CAIE FP2 2010 June — Question 10 13 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2010
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChi-squared test of independence
TypeStandard 2×3 contingency table
DifficultyStandard +0.3 This is a straightforward chi-squared test of independence with clear data organization (2×3 contingency table), standard expected frequency calculations, and routine hypothesis testing at 5% level. The second part requires only basic percentage comparison. While chi-squared tests are Further Maths content, this is a textbook application with no conceptual complications or unusual features.
Spec5.06a Chi-squared: contingency tables
10 Three new flu vaccines, \(A , B\) and \(C\), were tested on 500 volunteers. The vaccines were assigned randomly to the volunteers and 178 received \(A , 149\) received \(B\) and 173 received \(C\). During the following year, 30 of the volunteers given \(A\) caught flu, 29 of the volunteers given \(B\) caught flu, and 16 of the volunteers given \(C\) caught flu. Carry out a suitable test for independence at the 5\% significance level. Without using a statistical test, decide which of the vaccines appears to be most effective.
Question 10:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Observed totals: Flu: 30, 29, 16 = 75; No flu: 148, 120, 157 = 425; Totals: 178, 149, 173M1 Tabulate observed data with totals
Expected values — Flu: 26.7, 22.35, 25.95; No flu: 151.3, 126.65, 147.05M1 A2 Find expected values (lose A1 if 1 or 2 errors; lose A1 if rounded to integers)
\(H_0\): Catching flu independent of vaccineB1 State null hypothesis (A.E.F.)
\(\chi^2 = 7.30\)M1 A1 Calculate value of \(\chi^2\) (to 2 dp)
\(\chi^2 = 7.53\) if rounded to integers (earns max 8/10)(B1) S.R. if rounded to integers above allowed
\(\chi^2_{2,\,0.95} = 5.991\)B1 Compare with consistent tabular value (to 2 dp)
Reject \(H_0\) if \(\chi^2 >\) tabular valueM1 Valid method for reaching conclusion
Catching flu depends on vaccineA1 Correct conclusion (A.E.F., requires correct values)
Proportions for \(A, B, C\): 0.169, 0.195, 0.092 (to 2 dp)M1 A1 Find proportions (or complements)
\(C\) appears most effectiveA1 Correct conclusion (A.E.F., requires correct values)
Total: 13 marks
## Question 10:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Observed totals: Flu: 30, 29, 16 = 75; No flu: 148, 120, 157 = 425; Totals: 178, 149, 173 | M1 | Tabulate observed data with totals |
| Expected values — Flu: 26.7, 22.35, 25.95; No flu: 151.3, 126.65, 147.05 | M1 A2 | Find expected values (lose A1 if 1 or 2 errors; lose A1 if rounded to integers) |
| $H_0$: Catching flu independent of vaccine | B1 | State null hypothesis (A.E.F.) |
| $\chi^2 = 7.30$ | M1 A1 | Calculate value of $\chi^2$ (to 2 dp) |
| $\chi^2 = 7.53$ if rounded to integers (earns max 8/10) | (B1) | S.R. if rounded to integers above allowed |
| $\chi^2_{2,\,0.95} = 5.991$ | B1 | Compare with consistent tabular value (to 2 dp) |
| Reject $H_0$ if $\chi^2 >$ tabular value | M1 | Valid method for reaching conclusion |
| Catching flu depends on vaccine | A1 | Correct conclusion (A.E.F., requires correct values) |
| Proportions for $A, B, C$: 0.169, 0.195, 0.092 (to 2 dp) | M1 A1 | Find proportions (or complements) |
| $C$ appears most effective | A1 | Correct conclusion (A.E.F., requires correct values) |

**Total: 13 marks**

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10 Three new flu vaccines, $A , B$ and $C$, were tested on 500 volunteers. The vaccines were assigned randomly to the volunteers and 178 received $A , 149$ received $B$ and 173 received $C$. During the following year, 30 of the volunteers given $A$ caught flu, 29 of the volunteers given $B$ caught flu, and 16 of the volunteers given $C$ caught flu. Carry out a suitable test for independence at the 5\% significance level.

Without using a statistical test, decide which of the vaccines appears to be most effective.

\hfill \mbox{\textit{CAIE FP2 2010 Q10 [13]}}
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Q1 5 Q2 7 Q3 9 Q4 9 Q5 13 Q6 5 Q7 7 Q8 9 Q9 9 Q10 13 Q11 EITHER Q11 OR