| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2002 |
| Session | June |
| Marks | 12 |
| 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 requiring routine application of the binomial distribution to find expected frequencies, calculation of the test statistic, and comparison with critical values. Part (a) uses symmetry of the binomial distribution, part (b) follows a standard hypothesis testing procedure, and part (c) requires recall of degrees of freedom rules. While it has multiple parts worth 12 marks total, each component is straightforward application of learned techniques with no novel problem-solving required. |
| 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 |
| Answer | Marks |
|---|---|
| \(R = 43.76; S = 54.68; T = 43.76\) using tables (OR \(R = 43.75; S = 54.69; T = 43.75\) using calculator) | M1 A1; B1 B1 (4) |
| Answer | Marks |
|---|---|
| \(H_0:\) Binomial model with \(n = 8, p = 0.5\) is suitable | B1 (both) |
| \(H_1:\) Binomial model with \(n = 8, p = 0.5\) is not suitable | |
| Amalgamation of data | M1 |
| \(\sum \frac{(O-E)^2}{E} = 5.69\) (awrt) | M1 A1 |
| \(\alpha = 0.05, \nu = 6 \Rightarrow \text{CR: } \chi^2 > 12.592\) | B1 B1 |
| Since \(5.69\) is not in the critical region there is no evidence to reject \(H_0\). The binomial model with \(n = 8\) and \(p = 0.5\) is a suitable model. | A1ft (7) |
| Answer | Marks |
|---|---|
| Apart from the expected values and \(\sum \frac{(O-E)^2}{E}\) being different, the degrees of freedom would have been reduced by 1 (\(\nu = 5\)). | B1 (1) |
| Total: 12 marks |
## Part (a)
| $R = 43.76; S = 54.68; T = 43.76$ using tables (OR $R = 43.75; S = 54.69; T = 43.75$ using calculator) | M1 A1; B1 B1 (4) | |
## Part (b)
| $H_0:$ Binomial model with $n = 8, p = 0.5$ is suitable | B1 (both) | |
| $H_1:$ Binomial model with $n = 8, p = 0.5$ is not suitable | | |
| Amalgamation of data | M1 | |
| $\sum \frac{(O-E)^2}{E} = 5.69$ (awrt) | M1 A1 | |
| $\alpha = 0.05, \nu = 6 \Rightarrow \text{CR: } \chi^2 > 12.592$ | B1 B1 | |
| Since $5.69$ is not in the critical region there is no evidence to reject $H_0$. The binomial model with $n = 8$ and $p = 0.5$ is a suitable model. | A1ft (7) | |
## Part (c)
| Apart from the expected values and $\sum \frac{(O-E)^2}{E}$ being different, the degrees of freedom would have been reduced by 1 ($\nu = 5$). | B1 (1) | |
| **Total: 12 marks** | | |
---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{enumi}{2}
\item Explain how this would have affected the test. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2002 Q6 [12]}}