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Date May 2017 Marks available 5 Reference code 17M.2.SL.TZ1.S_9
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number S_9 Adapted from N/A

Question

A random variable X is normally distributed with mean, μ . In the following diagram, the shaded region between 9 and μ represents 30% of the distribution.

M17/5/MATME/SP2/ENG/TZ1/09

The standard deviation of X is 2.1.

The random variable Y is normally distributed with mean λ and standard deviation 3.5. The events X > 9 and Y > 9 are independent, and P ( ( X > 9 ) ( Y > 9 ) ) = 0.4 .

Find P ( X < 9 ) .

[2]
a.

Find the value of μ .

[3]
b.

Find λ .

[5]
c.

Given that Y > 9 , find P ( Y < 13 ) .

[5]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach     (M1)

eg P ( X < μ ) = 0.5 ,   0.5 0.3

P ( X < 9 ) = 0.2 (exact)     A1     N2

[2 marks]

a.

z = 0.841621 (may be seen in equation)     (A1)

valid attempt to set up an equation with their z     (M1)

eg 0.842 = μ X σ ,   0.842 = X μ σ ,   z = 9 μ 2.1

10.7674

μ = 10.8     A1     N3

[3 marks]

b.

P ( X > 9 ) = 0.8 (seen anywhere)     (A1)

valid approach     (M1)

eg P ( A ) × P ( B )

correct equation     (A1)

eg 0.8 × P ( Y > 9 ) = 0.4

P ( Y > 9 ) = 0.5     A1

λ = 9     A1     N3

[5 marks]

c.

finding P ( 9 < Y < 13 ) = 0.373450 (seen anywhere)     (A2)

recognizing conditional probability     (M1)

eg P ( A | B ) ,  P ( Y < 13 | Y > 9 )

correct working     (A1)

eg 0.373 0.5

0.746901

0.747     A1     N3

[5 marks]

d.

Examiners report

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

Syllabus sections

Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
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Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations
Topic 4—Statistics and probability

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