| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Poisson |
| Difficulty | Standard +0.3 This is a standard chi-squared goodness of fit test with a given Poisson distribution. Part (a) requires straightforward calculation of Poisson probabilities P(X=4) and P(X≥7) multiplied by 120. Part (b) follows the routine procedure: combine cells to ensure expected frequencies ≥5, calculate test statistic, find critical value, and conclude. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number per square metre | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Observed frequency | 12 | 20 | 36 | 32 | 13 | 6 | 1 | 0 |
| Expected frequency | 9.85 | 24.63 | 30.78 | 25.65 | \(p\) | 8.02 | 3.34 | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p = 16.03\) | B1 | |
| \(q = 1.70\) | B1 | Condone 1.7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Combine last two columns: O value \(= 1\), E value \(= 5.04\) | M1 | Add last 2 columns (or 3 columns: 13.06) |
| \(\frac{(O-E)^2}{E}\): \(0.4693\), \(0.8704\), \(0.8853\), \(1.5720\), \(0.5727\), \(0.5088\), \(3.2384\) | M1 | Calculate values |
| Test statistic \(= 8.12\) | A1 | |
| \(H_0\): data fits a Poisson distribution with mean 2.5; \(H_1\): data does not fit a Poisson distribution with mean 2.5 | B1 | Need data and distribution. e.g. Data fits the given distribution. Number of butterflies per square metre fits Po(2.5) |
| Critical value of chi-squared \(= 10.64\); Compare: \(8.12 < 10.64\), accept \(H_0\) | M1 | Compare their test statistic with 10.64. Note: allow 9.236 if 3 columns combined or 12.02 if none combined |
| There is sufficient evidence to support the scientist's claim / data fits a Poisson distribution with mean 2.5 | A1 | Correct conclusion, in context. Level of uncertainty in language is used |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p = 16.03$ | B1 | |
| $q = 1.70$ | B1 | Condone 1.7 |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Combine last two columns: O value $= 1$, E value $= 5.04$ | M1 | Add last 2 columns (or 3 columns: 13.06) |
| $\frac{(O-E)^2}{E}$: $0.4693$, $0.8704$, $0.8853$, $1.5720$, $0.5727$, $0.5088$, $3.2384$ | M1 | Calculate values |
| Test statistic $= 8.12$ | A1 | |
| $H_0$: data fits a Poisson distribution with mean 2.5; $H_1$: data does not fit a Poisson distribution with mean 2.5 | B1 | Need data and distribution. e.g. Data fits the given distribution. Number of butterflies per square metre fits Po(2.5) |
| Critical value of chi-squared $= 10.64$; Compare: $8.12 < 10.64$, accept $H_0$ | M1 | Compare their test statistic with 10.64. Note: allow 9.236 if 3 columns combined or 12.02 if none combined |
| There is sufficient evidence to support the scientist's claim / data fits a Poisson distribution with mean 2.5 | A1 | Correct conclusion, in context. Level of uncertainty in language is used |4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims that the numbers of these butterflies found per square metre can be modelled by a Poisson distribution with mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the number of these butterflies in each of these areas. The following table shows the observed frequencies together with some of the expected frequencies using the scientist's Poisson distribution.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Number per square metre & 0 & 1 & 2 & 3 & 4 & 5 & 6 & $\geqslant 7$ \\
\hline
Observed frequency & 12 & 20 & 36 & 32 & 13 & 6 & 1 & 0 \\
\hline
Expected frequency & 9.85 & 24.63 & 30.78 & 25.65 & $p$ & 8.02 & 3.34 & $q$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $p$ and $q$, correct to 2 decimal places.
\item Carry out a goodness of fit test, at the $10 \%$ significance level, to test the scientist's claim.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q4 [8]}}