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Date May 2019 Marks available 4 Reference code 19M.1.SL.TZ1.S_9
Level Standard Level Paper Paper 1 Time zone Time zone 1
Command term Find Question number S_9 Adapted from N/A

Question

A random variable Z is normally distributed with mean 0 and standard deviation 1. It is known that P(z < −1.6) = a and P(z > 2.4) = b. This is shown in the following diagram.

A second random variable X is normally distributed with mean m and standard deviation s.

It is known that P(x < 1) = a.

Find P(−1.6 < z < 2.4). Write your answer in terms of a and b.

[2]
a.

Given that z > −1.6, find the probability that z < 2.4 . Write your answer in terms of a and b.

[4]
b.

Write down the standardized value for x=1.

[1]
c.

It is also known that P(x > 2) = b.

Find s.

[6]
d.

Markscheme

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

recognizing area under curve = 1        (M1)

eg   a+x+b=1,  100ab,  1a+b

P(1.6<z<2.4)=1ab(=1(a+b))               A1  N2

[2 marks]

a.

P(z>1.6)=1a (seen anywhere)        (A1)

recognizing conditional probability        (M1)

eg   P(A|B),  P(B|A)

correct working       (A1)

eg   P(z<2.4z>1.6)P(z>1.6)P(1.6<z<2.4)P(z>1.6) 

P(z<2.4|z>1.6)=1ab1a      A1  N4

Note: Do not award the final A1 if correct answer is seen followed by incorrect simplification.

[4 marks]

b.

z=1.6 (may be seen in part (d))     A1  N1

Note: Depending on the candidate’s interpretation of the question, they may give 1ms as the answer to part (c). Such answers should be awarded the first (M1) in part (d), even when part (d) is left blank. If the candidate goes on to show z=1.6 as part of their working in part (d), the A1 in part (c) may be awarded.

[1 mark]

c.

attempt to standardize x (do not accept xμσ)       (M1)

eg   1ms (may be seen in part (c)), m2s,  xmσ

correct equation with each z-value      (A1)(A1)

eg   1.6=1ms,  2.4=2ms,  m+2.4s=2

valid approach (to set up equation in one variable)       M1

eg   2.4=2(1.6s+1)s,   1m1.6=2m2.4

correct working        (A1)

eg   1.6s+1=22.4s,  4s=1,  m=75

s=14       A1 N2

[6 marks]

d.

Examiners report

[N/A]
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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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