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DISCRETE RANDOM VARIABLES
REVIEW SET 25A
1 P(X = x) =
NON-CALCULATOR
a
,
x2 + 1
x = 0, 1, 2, 3 is a probability distribution function.
b Find P(X > 1).
a Find a.
2 A random sample of 120 toothbrushes is taken with replacement from a very large batch where
4% are known to be defective. Find the mean number and standard deviation of defectives in
the sample.
3 A random variable X has the probability
distribution function P (x) described in the
table.
a Find k.
b Find P(X > 3).
x
P (x)
0
0:10
1
0:30
2
0:45
3
0:10
4
k
c Find the expectation E(X) for the distribution.
4
d Find the standard deviation ¾ for the distribution.
¡ 3 2 ¢4
a Expand
5 + 5 .
b A tin contains 20 pens of which 12 have blue ink. Four pens are randomly selected, with
replacement, from the tin. Find the probability that:
i two of them have blue ink
ii at most two have blue ink.
5 Three green balls and two yellow balls are placed in a hat. When two balls are randomly drawn
without replacement, X is the number of green balls drawn. Find:
a P(X = 0)
b P(X = 1)
c P(X = 2)
d E(X)
6 Lakshmi rolls a normal six-sided die. She wins twice the number of pounds as the number
shown on the face.
a How much does Lakshmi expect to win from one roll of the die?
b If it costs $8 to play the game, would you advise Lakshmi to play several games? Explain
your answer.
¡ ¢x ¡ 2 ¢7¡x
7 A binomial distribution has probability distribution function P(X = x) = k 13
3
where x = 0, 1, 2, 3, ...., 7.
¡ ¢
a Write k in the form nr .
b Find the mean and variance of the distribution.
¡ 4 1 ¢5
8 a Expand
5 + 5 .
b With every attempt, Jack has an 80% chance of
kicking a goal. In one quarter of a match he has
5 kicks for goal.
Determine the probability that he scores:
i 3 goals then misses twice
ii 3 goals and misses twice.
9 At a social club fundraiser there is a dice game where, on a single roll of a six-sided die, the
following payouts are made:
$2 for an odd number, $3 for a 2, $6 for a 4, and $9 for a 6.
a Find the expected return for a single roll of the die.
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b If the club charges $5 for each roll, and 75 people play the game once each, how much
money will the club expect to make?
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10 Consider the two spinners illustrated:
square
spinner
pentagonal
spinner
Qr
Et
a Copy and complete the tree diagram which
shows all possible results when the two are
spun together.
R
R
R0
R0
R
R0
b Calculate the probability that exactly one red will occur.
c The pair of spinners is now spun 10 times and X is the number of times that exactly one
red occurs.
i Write down expressions for P(X = 1) and P(X = 9).
ii Hence determine which of these outcomes is more likely.
11 A biased tetrahedral die has the numbers 6, 12, and 24 marked on three of its faces. The fourth
number is x. The table shows the probability of each of the numbers occurring if the die is
rolled once.
a Find y, the probability of obtaining the number 24.
Number
6 12 x 24
b Find the fourth number if the average result when
Probability
rolling the die once is 14.
c Find the median and modal score for one roll of this die.
1
3
1
6
1
4
y
12 The random variable X has mean ¹ and standard deviation ¾.
Prove that the random variable Y = aX + b has mean a¹ + b and standard deviation j a j ¾.
REVIEW SET 25B
CALCULATOR
1 A binomial random variable X has probability distribution function P (x) = k
where x = 0, 1, 2, 3 and k is a constant. Find:
b P(X > 1)
a k
¡ 3 ¢x ¡ 1 ¢3¡x
4
4
c E(X)
d the standard deviation of the distribution.
2 A manufacturer finds that 18% of the items produced from its assembly lines are defective.
During a floor inspection, the manufacturer randomly selects ten items with replacement. Find
the probability that the manufacturer finds:
a one defective
b two defective
c at least two defective items.
