| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Binomial |
| Difficulty | Standard +0.8 This is a standard chi-squared goodness of fit test with binomial distribution, requiring calculation of sample statistics, verification of expected frequencies using binomial probabilities, and hypothesis testing. While it involves multiple steps and parameter estimation (reducing degrees of freedom), the procedures are routine for Further Maths statistics with no novel problem-solving required. The computational demands and multi-part structure place it slightly above average difficulty. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Observed frequency | 22 | 83 | 72 | 53 | 17 | 3 | 0 |
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Observed frequency | 22 | 83 | 72 | 53 | 17 | 3 | 0 |
| Expected frequency | 26.3 | 71.9 | 81.8 | 49.7 | 17.0 | 3.1 | 0.2 |
A scientist carries out an experiment to investigate the quantity $X$, which takes the values $0,1,2,3,4$, 5 or 6 . He believes that the values taken by $X$ follow a binomial distribution. He conducts 250 trials. His results are summarised in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Observed frequency & 22 & 83 & 72 & 53 & 17 & 3 & 0 \\
\hline
\end{tabular}
\end{center}
(i) Show that unbiased estimates of the mean and variance for these results are 1.876 and 1.266 respectively, correct to 3 decimal places. By evaluating the mean and variance of the distribution B(6, 0.313), explain why $X$ could have this distribution.\\
The expected frequencies corresponding to the distribution $\mathrm { B } ( 6,0.313 )$ are shown in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Observed frequency & 22 & 83 & 72 & 53 & 17 & 3 & 0 \\
\hline
Expected frequency & 26.3 & 71.9 & 81.8 & 49.7 & 17.0 & 3.1 & 0.2 \\
\hline
\end{tabular}
\end{center}
(ii) Show how the expected frequency for $x = 4$ is calculated.\\
(iii) Test at the $5 \%$ significance level whether the scientist's belief is correct.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q11 OR}}