| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 11 |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a straightforward twist requiring column combination due to small expected frequencies. While it tests understanding of the '5 rule' and requires calculating expected frequencies, test statistic, and interpretation, these are routine procedures for Further Statistics students with no novel problem-solving required. The multi-part structure guides students through each step clearly. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | Subject | Mathematics | English | Physics | ||
\multirow{3}{*}{
| Year 7 | 17 | 16 | 7 | ||
| \cline { 2 - 5 } | Year 12 | 13 | 2 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequencies 20, 12, 8; 10, 6, 4 | B1\* | 1.1 |
| One expected frequency is less than 5 | depB1 | 2.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Mathematics and Physics are the more closely related subjects | B1 | 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): No association between subject and year | B1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Observed | Expected | |
| 24, 16 | 28, 12 | |
| 18, 2 | 14, 6 | |
| \(\sum\frac{(O-E)-0.5)^2}{E} = 0.4735 + 1.0208 + 0.875 + 2.042 = 4.375\) | M1 | 1.1a |
| \(= 4.375\) | A1 | 1.1 |
| \(> 2.706\) | B1 | 1.1a |
| Reject \(H_0\) | M1FT | 2.2b |
| There is significant evidence of association between choice of subject and year of entry | A1 FT | 3.5a |
| Answer | Marks | Guidance |
|---|---|---|
| English/Year 12 entry | B1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| Significantly fewer students who enter in Year 12 choose English than would be expected if there were no association. | B1 FT | 3.2a |
### Part (i)
Expected frequencies 20, 12, 8; 10, 6, 4 | **B1\*** | 1.1
One expected frequency is less than 5 | **depB1** | 2.3
### Part (ii)
Mathematics and Physics are the more closely related subjects | **B1** | 2.4
### Part (iii)
$H_0$: No association between subject and year | **B1** | 1.1
$H_1$: there is association between subject & year
Combine cells to obtain:
| Observed | Expected |
|----------|----------|
| 24, 16 | 28, 12 |
| 18, 2 | 14, 6 |
$\sum\frac{(O-E)-0.5)^2}{E} = 0.4735 + 1.0208 + 0.875 + 2.042 = 4.375$ | **M1** | 1.1a | No Yates (5.714): M1 A0
$= 4.375$ | **A1** | 1.1
$> 2.706$ | **B1** | 1.1a
Reject $H_0$ | **M1FT** | 2.2b | Correct method and first conclusion | FT on TS only
There is significant evidence of association between choice of subject and year of entry | **A1 FT** | 3.5a | Contextualised, not too definite
### Part (iv)
English/Year 12 entry | **B1** | 2.2a | FT on Physics & English makes no sense here
### Part (v)
Significantly fewer students who enter in Year 12 choose English than would be expected if there were no association. | **B1 FT** | 3.2a
---7 The numbers of students taking A levels in three subjects at a school were classified by the year in which they entered the school as follows.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & Subject & Mathematics & English & Physics \\
\hline
\multirow{3}{*}{\begin{tabular}{ c }
Year of \\
Entry \\
\end{tabular}} & Year 7 & 17 & 16 & 7 \\
\cline { 2 - 5 }
& Year 12 & 13 & 2 & 5 \\
\hline
\end{tabular}
\end{center}
The Head of the school carries out a significance test at the $10 \%$ level to test whether subjects taken are independent of year of entry.\\
(i) Show that in carrying out the test it is necessary to combine columns.\\
(ii) Suggest a reason why it is more sensible to combine the columns for Mathematics and Physics than the columns for Physics and English.\\
(iii) Carry out the test.\\
(iv) State which cell gives the largest contribution to the test statistic.\\
(v) Interpret your answer to part (iv).
\hfill \mbox{\textit{OCR FS1 AS 2018 Q7 [11]}}