| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Moderate -0.3 This is a standard chi-squared test of independence with all calculations scaffolded. Parts (i) and (ii) are routine verification of given values using formulas, (iii) is a standard hypothesis test comparing to critical values, and (iv) requires basic interpretation of a provided table. The question requires only recall and application of standard procedures with no problem-solving or novel insight. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{Observed Frequency} | Place | ||||
| Farm | Garden | Woodland | Totals | ||
| \multirow{4}{*}{Species} | Thrushes | 11 | 74 | 7 | 92 |
| Tits | 70 | 26 | 88 | 184 | |
| Finches | 17 | 2 | 10 | 29 | |
| Totals | 98 | 102 | 105 | 305 | |
| \multirow{2}{*}{Expected Frequency} | Place | |||
| Farm | Garden | Woodland | ||
| \multirow{3}{*}{Species} | Thrushes | 29.5607 | 30.7672 | 31.6721 |
| Tits | 59.1213 | 61.5344 | 63.3443 | |
| Finches | 9.3180 | 9.6984 | 9.9836 | |
| \multirow{2}{*}{Contribution} | Place | |||
| Farm | Garden | Woodland | ||
| \multirow{3}{*}{Species} | Thrushes | 11.6539 | 60.7489 | 19.2192 |
| Tits | 2.0017 | 20.5201 | 9.5969 | |
| Finches | 6.3332 | 6.1108 | 0.0000 | |
| Answer | Marks |
|---|---|
| Marks: M1 | A1 |
| Answer | Marks |
|---|---|
| Marks: M1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Marks: B1 | M1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Marks: E2, 1.0 | E2,1,0 | E2,1,0 |
| Answer | Marks | Guidance |
|---|---|---|
| Farm | Garden | Woodland |
| Thrushes | 11.6539 (far) fewer than expected | 60.7489 Far more than expected |
| Tits | 2.0017 As expected/ more than expected | 20.5201 (far) fewer than expected |
| Finches | 6.3332 (far) more than expected | 6.1108 fewer than expected |
## (i)
**Answer:** Expected frequency = $\frac{29}{305} \times 98 = 9.3180$
**Marks:** M1 | A1
**Guidance:**
- M1 for row total × column total ÷ grand total
- **NB Answer given**
---
## (ii)
**Answer:** Contribution = $(17 - 9.3180)^2 / 9.3180 = 6.3332$
**Marks:** M1 | A1
**Guidance:**
- M1 for valid attempt at (fo − fe)² ÷ fe
- **NB Answer given**
---
## (iii)
**Answer:**
$$H_0: \text{no association between place and species.}$$
$$H_1: \text{some association between place and species.}$$
Test statistic $X^2 = 136.185$
Refer to $X_{.} ^2$
Critical value at 1% level is 13.28
Result is significant
There is (strong) evidence to suggest association between place and species.
**Marks:** B1 | M1 | A1 | B1 | B1 | B1 | E1
**Guidance:**
- B1 for hypotheses in context
- NB if $H_0$ $H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final E1
- M1 for sum of contributions
- A1 for test statistic. Allow 136.1847
- B1 For 4 deg of freedom (seen)
- CAO For cv – no FT from here if wrong/omitted
- B1 For significant
- E1 for correct conclusion in context
---
## (iv)
**Answer:**
**Farm**
Contribution of 11.6539 implies that there are (far) fewer thrushes than expected Also more finches than expected and more tits.
**Garden**
Contribution of 60.7489 implies that there are far more thrushes than expected. Contribution of 20.5201 implies that there are (far) fewer tits than expected. Also fewer finches than expected.
**Woodland**
Contribution of 19.2192 implies that there are (far) fewer thrushes than expected. Contribution of 9.5969 implies that there are (far) more tits than expected. Number of finches is as expected.
**Marks:** E2, 1.0 | E2,1,0 | E2,1,0 | [6]
**Guidance:**
- **For each place** - award E2 for an explicit reference to a value from the contributions table with a correct* corresponding comment, provided that no further incorrect statements are made regarding that place.
- *See table below
- Award E1 either for an explicit/implicit reference to a value from the contributions table with a correct corresponding comment accompanied by one or more incorrect comment for that place
- Allow alternative wording. e.g. 'more thrushes were expected' in place of 'there were fewer thrushes than expected'
| | Farm | Garden | Woodland |
|---|---|---|---|
| Thrushes | 11.6539 (far) fewer than expected | 60.7489 Far more than expected | 19.2192 (far) fewer than expected |
| Tits | 2.0017 As expected/ more than expected | 20.5201 (far) fewer than expected | 9.5969 (far) more than expected |
| Finches | 6.3332 (far) more than expected | 6.1108 fewer than expected | 0.0000 as expected |4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Observed Frequency}} & \multicolumn{4}{|c|}{Place} \\
\hline
& & Farm & Garden & Woodland & Totals \\
\hline
\multirow{4}{*}{Species} & Thrushes & 11 & 74 & 7 & 92 \\
\hline
& Tits & 70 & 26 & 88 & 184 \\
\hline
& Finches & 17 & 2 & 10 & 29 \\
\hline
& Totals & 98 & 102 & 105 & 305 \\
\hline
\end{tabular}
\end{center}
The expected frequencies under the null hypothesis for the usual $\chi ^ { 2 }$ test are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Expected Frequency}} & \multicolumn{3}{|c|}{Place} \\
\hline
& & Farm & Garden & Woodland \\
\hline
\multirow{3}{*}{Species} & Thrushes & 29.5607 & 30.7672 & 31.6721 \\
\hline
& Tits & 59.1213 & 61.5344 & 63.3443 \\
\hline
& Finches & 9.3180 & 9.6984 & 9.9836 \\
\hline
\end{tabular}
\end{center}
(i) Verify that the entry 9.3180 is correct.
The corresponding contributions to the test statistic are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|r|}{\multirow{2}{*}{Contribution}} & \multicolumn{3}{|c|}{Place} \\
\hline
& & Farm & Garden & Woodland \\
\hline
\multirow{3}{*}{Species} & Thrushes & 11.6539 & 60.7489 & 19.2192 \\
\hline
& Tits & 2.0017 & 20.5201 & 9.5969 \\
\hline
& Finches & 6.3332 & 6.1108 & 0.0000 \\
\hline
\end{tabular}
\end{center}
(ii) Verify that the entry 6.3332 is correct.\\
(iii) Carry out the test at the $1 \%$ level of significance.\\
(iv) For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.
\hfill \mbox{\textit{OCR MEI S2 2012 Q4 [17]}}