3 From data over the last fifteen years it is known that the chance of a netballer with a knee injury
needing major knee surgery in any one season is 0:0132 . In 2007 there were 487 knee injuries
in netball games throughout the country. Find the expected number of major knee surgeries
required.
4 24% of visitors to a museum make voluntary donations. On a certain day the museum has
175 visitors. Find:
a the expected number of donations
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b the probability that less than 40 visitors make a donation.
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5 An X-ray has probability of 0:96 of showing a fracture
in the arm. If four different X-rays are taken of a
particular fracture, find the probability that:
a all four show the fracture
b the fracture does not show up
c at least three X-rays show the fracture
d only one X-ray shows the fracture.
6 A school basketball team has 8 players, each of whom has a 75% chance of turning up for any
given game. The team needs at least 5 players to avoid forfeiting the game.
a Find the probability that for a randomly chosen game, the team will:
i have all of its players
ii have to forfeit the game.
b The team plays 30 games for the season. How many games would you expect the team to
forfeit?
7 The binomial distribution X » B(n, p) has mean 30 and variance 22:5 .
a Find n and p.
b Hence find:
ii P(X > 25)
i P(X = 25)
iii P(15 6 X 6 25)
8 The discrete random variable X has the probability distribution function
¡ ¢x
for x = 0, 1, 2, 3, ....
P(X = x) = a 56
Find the value of a.
9 When X plays Y at table tennis, we know from past experience that X wins 3 sets in every 5
played.
a If they play 6 sets, write down the probability generator.
b Hence determine the probability that:
i Y wins 3 of them
ii Y wins at least 5 of them.
10 One glass blower was known to break one out of every 200 objects he attempts. A second glass
blower was known to break three out of every 200 objects she attempts. During one period, the
first glass blower produced 20 objects and the second glass blower produced 40 objects. Use
the Poisson distribution to find the probability that the glass blowers broke 2 or more objects
between them.
11 For a given binomial random variable X with 7 independent trials, we know that
P(X = 3) = 0:226 89 .
a Find the smallest possible value of p, the probability of obtaining a success in one trial.
b Hence calculate the probability of getting at most 4 successes in 10 trials.
12 A discrete random variable has the probability distribution function
P(X = x) = k(x + x¡1 ) where x = 1, 2, 3, 4. Find:
a the exact value of k
b E(X) and Var(X)
c the median and mode of X.
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13 The random variable Y has a Poisson distribution with P(Y > 3) ¼ 0:033 768 97 .
Find P(Y < 3).
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REVIEW SET 25C
1 Find k for the following probability distribution functions:
a P(X = x) =
k
,
2x
b
x = 1, 2, 3
x
0
P (x)
k
2
1
0:2
2
k
2 A random variable X has probability distribution function P(X = x) =
for x = 0, 1, 2, 3, 4.
a Find P(X = x) for x = 0, 1, 2, 3, 4.
3
2
0:3
¡ 4 ¢ ¡ 1 ¢x ¡ 1 ¢4¡x
x
2
2
b Find the mean ¹ for the distribution.
c Find the standard deviation ¾ for the distribution.
3 A die is biased such that the probability of obtaining a 6 is 25 . The die is rolled 1200 times. Let
X be the number of sixes obtained. Find the mean and standard deviation of X.
4 Only 40% of young trees that are planted will survive the first year. The Botanical Gardens
buys five young trees. Assuming independence, find the probability that during the first year:
a exactly one tree will survive
b at most one tree will survive
c at least one tree will survive.
5 In a game, the numbers from 1 to 20 are written on tickets and placed in a bag. A player draws
out a number at random. He or she wins $3 if the number is even, $6 if the number is a square
number, and $9 if the number is both even and square.
a Calculate the probability that the player wins:
i $3
ii $6
iii $9
b How much should be charged to play the game so that it is a fair game?
6 A fair die is rolled 360 times. Find the probability that:
a less than 50 results are a 6
b between 55 and 65 results (inclusive) are a 6.
