| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2018 |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Percentages given, table construction required |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly stated hypotheses, straightforward conversion of percentages to frequencies, and routine calculation of expected values and test statistic. While it requires multiple computational steps, it follows a well-practiced procedure with no conceptual challenges beyond applying the standard formula and comparing to critical values. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 3 - 4 } \multicolumn{2}{c|}{} | Male | Female | |
| \multirow{3}{*}{Grade} | Distinction | \(18.5 \%\) | \(27.5 \%\) |
| \cline { 2 - 4 } | Merit | \(63.5 \%\) | \(60.0 \%\) |
| \cline { 2 - 4 } | Unsatisfactory | \(18.0 \%\) | \(12.5 \%\) |
| 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 | For use of \(\frac{\text{Row Total} \times \text{Col. Total}}{\text{Grand Total}}\); may be implied by correct \(E_i\) |
| All expected frequencies correct | A1 | |
| \(H_0\): State of finances and income are independent (not associated) \(H_1\): State of finances and income are not independent (associated) | B1 | Must mention "state" or "finances" and "income"; use of "relationship"/"correlation"/"connection" is B0 |
| At least two correct \(\frac{(O_i-E_i)^2}{E_i}\) or \(\frac{O_i^2}{E_i}\) terms | M1 | |
| All correct terms | A1 | May be implied by correct answer (2 dp or better) |
| \(\sum \frac{(O_i-E_i)^2}{E_i} = 3.553...\) awrt 3.55 | A1 | |
| \(\nu = (3-1)(2-1) = 2\) | B1 | |
| cv is 5.991 | B1 | |
| \(3.553 < 5.991\) so insufficient evidence to reject \(H_0\) | M1 | Must be \(\chi^2\) not normal |
| There is no evidence of association between state of finances and income | A1 | Must mention "state"/"finances" and "income"; condone "relationship"/"connection" but not "correlation" |
# 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 | For use of $\frac{\text{Row Total} \times \text{Col. Total}}{\text{Grand Total}}$; may be implied by correct $E_i$ |
| All expected frequencies correct | A1 | |
| $H_0$: State of finances and income are independent (not associated) $H_1$: State of finances and income are not independent (associated) | B1 | Must mention "state" or "finances" and "income"; use of "relationship"/"correlation"/"connection" is B0 |
| At least two correct $\frac{(O_i-E_i)^2}{E_i}$ or $\frac{O_i^2}{E_i}$ terms | M1 | |
| All correct terms | A1 | May be implied by correct answer (2 dp or better) |
| $\sum \frac{(O_i-E_i)^2}{E_i} = 3.553...$ awrt **3.55** | A1 | |
| $\nu = (3-1)(2-1) = 2$ | B1 | |
| cv is 5.991 | B1 | |
| $3.553 < 5.991$ so insufficient evidence to reject $H_0$ | M1 | Must be $\chi^2$ not normal |
| There is no evidence of association between state of finances and income | A1 | Must mention "state"/"finances" and "income"; condone "relationship"/"connection" but **not** "correlation" |
---\begin{enumerate}
\item A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.
\end{enumerate}
The results are summarised below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\cline { 3 - 4 }
\multicolumn{2}{c|}{} & Male & Female \\
\hline
\multirow{3}{*}{Grade} & Distinction & $18.5 \%$ & $27.5 \%$ \\
\cline { 2 - 4 }
& Merit & $63.5 \%$ & $60.0 \%$ \\
\cline { 2 - 4 }
& Unsatisfactory & $18.0 \%$ & $12.5 \%$ \\
\hline
\end{tabular}
\end{center}
Stating your hypotheses clearly, test the Head of Department's belief using a $5 \%$ level of significance. Show your working clearly.
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel S3 2018 Q5 [12]}}