| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Test statistic given, complete the test |
| Difficulty | Moderate -0.3 This is a standard chi-squared test of independence with straightforward calculations. Parts (a)-(c) are routine bookwork requiring only recall of hypothesis format, expected frequency formula (row total × column total / grand total), and degrees of freedom formula (r-1)(c-1). Part (d) requires comparing a given test statistic to critical values from tables. No problem-solving or novel insight required, though it's slightly above trivial recall questions. |
| Spec | 5.06a Chi-squared: contingency tables |
| Computer game | \(A\) | \(B\) | \(C\) | |
| \multirow{3}{*}{Age range} | \(< 20\) | 8 | 15 | 6 |
| \cline { 2 - 5 } | \(20 - 30\) | 21 | 12 | 9 |
| \cline { 2 - 5 } | \(> 30\) | 6 | 10 | 13 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): There is no association between age range and preferred game | B1 | Both hypotheses required in context; must mention independence/dependence or no association/association; must mention age and game at least once. Do not allow e.g. '\(H_0\): games sold appeal equally to all age ranges'. Use of 'correlation', 'link', 'relationship', 'connection' is B0 but allow for \(2^{nd}\) B1 in (d) |
| \(H_1\): There is an association between age range and preferred game | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (i) \(\frac{29 \times 28}{100} = \mathbf{8.12}\) | M1 | For one correct expression or correct value |
| (ii) \(\frac{42 \times 35}{100} = \mathbf{14.7}\) | A1 | For both correct values (correct to 1 dp). Accept 8.1 |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Since (b)(i) \(> 5\) there is no pooling so \(df = (3-1)\times(3-1) = \mathbf{4}\) | B1 | For 4 cao |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\chi^2_4(0.05) = \mathbf{9.488}\) | B1ft | For awrt 9.488 or ft their df for a 5% critical value. May see \(\chi^2_3(0.05)=7.815\), \(\chi^2_2(0.05)=5.991\), \(\chi^2_7(0.05)=14.067\), \(\chi^2_8(0.05)=15.507\) |
| The test is significant so reject \(H_0\): Sharma's belief is not supported / there is significant evidence of an association between age range and computer game preference. | B1 | Indep of hypotheses but dep on cv \(< 11.542\). For correct statement in context about Sharma's belief. Condone e.g. 'games sold do not appeal equally to all age ranges'. For a contextual statement must mention age and game. B0 for contradictory statements e.g. "test is not significant so belief is not supported" |
| (2) | ||
| (6 marks) |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: There is no association between age range and preferred game | B1 | Both hypotheses required in context; must mention independence/dependence or no association/association; must mention age and game at least once. Do not allow e.g. '$H_0$: games sold appeal equally to all age ranges'. Use of 'correlation', 'link', 'relationship', 'connection' is B0 but allow for $2^{nd}$ B1 in (d) |
| $H_1$: There is an association between age range and preferred game | (1) | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (i) $\frac{29 \times 28}{100} = \mathbf{8.12}$ | M1 | For one correct expression or correct value |
| (ii) $\frac{42 \times 35}{100} = \mathbf{14.7}$ | A1 | For both correct values (correct to 1 dp). Accept 8.1 |
| | (2) | |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Since (b)(i) $> 5$ there is no pooling so $df = (3-1)\times(3-1) = \mathbf{4}$ | B1 | For 4 cao |
| | (1) | |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\chi^2_4(0.05) = \mathbf{9.488}$ | B1ft | For awrt 9.488 or ft their df for a 5% critical value. May see $\chi^2_3(0.05)=7.815$, $\chi^2_2(0.05)=5.991$, $\chi^2_7(0.05)=14.067$, $\chi^2_8(0.05)=15.507$ |
| The test is significant so reject $H_0$: Sharma's belief is not supported / there is significant evidence of an association between age range and computer game preference. | B1 | **Indep of hypotheses but dep on cv $< 11.542$**. For correct statement in context about Sharma's belief. Condone e.g. 'games sold do **not** appeal equally to all age ranges'. For a contextual statement must mention age and game. B0 for contradictory statements e.g. "test is not significant so belief is not supported" |
| | (2) | |
| | **(6 marks)** | |\begin{enumerate}
\item Sharma believes that each computer game he sells appeals equally to all age ranges.
\end{enumerate}
To investigate this, he takes a random sample of 100 people who play these games and asks them which of the games $A , B$ or $C$ they prefer.\\
The results are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{Computer game} & $A$ & $B$ & $C$ \\
\hline
\multirow{3}{*}{Age range} & $< 20$ & 8 & 15 & 6 \\
\cline { 2 - 5 }
& $20 - 30$ & 21 & 12 & 9 \\
\cline { 2 - 5 }
& $> 30$ & 6 & 10 & 13 \\
\hline
\end{tabular}
\end{center}
(a) Write down hypotheses for a suitable test to assess Sharma's belief.\\
(b) For the test, calculate the expected frequency for\\
(i) those players aged under 20 who prefer game $C$\\
(ii) those players aged between 20 and 30 who prefer game $A$\\
(c) State the degrees of freedom of the test statistic for this test.
Sharma correctly calculates the test statistic for this test to be 11.542 (to 3 decimal places).\\
(d) Using a $5 \%$ significance level, and stating your critical value, comment on Sharma's belief.
\hfill \mbox{\textit{Edexcel FS1 AS 2024 Q1 [6]}}