above average difficulty, but all components are standard Further Statistics fare.
8 A team of researchers have reason to believe that the number of calls received in randomly chosen 10-minute intervals to a call centre can be well modelled by a Poisson distribution. To test this belief the researchers record the number of telephone calls received in 60 randomly chosen 10-minute intervals. The results, together with relevant calculations, are shown in the following table.
Total
Number of calls, \(r\)
0
1
2
3
4
\(\geqslant 5\)
Observed frequency, \(f\)
18
13
12
9
8
0
60
rf
0
13
24
27
32
0
96
\(\mathrm { r } ^ { 2 } \mathrm { f }\)
0
13
48
81
128
0
270
Expected frequency
12.114
19.382
15.506
8.270
3.308
1.421
60
Contribution to test statistic
2.860
2.101
0.793
1.232
6.99
Calculate the mean of the observed number of calls received.
Calculate the variance of the observed number of calls received.
Comment on what your answers to parts (a) and (b) suggest about the proposed model.
Explain why it is necessary to combine some cells in the table.
Show how the values 15.506 and 0.793 in the table were obtained.
Carry out the test, at the \(5 \%\) significance level.
In the light of the result of the test, the team consider that a different model is appropriate. They propose the following improved model:
$$P ( R = r ) = \begin{cases} \frac { 1 } { 60 } ( a + ( 2 - r ) b ) & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$
where \(a\) and \(b\) are integers.
Use at least three of the observed frequencies to suggest appropriate values for \(a\) and \(b\). You should consider more than one possible pair of values, and explain which pair of values you consider best. (Do not carry out a goodness-of-fit test.)
8 A team of researchers have reason to believe that the number of calls received in randomly chosen 10-minute intervals to a call centre can be well modelled by a Poisson distribution. To test this belief the researchers record the number of telephone calls received in 60 randomly chosen 10-minute intervals. The results, together with relevant calculations, are shown in the following table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& & & & & & & Total \\
\hline
Number of calls, $r$ & 0 & 1 & 2 & 3 & 4 & $\geqslant 5$ & \\
\hline
Observed frequency, $f$ & 18 & 13 & 12 & 9 & 8 & 0 & 60 \\
\hline
rf & 0 & 13 & 24 & 27 & 32 & 0 & 96 \\
\hline
$\mathrm { r } ^ { 2 } \mathrm { f }$ & 0 & 13 & 48 & 81 & 128 & 0 & 270 \\
\hline
Expected frequency & 12.114 & 19.382 & 15.506 & 8.270 & 3.308 & 1.421 & 60 \\
\hline
Contribution to test statistic & 2.860 & 2.101 & 0.793 & \multicolumn{3}{|c|}{1.232} & 6.99 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the mean of the observed number of calls received.
\item Calculate the variance of the observed number of calls received.
\item Comment on what your answers to parts (a) and (b) suggest about the proposed model.
\item Explain why it is necessary to combine some cells in the table.
\item Show how the values 15.506 and 0.793 in the table were obtained.
\item Carry out the test, at the $5 \%$ significance level.
In the light of the result of the test, the team consider that a different model is appropriate. They propose the following improved model:
$$P ( R = r ) = \begin{cases} \frac { 1 } { 60 } ( a + ( 2 - r ) b ) & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are integers.
\item Use at least three of the observed frequencies to suggest appropriate values for $a$ and $b$. You should consider more than one possible pair of values, and explain which pair of values you consider best. (Do not carry out a goodness-of-fit test.)
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2023 Q8 [16]}}