| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with straightforward hypotheses, routine expected frequency calculations, and table lookup. The only slightly non-routine element is part (b)(i) requiring identification of the minimum expected frequency cell, but this is a simple logical step (smallest row total × smallest column total). Overall, this is easier than average as it's a textbook application with no conceptual challenges. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Colour | ||||||
| Red | Blue | Green | Yellow | Black | Total | ||
| \multirow{3}{*}{Year group} | 1-5 | 34 | 15 | 14 | 22 | 3 | 88 |
| 6-9 | 23 | 32 | 12 | 9 | 8 | 84 | |
| 10-12 | 5 | 28 | 19 | 8 | 8 | 68 | |
| Total | 62 | 75 | 45 | 39 | 19 | 240 | |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(H_0:\) There is no association between the colour chosen and year group (allow use of "independence" instead of association) | B1 |
| \(H_1:\) There is some association between colour and year group | ||
| (b)(i) | Need lowest row total and column total so "Black" and "10-12" | B1 |
| (b)(ii) | With expected frequency \(\frac{68 \times 19}{240} \approx 5.3833\ldots\) | B1 |
| (c) | \(v = (5-1) \times (3-1) = \mathbf{8}\) | B1 |
| cv of \(\chi_8^2(1\%)\) = \(\mathbf{20.090}\) | B1ft | 1.1b |
| (significant) evidence of an association between colour chosen and year group | B1ft | 2.2b |
(a) | $H_0:$ There is no association between the colour chosen and year group (allow use of "independence" instead of association) | B1 | 2.5 |
| $H_1:$ There is some association between colour and year group | | |
(b)(i) | Need lowest row total and column total so "Black" and "10-12" | B1 | 2.2a |
(b)(ii) | With expected frequency $\frac{68 \times 19}{240} \approx 5.3833\ldots$ | B1 | 1.1b |
(c) | $v = (5-1) \times (3-1) = \mathbf{8}$ | B1 | 3.4 |
| cv of $\chi_8^2(1\%)$ = $\mathbf{20.090}$ | B1ft | 1.1b |
| (significant) evidence of an association between colour chosen and year group | B1ft | 2.2b |
**Notes:**
- (a) B1 for both hypotheses in context. Must mention colour and year group at least once
- (a) **NB:** condone use of related/correlated/linked etc if recovered with independent/associated
- (b)(i) B1 for a choosing "Black" and "10-12" with some correct reasoning mentioning row and column totals (allow clear equivalent, e.g. 'lowest total frequencies')
- (b)(ii) B1 for a correct expression or awrt 5.38
- (c) 1st B1 for a correct calculation or answer of 8. May be implied by sight of cv of 20.09
- (c) 2nd B1ft for 20.090 (accept 20.09) or a correct ft 1% cv using their df and correct to 2 d.p.
- (c) 3rd B1ft for a correct conclusion in context (ft their cv)
- (c) E.g. "students in different years (tend to) have different favourite colours"
- (c) Do not allow contradictory statements. Condone 'related/relationship' in part (c)
- **NB: We ignore their hypotheses when marking (c)**
---\begin{enumerate}
\item Tisam took a survey of students' favourite colours. The results are summarised in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{5}{|c|}{Colour} & \\
\hline
& & Red & Blue & Green & Yellow & Black & Total \\
\hline
\multirow{3}{*}{Year group} & 1-5 & 34 & 15 & 14 & 22 & 3 & 88 \\
\hline
& 6-9 & 23 & 32 & 12 & 9 & 8 & 84 \\
\hline
& 10-12 & 5 & 28 & 19 & 8 & 8 & 68 \\
\hline
& Total & 62 & 75 & 45 & 39 & 19 & 240 \\
\hline
\end{tabular}
\end{center}
Tisam carries out a suitable test to see if there is any association between favourite colour and year group.\\
(a) Write down the hypotheses for a suitable test.
For her table, Tisam only needs to check one cell to show that none of the expected frequencies are less than 5\\
(b) (i) Identify this cell, giving your reason.\\
(ii) Calculate the expected frequency for this cell.
The test statistic for Tisam's test is 38.449\\
(c) Using a $1 \%$ level of significance, complete the test.
You should state your critical value and conclusion clearly.
\hfill \mbox{\textit{Edexcel FS1 2024 Q3 [6]}}