| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Larger contingency table (4+ categories) |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly presented contingency table data. Students must calculate expected frequencies, compute the test statistic, find critical value from tables, and state conclusion. While it requires multiple computational steps and understanding of the independence test framework, it follows a routine procedure taught explicitly in Further Statistics with no novel insight required. The 4×2 table is straightforward, and all necessary information is provided. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Beach location | |||||
| A | \(B\) | C | D | Total | ||
| \multirow{2}{*}{Size of shell} | Large | 68 | 69 | 96 | 81 | 314 |
| Small | 28 | 55 | 64 | 39 | 186 | |
| Total | 96 | 124 | 160 | 120 | 500 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): the size of shell is independent of the location; \(H_1\): the size of shell is not independent of the location | B1 | |
| Expected values: \(E\): 60.29, 77.87, 100.48, 75.36 (top row) and 35.71, 46.13, 59.52, 44.64 (bottom row) | M1 A1 | Attempt at calculation of E values |
| \(\frac{(O-E)^2}{E}\) values: 0.9859, 1.0104, 0.1997, 0.4221, 1.6646, 1.7055, 0.3372, 0.7126 | M1 | Attempt at Chi-squared contributions. Accuracy of \(\pm 0.1\) |
| Test statistic \(= 7.04\) | A1 | Correct total to 3sf |
| Tabular value for \(3df = 6.251\); \(7.04 > 6.251\); Reject \(H_0\) | M1 | Compare with 6.251 and correct FT conclusion. May be implied by FT correct conclusion in words |
| Insufficient evidence to suggest size of shell is independent of location / Evidence suggests that size of shell is dependent on (beach) location | A1 | Correct conclusion, in context. Level of uncertainty in language is used |
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: the size of shell is independent of the location; $H_1$: the size of shell is not independent of the location | B1 | |
| Expected values: $E$: 60.29, 77.87, 100.48, 75.36 (top row) and 35.71, 46.13, 59.52, 44.64 (bottom row) | M1 A1 | Attempt at calculation of E values |
| $\frac{(O-E)^2}{E}$ values: 0.9859, 1.0104, 0.1997, 0.4221, 1.6646, 1.7055, 0.3372, 0.7126 | M1 | Attempt at Chi-squared contributions. Accuracy of $\pm 0.1$ |
| Test statistic $= 7.04$ | A1 | Correct total to 3sf |
| Tabular value for $3df = 6.251$; $7.04 > 6.251$; Reject $H_0$ | M1 | Compare with 6.251 and correct FT conclusion. May be implied by FT correct conclusion in words |
| Insufficient evidence to suggest size of shell is independent of location / Evidence suggests that size of shell is dependent on (beach) location | A1 | Correct conclusion, in context. Level of uncertainty in language is used |
---2 A scientist is investigating the size of shells at various beach locations. She selects four beach locations and takes a random sample of shells from each of these beaches. She classifies each shell as large or small. Her results are summarised in the following table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{Beach location} & \\
\hline
& & A & $B$ & C & D & Total \\
\hline
\multirow{2}{*}{Size of shell} & Large & 68 & 69 & 96 & 81 & 314 \\
\hline
& Small & 28 & 55 & 64 & 39 & 186 \\
\hline
& Total & 96 & 124 & 160 & 120 & 500 \\
\hline
\end{tabular}
\end{center}
Test, at the 10\% significance level, whether the size of shell is independent of the beach location.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q2 [7]}}