| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a straightforward chi-squared test of independence with a 2×3 contingency table. Students need to calculate expected frequencies, compute the test statistic, find critical value from tables, and state a conclusion. All steps are routine and mechanical with no conceptual challenges beyond standard application of the test procedure. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 2 - 4 } \multicolumn{1}{c|}{} | Grade awarded | ||
| \cline { 2 - 4 } \multicolumn{1}{c|}{} | A | B | C |
| Background music | 49 | 51 | 40 |
| Silence | 93 | 68 | 49 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Expected values table: \(49 \Rightarrow E=56.8\), \(51 \Rightarrow E=47.6\), \(40 \Rightarrow E=35.6\), \(93 \Rightarrow E=85.2\), \(68 \Rightarrow E=71.4\), \(49 \Rightarrow E=53.4\) | M1A1 | |
| \(\frac{(49-56.8)^2}{56.8}+\frac{(51-47.6)^2}{47.6}+\frac{(40-35.6)^2}{35.6}+\frac{(93-85.2)^2}{85.2}+\frac{(68-71.4)^2}{71.4}+\frac{(49-53.4)^2}{53.4}\) | M1 | |
| \(= 3.096\) \((3.10)\) | A1 | |
| Use appropriate tabular value \(= 4.605\) | M1 | |
| \(3.096 < 4.605\) so the grades awarded are independent of the background | A1 | |
| Total: 6 |
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expected values table: $49 \Rightarrow E=56.8$, $51 \Rightarrow E=47.6$, $40 \Rightarrow E=35.6$, $93 \Rightarrow E=85.2$, $68 \Rightarrow E=71.4$, $49 \Rightarrow E=53.4$ | M1A1 | |
| $\frac{(49-56.8)^2}{56.8}+\frac{(51-47.6)^2}{47.6}+\frac{(40-35.6)^2}{35.6}+\frac{(93-85.2)^2}{85.2}+\frac{(68-71.4)^2}{71.4}+\frac{(49-53.4)^2}{53.4}$ | M1 | |
| $= 3.096$ $(3.10)$ | A1 | |
| Use appropriate tabular value $= 4.605$ | M1 | |
| $3.096 < 4.605$ so the grades awarded are independent of the background | A1 | |
| | **Total: 6** | |1 Two randomly selected groups of students, with similar ranges of abilities, take the same examination in different rooms. One group of 140 students takes the examination with background music playing. The other group of 210 students takes the examination in silence. Each student is awarded a grade for their performance in the examination and the numbers from each group gaining each grade are shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{Grade awarded} \\
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & A & B & C \\
\hline
Background music & 49 & 51 & 40 \\
\hline
Silence & 93 & 68 & 49 \\
\hline
\end{tabular}
\end{center}
Test at the 10\% significance level whether grades awarded are independent of whether background music is playing during the examination.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q1 [6]}}