| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Uniform |
| Difficulty | Moderate -0.3 This is a straightforward chi-squared goodness of fit test with equal expected frequencies. Parts (a) and (b) are trivial recall (discrete uniform distribution), and part (c) follows a standard template: state hypotheses, calculate expected frequencies (all 50), compute chi-squared statistic, compare to critical value, and conclude. The calculation is routine with no conceptual challenges beyond applying the standard procedure. |
| Spec | 5.02a Discrete probability distributions: general5.06c Fit other distributions: discrete and continuous |
| Answer | Marks |
|---|---|
| \(P(X = x) = \frac{1}{6}; x = 1, 2, \ldots, 6\) | B1 B1 |
| (2 marks) |
| Answer | Marks |
|---|---|
| Discrete uniform distribution | B1 |
| (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\) : Discrete uniform distribution is a suitable model | B1 | |
| \(H_1\) : Discrete uniform distribution is not a suitable model | B1 | |
| \(\alpha = 0.05\) \(\nu = 5\); CR: \(\chi^2 > 11.070\) | B1 B1 | |
| \(\sum \frac{(O-E)^2}{E} = \frac{1}{50}\{9^2 + 1^2 + 2^2 + 8^2 + 13^2 + 13^2\}\) | B1 | All E's = 50 |
| \(= \frac{448}{50} = 9.76\) | M1 A1 | \(\sum \frac{(O-E)^2}{E}\) |
| Since 9.76 is not in the critical region there is no evidence to reject \(H_0\) and thus the data is compatible with the assumption. | A1 ft | |
| (8 marks) |
## (a)
$P(X = x) = \frac{1}{6}; x = 1, 2, \ldots, 6$ | B1 B1 |
| (2 marks) |
## (b)
Discrete uniform distribution | B1 |
| (1 mark) |
## (c)
$H_0$ : Discrete uniform distribution is a suitable model | B1 |
$H_1$ : Discrete uniform distribution is not a suitable model | B1 |
$\alpha = 0.05$ $\nu = 5$; CR: $\chi^2 > 11.070$ | B1 B1 |
$\sum \frac{(O-E)^2}{E} = \frac{1}{50}\{9^2 + 1^2 + 2^2 + 8^2 + 13^2 + 13^2\}$ | B1 | All E's = 50
$= \frac{448}{50} = 9.76$ | M1 A1 | $\sum \frac{(O-E)^2}{E}$
Since 9.76 is not in the critical region there is no evidence to reject $H_0$ and thus the data is compatible with the assumption. | A1 ft |
| (8 marks) |For a six-sided die it is assumed that each of the sides has an equal chance of landing uppermost when the die is rolled.
\begin{enumerate}[label=(\alph*)]
\item Write down the probability function for the random variable $X$, the number showing on the uppermost side after the die has been rolled. [2]
\item State the name of the distribution. [1]
\end{enumerate}
A student wishing to check the above assumption rolled the die 300 times and for the sides 1 to 6, obtained the frequencies 41, 49, 52, 58, 37 and 63 respectively.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Analyse these data and comment on whether or not the assumption is valid for this die. Use a 5\% level of significance and state your hypotheses clearly. [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q5 [11]}}