| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | June |
| Marks | 18 |
| 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 calculations. Part (i) requires writing standard hypotheses, (ii) involves basic expected frequency calculation and verification, (iii) is routine hypothesis testing with given test statistic, (iv) requires interpreting contributions (accessible from the table), and (v) is a simple probability calculation. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.06a Chi-squared: contingency tables |
| Location | ||||
| \cline { 3 - 5 } \multicolumn{2}{|c|}{} | Exposed | Sheltered | Pool | |
| \multirow{3}{*}{Species} | Limpet | 24 | 32 | 16 |
| \cline { 2 - 5 } | Mussel | 24 | 11 | 3 |
| \cline { 2 - 5 } | Other | 5 | 22 | 23 |
| Contribution | Location | |||
| \cline { 3 - 5 } | Exposed | Sheltered | Pool | |
| \multirow{3}{*}{Species} | Limpet | 0.0009 | 0.2585 | 0.4450 |
| \cline { 2 - 5 } | Mussel | 10.3472 | 1.2756 | 4.8773 |
| \cline { 2 - 5 } | Other | 8.0719 | 0.1402 | 7.4298 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): no association between location and species | B1 for both | |
| \(H_1\): some association between location and species | — | 1 mark total |
| Answer | Marks | Guidance |
|---|---|---|
| Expected frequency \(= \frac{38}{160}\times 42 = 9.975\) | M1 A1 — M1 for valid attempt at \((O-E)^2/E\) | |
| Contribution \(= \frac{(3-9.975)^2}{9.975} = 4.8773\) | A1 NB Answer given | 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Refer to \(\chi^2_4\); critical value at 5% level \(= 9.488\) | B1 for 4 deg of f (seen), B1 CAO for cv | |
| Test statistic \(X^2 = 32.85\) | M1 sensible comparison, using 32.85, leading to a conclusion | |
| Result is significant | A1 for correct conclusion (FT their c.v.) | |
| There appears to be some association between location and species | E1 conclusion in context | |
| NB if \(H_0\ H_1\) reversed, or 'correlation' mentioned, do not award first B1 or final E1 | — | 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| - Limpets appear to be distributed as expected throughout all locations | E1 | |
| - Mussels are much more frequent in exposed locations and much less in pools than expected | E1, E1 | |
| - Other shellfish are less frequent in exposed locations and more frequent in pools than expected | E1, E1 | 5 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{24}{53}\times\frac{32}{65}\times\frac{16}{42} = 0.0849\) | M1 for one fraction, M1 for product of all 3, A1 CAO | 3 marks total |
# Question 4:
## Part (i)
$H_0$: no association between location and species | B1 for both |
$H_1$: some association between location and species | — | 1 mark total
## Part (ii)
Expected frequency $= \frac{38}{160}\times 42 = 9.975$ | M1 A1 — M1 for valid attempt at $(O-E)^2/E$ |
Contribution $= \frac{(3-9.975)^2}{9.975} = 4.8773$ | A1 **NB Answer given** | 4 marks total
## Part (iii)
Refer to $\chi^2_4$; critical value at 5% level $= 9.488$ | B1 for 4 deg of f (seen), B1 CAO for cv |
Test statistic $X^2 = 32.85$ | M1 sensible comparison, using 32.85, leading to a conclusion |
Result is significant | A1 for correct conclusion (FT their c.v.) |
There appears to be some association between location and species | E1 conclusion in context |
NB if $H_0\ H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final E1 | — | 5 marks total
## Part (iv)
- Limpets appear to be distributed as expected throughout all locations | E1 |
- Mussels are much more frequent in exposed locations and much less in pools than expected | E1, E1 |
- Other shellfish are less frequent in exposed locations and more frequent in pools than expected | E1, E1 | 5 marks total
## Part (v)
$\frac{24}{53}\times\frac{32}{65}\times\frac{16}{42} = 0.0849$ | M1 for one fraction, M1 for product of all 3, A1 CAO | 3 marks total4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.
\begin{center}
\begin{tabular}{ | c | l | c | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{3}{|c|}{Location} \\
\cline { 3 - 5 }
\multicolumn{2}{|c|}{} & Exposed & Sheltered & Pool \\
\hline
\multirow{3}{*}{Species} & Limpet & 24 & 32 & 16 \\
\cline { 2 - 5 }
& Mussel & 24 & 11 & 3 \\
\cline { 2 - 5 }
& Other & 5 & 22 & 23 \\
\hline
\end{tabular}
\end{center}
(i) Write down null and alternative hypotheses for a test to examine whether there is any association between species and location.
The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | r | c | c | }
\hline
\multicolumn{2}{|c|}{Contribution} & \multicolumn{3}{|c|}{Location} \\
\cline { 3 - 5 }
& Exposed & Sheltered & Pool & \\
\hline
\multirow{3}{*}{Species} & Limpet & 0.0009 & 0.2585 & 0.4450 \\
\cline { 2 - 5 }
& Mussel & 10.3472 & 1.2756 & 4.8773 \\
\cline { 2 - 5 }
& Other & 8.0719 & 0.1402 & 7.4298 \\
\hline
\end{tabular}
\end{center}
The sum of these contributions is 32.85 .\\
(ii) Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.\\
(iii) Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.\\
(iv) For each species, comment briefly on how its distribution compares with what would be expected if there were no association.\\
(v) If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.
\hfill \mbox{\textit{OCR MEI S2 2008 Q4 [18]}}