Standard +0.3 This is a standard chi-squared test of independence with clearly defined categories and straightforward calculations. The contingency table setup is routine, and the second part requires only basic comparison of proportions. While it's a 13-mark question requiring careful arithmetic and interpretation, it involves no novel insight or complex reasoning—just methodical application of a standard statistical test taught in Further Maths.
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 winter, 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. [10]
Without using a statistical test, decide which of the vaccines appears to be most effective. [3]
Differentiate to find prob. density fn: g(y) = 2y M1 A1
Answer
Marks
Give complete statement of g(y): g(y) = 2y (0 ≤ y ≤ 1), 0 otherwise A1
3
3
Answer
Marks
8
[14]
Question 10:
10 | Tabulate observed data with totals: A B C
Flu 30 29 16 75
No flu 148 120 157 425
178 149 173 M1
Find expected values: A B C
(lose A1 if 1 or 2 errors; Flu 26⋅7 22⋅35 25⋅95
lose A1 if rounded to integers) No flu 151⋅3 126⋅65 147⋅05 M1 A2
State (at least) null hypothesis (A.E.F.): H 0: Catching flu indep of vaccine B1
Calculate value of χ2 (to 2 dp): χ2 = 7⋅30 M1 A1
S.R. If rounded to integers above allow (to 2 dp): χ2 = 7⋅53 (earns max 8/10) (B1)
Compare with consistent tabular value (to 2 dp): χ 2, 0.95 2 = 5⋅991 B1
χ2
Valid method for reaching conclusion: Reject H0 if > tabular value M1
Correct conclusion (A.E.F., requires correct values): Catching flu depends on vaccine A1
Find proportions (or complements) for A,B,C: 0⋅169, 0⋅195, 0⋅092 (to 2 dp) M1 A1
Correct conclusion (A.E.F., requires correct values): C appears most effective A1 | 10
3 | [13]
11
EITHER | 2 2 2
Find MI of disc about O [or A]: I = ½ 4ma [= 2ma or 18ma ] B1
disc
2 2 2
Find MI of ring about O [or A]: I = m(2a) [= 4ma or 8ma ] B1
ring
2 2
Find MI of AO about O [or BO about A]: I = (4/3)ma [or 22ma /3] B1
rod
2 2
Find MI of wheel about A: I = 10ma + 8m(2a) M1
wheel
= 42ma 2 A.G. A1
Find angular speed ω using energy: ½ I ω 2 = 8mg × 2a sin 30° M1 A1
wheel
2 2 2
ω = 8mga /21ma
ω = √(8g/21a) or 1⋅95/√a (A.E.F.) A1
2 2 2
Find new MI about A: I = 8ma + 4m(2a) = 24ma M1 A1
new
Find reqd. angle θ using energy: ½ I ω 2 = M g × 2a sin θ M1
new new
Find and use new mass: M = m + 3m = 4m A1
new
2
Substitute for I , M , ω : (32/7)mga = 8mga sin θ (A.E.F.) A1
new new
Solve for θ: θ = sin -1 (4/7) = 0⋅608 rad or 34⋅8° A1 | 5
3
6 | [14]
Page 7 | Mark Scheme: Teachers’ version | Syllabus | Paper
GCE A LEVEL – May/June 2010 | 9231 | 22
11 OR (i)
(ii)
(iii) | Integrate to find F(t) for t ≥ 2 [c needed]: F(t) = c – (t – 1) -2 M1
-2
Use F(2) = 0 to find c: F(t) = 1 – (t – 1) A1
-2
Find p = P(T > 5): p = 1 – F(5) = 1 – (1 – 4 ) = 1/16 B1
p)n-1
State or imply distribution: P(N > n) = p (1 –
or geometric distn. with par. p M1
Find P(N > E(N)): (1 – p) 1/p = (15/16) 16 = 0⋅356 M1 A1
Relate dist. fn. G(y) of Y to T: G(y) = P(Y < y) = P(1/(T – 1) < y) M1
Rearrange : = P(T > 1 + 1/y) A1
Relate to dist. fn. F: = 1 – F(1 + 1/y) M1
-2
Substitute expression for F: = 1 – {1 – (1 + 1/y – 1) } A1
2
Simplify: = y A1
Differentiate to find prob. density fn: g(y) = 2y M1 A1
Give complete statement of g(y): g(y) = 2y (0 ≤ y ≤ 1), 0 otherwise A1 | 3
3
8 | [14]
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 winter, 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. [10]
Without using a statistical test, decide which of the vaccines appears to be most effective. [3]
\hfill \mbox{\textit{CAIE FP2 2010 Q10 [13]}}