| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×3 contingency table, requiring routine calculation of expected frequencies, test statistic, and comparison with critical value. The question is slightly easier than average because it's a straightforward application of a well-defined procedure with clear data, though it requires careful arithmetic across multiple steps. |
| Spec | 5.06a Chi-squared: contingency tables |
| \backslashbox{Annual income}{Finances} | Worse | Same | Better |
| Under £15 000 | 14 | 11 | 9 |
| £15000 and above | 17 | 20 | 29 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Expected frequencies: Under £15000: \(10.54, 10.54, 12.92\); £15000 and above: \(20.46, 20.46, 25.08\) | M1, A1 | For some use of \(\frac{\text{Row Total}\times\text{Col.Total}}{\text{Grand Total}}\); all expected frequencies correct |
| \(H_0\): State of finances and income are independent; \(H_1\): State of finances and income are not independent | B1 | Must mention "state" or "finances" and "income"; use of "relationship"/"correlation"/"connection" is B0 |
| \(\sum\frac{(O_i-E_i)^2}{E_i} = 3.553...\) | M1, A1 | At least two correct terms; all correct terms (awrt \(3.55\)) |
| \(v = (3-1)(2-1) = 2\) | B1 | |
| cv is \(5.991\) | B1 | |
| \(3.553 < 5.991\) so insufficient evidence to reject \(H_0\) | M1 | Correct statement linking test statistic and cv; must be \(\chi^2\) not normal |
| There is no evidence of association between state of finances and income | A1 | Correct comment in context; "correlation" is not acceptable; no follow through |
# Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Expected frequencies: Under £15000: $10.54, 10.54, 12.92$; £15000 and above: $20.46, 20.46, 25.08$ | M1, A1 | For some use of $\frac{\text{Row Total}\times\text{Col.Total}}{\text{Grand Total}}$; all expected frequencies correct |
| $H_0$: State of finances and income are independent; $H_1$: State of finances and income are not independent | B1 | Must mention "state" or "finances" and "income"; use of "relationship"/"correlation"/"connection" is B0 |
| $\sum\frac{(O_i-E_i)^2}{E_i} = 3.553...$ | M1, A1 | At least two correct terms; all correct terms (awrt $3.55$) |
| $v = (3-1)(2-1) = 2$ | B1 | |
| cv is $5.991$ | B1 | |
| $3.553 < 5.991$ so insufficient evidence to reject $H_0$ | M1 | Correct statement linking test statistic and cv; must be $\chi^2$ not normal |
| There is no evidence of association between state of finances and income | A1 | Correct comment in context; "correlation" is not acceptable; no follow through |
---5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\backslashbox{Annual income}{Finances} & Worse & Same & Better \\
\hline
Under £15 000 & 14 & 11 & 9 \\
\hline
£15000 and above & 17 & 20 & 29 \\
\hline
\end{tabular}
\end{center}
Test, at the $5 \%$ level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.\\
\includegraphics[max width=\textwidth, alt={}, center]{304e58fa-eb82-4e2d-83f4-848f3eb461c8-15_2576_1774_141_159}\\
\hfill \mbox{\textit{Edexcel S3 Q5 [10]}}