User interface language: English | Español

Date May 2007 Marks available 6 Reference code 07M.1.hl.TZ0.6
Level HL only Paper 1 Time zone TZ0
Command term Find Question number 6 Adapted from N/A

Question

The weights, \(X\) kg , of male birds of a certain species are normally distributed with mean \(4.5\) kg and standard deviation \(0.2\) kg . The weights, \(Y\) kg , of female birds of this species are normally distributed with mean \(2.5\) kg and standard deviation \(0.15\) kg .

(i)     Find the mean and variance of \(2Y - X\) .

(ii)     Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird.

[6]
a.

Two randomly chosen male birds and three randomly chosen female birds are placed together on a weighing machine for which the recommended maximum weight is \(16\) kg . Find the probability that this maximum weight is exceeded.

[5]
b.

Markscheme

(i)     \({\rm{E}}(2Y - X) = 2 \times 2.5 - 4.5 = 0.5\)     A1

\(Var(2Y - X) = 4 \times 0.1{5^2} + {0.2^2} = 0.13\)     M1A1

 

(ii)     We require \({\rm{P}}(X > 2Y) = {\rm{P}}(2Y - X < 0)\)     M1

\(0.0828\)     A2

Note: Using tables, answer is \(0.0823\).

 

[6 marks]

a.

Let \(S\) denote the total weight of the \(5\) birds.

Then,

\({\rm{E}}(S) = 2 \times 4.5 + 3 \times 2.5 = 16.5\)     A1

\(Var(S) = 2 \times 0.{2^2} + 3 \times 0.1{5^2} = 0.1475\)    M1A1

\({\rm{P}}(S > 16) = 0.904\)     A2

Note: Using tables, answer is \(0.903\).

[5 marks]

b.

Examiners report

[N/A]
a.
[N/A]
b.

Syllabus sections

Topic 3 - Statistics and probability » 3.4 » A linear combination of independent normal random variables is normally distributed. In particular, \(X{\text{ ~ N}}\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ N}}\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) .

View options