7 An unbiased coin is tossed n times. Find the smallest value of n for which the probability of
getting at least two heads is greater than 99%.
8 A hot water unit relies on 20 solar components for its power and will operate provided at least
one of its 20 components is working. The probability that an individual solar component will
fail in a year is 0:85, and the failure of each individual component is independent of the others.
a Find the probability that the hot water unit will fail within one year.
b Find the smallest number of solar components required to ensure that a hot water service
like this one is operating at the end of one year with a probability of at least 0:98 .
9 A dart thrower has a one in three chance of hitting the correct number with any throw. He
throws 5 darts at the board and X is the number of successful hits.
a Find the probability generator for X.
b Calculate the probability of the thrower scoring an odd number of successful hits given
that he has at least two successful hits.
10 A Poisson random variable X is such that P(X = 1) = P(2 6 X 6 4).
a Find the mean and standard deviation of:
i X
ii Y =
X +1
.
2
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b Find P(X > 2).
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11 A Poisson random variable X satisfies the rule 5Var(X) = 2[E(X)]2 ¡ 12.
a Find the mean of X.
b Find P(X < 3).
12 The random variable X has a binomial distribution for which P(X > 2) ¼ 0:070 198 for
10 independent trials. Find P(X < 2).
13 During peak period, customers arrive at random at a fish and chip shop at the rate of 20 customers
every 15 minutes.
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a Find the probability that during peak period, 15 customers will arrive in the next quarter
of an hour.
b If the probability that more than 10 customers will arrive at the fish and chip shop during
10 minutes of peak period is greater than 80%, the manager will employ an extra shop
assistant. Will the manager hire an extra shop assistant?
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CONTINUOUS RANDOM VARIABLES
REVIEW SET 26A
NON-CALCULATOR
1 The average height of 17 year old boys is normally distributed with mean 179 cm and standard
deviation 8 cm. Calculate the percentage of 17 year old boys whose heights are:
a more than 195 cm
c between 171 cm and 187 cm.
b between 163 cm and 195 cm
2 The contents of cans of a certain brand of soft drink are normally distributed with mean 377 mL
and standard deviation 4:2 mL.
a Find the percentage of cans with contents:
i less than 368:6 mL
ii between 372:8 mL and 389:6 mL.
b Find the probability that a randomly selected can contains between 377 mL and 381:2 mL.
3 The edible part of a batch of Coffin Bay oysters is normally
distributed with mean 38:6 grams and standard deviation
6:3 grams.
Let the random variable X be the mass of a Coffin Bay
oyster.
a Find a if P(38:6 ¡ a 6 X 6 38:6 + a) = 0:6826 .
b Find b if P(X > b) = 0:8413 .
4 A random variable X has probability density function f (x) = a(x + 1)x(x ¡ 1)(x ¡ 2)
for 0 < x < 1.
b Find the mode of X.
a Show that a = 30
11 .
c Show that f ( 12 ¡ x) = f ( 12 + x).
d Hence find the median of X.
5 The results of a test are normally distributed. Harri gained a z-score equal to ¡2.
a Interpret this z-score with regard to the mean and standard deviation of the test scores.
b What proportion of students obtained a better score than Harri?
c The mean test score was 151 and Harri’s actual score was 117. Find the standard deviation
of the test scores.
6 The continuous random variable Z is distributed such that Z » N(0, 1).
Find the value of k if P(¡k 6 Z 6 k) = 0:95 .
7 A continuous random variable X has the probability density function
f (x) = ax(4 ¡ x2 ), 0 6 x 6 2.
a Find a.
b Find the mode of X.
p
p
c Show that the median of X is
4 ¡ 2 2.
d Find the mean of X.
8 The distance that a 15 year old boy can throw a tennis ball is normally distributed with mean
35 m and standard deviation 4 m.
The distance that a 10 year old boy can throw a tennis ball is normally distributed with mean
25 m and standard deviation 3 m.
