| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | 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 requiring routine calculations: finding expected frequencies using binomial probabilities (with symmetry simplifying the work), computing the test statistic, and applying the standard hypothesis testing framework. Part (c) tests basic understanding of degrees of freedom. While it's a multi-part question worth 12 marks, all steps follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous |
| Number of females | Observed number of litters | Expected number of litters |
| 0 | 1 | 0.78 |
| 1 | 9 | 6.25 |
| 2 | 27 | 21.88 |
| 3 | 46 | \(R\) |
| 4 | 49 | \(S\) |
| 5 | 35 | \(T\) |
| 6 | 26 | 21.88 |
| 7 | 5 | 6.25 |
| 8 | 2 | 0.78 |
Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters $n = 8$ and $p = 0.5$ is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Number of females & Observed number of litters & Expected number of litters \\
\hline
0 & 1 & 0.78 \\
1 & 9 & 6.25 \\
2 & 27 & 21.88 \\
3 & 46 & $R$ \\
4 & 49 & $S$ \\
5 & 35 & $T$ \\
6 & 26 & 21.88 \\
7 & 5 & 6.25 \\
8 & 2 & 0.78 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the values of $R$, $S$ and $T$. [4]
\item Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a 5\% level of significance. [7]
\end{enumerate}
An alternative test might have involved estimating $p$ rather than assuming $p = 0.5$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Explain how this would have affected the test. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q6 [12]}}