| 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 standard chi-squared test of independence followed by a more challenging extension requiring understanding of how sample size affects test statistics. The first part is routine application of the test (calculating expected frequencies, test statistic, comparing to critical value), but the second part requires insight that χ² scales linearly with N and working backwards from a critical value to find the minimum N, which elevates this above a typical A-level question into Further Maths territory with genuine problem-solving required. |
| 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{tabular}{|c|c|c|c|}
\hline
& Cappuccino & Latte & Ground \\
\hline
Company $A$ & 60 & 52 & 32 \\
\hline
Company $B$ & 35 & 40 & 31 \\
\hline
\end{tabular}
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]}}