Jarrod is 15 years old and can throw a tennis ball 41 m. How far does his 10 year old brother
Paul need to throw a tennis ball to perform as well as Jarrod?
9 State the probability that a randomly selected, normally distributed value lies between:
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a ¾ above the mean and 2¾ above the mean
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10
A bottle shop sells on average 2500 bottles per day
with a standard deviation of 300 bottles. Assuming
that the number of bottles sold per day is normally
distributed, calculate the percentage of days when:
a less than 1900 bottles are sold
b more than 2200 bottles are sold
c between 2200 and 3100 bottles are sold.
11 The continuous random variable X has probability density function f (x) = 2e¡x , 0 6 x 6 k.
a Find the exact value of k.
b Find the probability that X lies between ln 43 and ln 53 .
d Show that the variance of X is 1 ¡ 2(ln 2)2 .
c Find the mean of X.
REVIEW SET 26B
CALCULATOR
1 The mean and standard deviation of a normal distribution are 150 and 12 respectively. What
percentage of values lie between:
a 138 and 162
b 126 and 174
c 126 and 162
d 162 and 174?
2 The arm lengths of 18 year old females are normally distributed with mean 64 cm and standard
deviation 4 cm.
a Find the percentage of 18 year old females whose arm lengths are:
i between 60 cm and 72 cm
ii greater than 60 cm.
b Find the probability that a randomly chosen 18 year old female has an arm length in the
range 56 cm to 64 cm.
c The arm lengths of 70% of the 18 year old females are more than x cm. Find the value
of x.
3 The length of steel rods produced by a machine is normally distributed with a standard deviation
of 3 mm. It is found that 2% of all rods are less than 25 mm long. Find the mean length of
rods produced by the machine.
4 The continuous random variable Z is distributed such that Z » N(0, 1).
Find the value of k if P(jZj > k) = 0:376 .
5 f(x) = ax(x ¡ 3), 0 6 x 6 2 is a continuous probability density function.
a Find a.
b Sketch the graph of y = f (x).
c For this distribution, find the:
i mean
ii mode
iii median
d Find P(1 6 x 6 2).
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6 The distribution curve shown corresponds to
X » N(¹, ¾ 2 ).
Area A = Area B = 0:2 .
a Find ¹ and ¾.
b Calculate:
i P(X 6 35)
ii P(23 6 X 6 30)
iv variance.
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7 Let X be the weight in grams of bags of sugar filled by a machine. Bags less than 500 grams
are considered underweight.
Suppose that X » N(503, 22 ).
a What proportion of bags are underweight?
b If a quality inspector randomly selects 20 bags, what is the probability that at most 2 bags
are underweight?
8 The marks of 2376 candidates in an IB examination are normally distributed with mean 49 marks
and variance 225.
a If the pass mark is 45, estimate the number of candidates who passed the examination.
b If the top 7% of the candidates are awarded a ‘7’, find the minimum mark required to
obtain a ‘7’.
c Find the interquartile range of the distribution of marks obtained.
9 The life of a Xenon-brand battery is normally distributed with mean 33:2 weeks and standard
deviation 2:8 weeks.
a Find the probability that a randomly selected battery will last at least 35 weeks.
b For how many weeks can the manufacturer expect the batteries to last before 8% of them
fail?
10 The random variable X is normally distributed with P(X 6 30) = 0:0832 and
P(X > 90) = 0:101 .
a Find the mean ¹ and standard deviation ¾.
b Hence find P(¡7 6 X ¡ ¹ 6 7).
11 Kerry’s marks for an English essay and a Chemistry test were 26 out of 40, and 82% respectively.
a Explain briefly why the information given is not sufficient to determine whether Kerry’s
results are better in English than in Chemistry.
b Suppose that the marks of all students in the English essay and the Chemistry test were
normally distributed as N(22, 42 ) and N(75, 72 ) respectively. Use this information to
determine which of Kerry’s two marks is better.
REVIEW SET 26C
1 The middle 68% of a normal distribution lies between 16:2 and 21:4 .
a What is the mean and standard deviation of the distribution?
b Over what range of values would you expect the middle 95% of the data to spread?
