| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared with algebraic frequencies |
| Difficulty | Challenging +1.2 This question requires understanding chi-squared test mechanics with algebraic frequencies, finding constraints from the critical value, and optimizing over integer values. While it involves multiple steps and algebraic manipulation beyond routine application, the conceptual framework is standard S3 material with clear signposting through parts (a) and (b). |
| Spec | 5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous |
| Number on die | 1 | 2 | 3 | 4 | 5 | 6 |
| Observed frequency | \(x + 6\) | \(x - 8\) | \(x + 8\) | \(x - 5\) | \(x + 4\) | \(x - 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(H_0\): The observed distribution can be modelled by a discrete uniform distribution | B1 | Allow \(H_0\): the die is not biased; \(H_1\): the die is biased |
| \(H_1\): The observed distribution cannot be modelled by a discrete uniform distribution | ||
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Expected frequency \(= x\) for each category | B1 | for expected frequency \(= x\) |
| One correct \(\frac{(O-E)^2}{E}\) or \(\frac{O^2}{E}\) ft their expected frequency | M1 | |
| \(X^2 = \sum\frac{(O-E)^2}{E}\) or \(\sum\frac{O^2}{E} - 6x\); \(= \frac{230}{x}\) or \(\frac{6x^2+230}{x} - 6x\) | M1; A1 | for attempt at \(X^2\) ft their values (at least 4 must be seen and added) |
| \(\nu = 6-1=5\); \(c^2_5(0.05) = 11.070 \Rightarrow\) CR: \(X^2 > 11.070\) | B1; B1 | for \(\nu=5\) (may be implied by correct critical value); for correct critical value ft DOF |
| Do not reject \(H_0\) if \(\frac{230}{x} \leq 11.070\) or \(\frac{6x^2+230}{x} - 6x \leq 11.070\) | M1 | for either expression compared with their CV |
| \(x \geq 20.7768...\), so \(x = 21\) | A1 | for \(x=21\) provided previous M mark awarded |
| (8) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Hence the die was rolled \(21 \times 6 = 126\) times | M1 A1 | M1 for their \(21 \times 6\); allow \(6 \times x\) or \(6 \times\) their value for \(x\); A1 cao |
| (2) |
## Question 7:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $H_0$: The observed distribution can be modelled by a discrete uniform distribution | B1 | Allow $H_0$: the die is not biased; $H_1$: the die is biased |
| $H_1$: The observed distribution cannot be modelled by a discrete uniform distribution | | |
| | **(1)** | |
### Part (b)(i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Expected frequency $= x$ for each category | B1 | for expected frequency $= x$ |
| One correct $\frac{(O-E)^2}{E}$ or $\frac{O^2}{E}$ ft their expected frequency | M1 | |
| $X^2 = \sum\frac{(O-E)^2}{E}$ or $\sum\frac{O^2}{E} - 6x$; $= \frac{230}{x}$ or $\frac{6x^2+230}{x} - 6x$ | M1; A1 | for attempt at $X^2$ ft their values (at least 4 must be seen and added) |
| $\nu = 6-1=5$; $c^2_5(0.05) = 11.070 \Rightarrow$ CR: $X^2 > 11.070$ | B1; B1 | for $\nu=5$ (may be implied by correct critical value); for correct critical value ft DOF |
| Do not reject $H_0$ if $\frac{230}{x} \leq 11.070$ or $\frac{6x^2+230}{x} - 6x \leq 11.070$ | M1 | for either expression compared with their CV |
| $x \geq 20.7768...$, so $x = 21$ | A1 | for $x=21$ provided previous M mark awarded |
| | **(8)** | |
### Part (b)(ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Hence the die was rolled $21 \times 6 = 126$ times | M1 A1 | M1 for their $21 \times 6$; allow $6 \times x$ or $6 \times$ their value for $x$; A1 cao |
| | **(2)** | |7 The following table shows observed frequencies, where $x$ is an integer, from an experiment to test whether or not a six-sided die is biased.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
Number on die & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Observed frequency & $x + 6$ & $x - 8$ & $x + 8$ & $x - 5$ & $x + 4$ & $x - 5$ \\
\hline
\end{tabular}
\end{center}
A goodness of fit test is conducted to determine if there is evidence that the die is biased.
\begin{enumerate}[label=(\alph*)]
\item Write down suitable null and alternative hypotheses for this test.
It is found that the null hypothesis is not rejected at the $5 \%$ significance level.
\item Hence
\begin{enumerate}[label=(\roman*)]
\item find the minimum value of $x$
\item determine the minimum number of times the die was rolled.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2022 Q7 [11]}}