| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Scaled sample, find minimum N |
| Difficulty | Standard +0.8 This is a two-part chi-squared test question requiring standard contingency table analysis followed by a reverse problem finding the scaling factor N. The first part is routine A-level statistics (calculate expected frequencies, compute χ², compare to critical value), but the second part requires understanding that χ² scales linearly with sample size and working backwards from the critical value, which demands deeper conceptual insight beyond standard textbook exercises. |
| Spec | 5.06a Chi-squared: contingency tables |
| Cappuccino | Latte | Ground | |
| Company \(A\) | 60 | 52 | 32 |
| Company \(B\) | 35 | 40 | 31 |
| Answer | Marks |
|---|---|
| 10 | Find expected values (to 1 d.p.): 54⋅72 52⋅992 36⋅288 |
| Answer | Marks |
|---|---|
| Find Nmin: N > 9⋅21/2⋅45, Nmin = 4 M1 A1 | 7 |
| 4 | [11] |
Question 10:
10 | Find expected values (to 1 d.p.): 54⋅72 52⋅992 36⋅288
(lose A1 if rounded to integers) 40⋅28 39⋅008 26⋅712 M1 A1
State (at least) null hypothesis (A.E.F.): H0: Preferences are independent B1
Calculate value of χ2 : χ2 = 0⋅5095 + 0⋅0186 + 0⋅5067
+ 0⋅6921 + 0⋅0252 + 0⋅6883
= 2⋅44 or 2⋅45 M1 A1
χ2
State or use correct tabular value (to 3
sf): χ 2, 0.95 2 = 5⋅99[1] B1
Conclusion consistent with values (A.E.F): Preferences are independent A1 √
C S a ta lc te u l o a r te u n se e w co v r a re lu c e t t χ a n b e u w l 2 a o r f χ χ 2 2 v : alue: χ χ n 2, e w 0. 2 9 9 = 2 = N × 9 χ ⋅2 2 1 M B 1 1
Find Nmin: N > 9⋅21/2⋅45, Nmin = 4 M1 A1 | 7
4 | [11]Random samples of employees are taken from two companies, $A$ and $B$. Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
& Cappuccino & Latte & Ground \\
\hline
Company $A$ & 60 & 52 & 32 \\
\hline
Company $B$ & 35 & 40 & 31 \\
\hline
\end{tabular}
\end{center}
Test, at the 5% significance level, whether coffee preferences of employees are independent of their company. [7]
Larger random samples, consisting of $N$ times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of $N$ that would lead to the conclusion, at the 1% significance level, that coffee preferences of employees are not independent of their company. [4]
\hfill \mbox{\textit{CAIE FP2 2012 Q10 [11]}}