2 A random variable X is normally distributed with mean 20:5 and standard deviation 4:3 . Find:
a P(X > 22)
b P(18 6 X 6 22)
c k such that P(X 6 k) = 0:3 .
3 X is a continuous random variable where X » N(¹, 22 ).
Find P(¡0:524 < X ¡ ¹ < 0:524).
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4 The lengths of metal rods produced in a manufacturing process are normally distributed with
mean ¹ cm and standard deviation 6 cm. 5:63% of the rods have length greater than 89:52 cm.
Find the mean, median, and modal length of the metal rods.
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5 The random variable T represents the lifetime in years of a component of a solar cell.
Its probability density function is F (t) = 0:4e¡0:4t , t > 0.
a Find the probability that this component of the solar cell fails within 1 year.
b Each solar cell has 5 of these components which operate independently of each other. The
cell will work provided at least 3 of the components continue to work. Find the probability
that a solar cell will still operate after 1 year.
6 The curve shown is the probability density function
for a normally distributed random variable X. Its
mean is 50, and P(X < 90) ¼ 0:975 .
Find the shaded area.
80
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7 The weight of an apple in an apple harvest is normally
distributed with mean 300 grams and standard deviation
50 grams. Only apples with weights between 250 and
350 grams are considered fit for sale.
a Find the proportion of apples fit for sale.
b In a sample of 100 apples, what is the probability that
at least 75 are fit for sale?
8 It is claimed that the continuous random variable X has probability density function
f(x) =
4
,
1 + x2
0 6 x 6 1.
a Show that this is not possible.
b Use your working from a to find an exact value of k for which F (x) = k f (x) would
be a well-defined probability density function.
c Hence, find the exact values of the mean and variance of X.
x2
1
=1¡
.
1 + x2
1 + x2
Hint:
9 A factory has a machine designed to fill bottles of drink with volume 375 mL of liquid. It is
found that the average amount of drink in each bottle is 376 mL, and that 2:3% of the drink
bottles have a volume smaller than 375 mL. Assuming that the amount of drink in each bottle
is normally distributed, find the standard deviation.
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10 The height of an 18 year old boy is normally distributed with mean 187 cm. Fifteen percent of
18 year old boys have heights greater than 193 cm. Find the probability that two 18 year old
boys, chosen at random, will have heights greater than 185 cm.
8 x
11 The continuous random variable X has probability
>
for 0 6 x < 2
>
>
density function defined by:
< 5
8
f(x) =
for 2 6 x 6 k
>
>
5x2
>
:
0
elsewhere.
a Find the value of k.
b Find the exact value of the median of X.
c Find the mean and variance of X.
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ANSWERS
REVIEW SET 25B
1 a k = 85
b 0:984
REVIEW SET 25A
1 a a = 59
3
a k = 0:05
4
a
¡3
5
b
2 ¹ = 4:8 defectives, ¾ ¼ 2:15
b 0:15
¢
2 4
5
+
4
9
¡ 3 ¢4
=
c 1:7
+4
5
¡ 3 ¢3 ¡ 2 ¢
5
5
+4
b
i
5
a
1
10
6
a $7
216
625
8
a
¡4
5
328
625
b
3
5
9
¡7¢
x
+
b ¹=
¢
1 5
5
i
64
3125
5
5
+
3
10
5
2 4
5
¡ ¢
d
=
¡ 4 ¢5
5
7
3
¾2
¼ 2:33,
+5
¡ 4 ¢4 ¡ 1 ¢
¡ 4 ¢2 ¡ 51 ¢3
5
5
5
+5
=
+ 10
5
Pentagonal
spinner
Qr
a n = 120, p =
5
b
d 0:75
c 0:561
3 6:43 surgeries
c 0:991
d 0:000 246
b 3:41 games
1
4
ii ¼ 0:878
iii ¼ 0:172
a (0:6 + 0:4)6
= (0:6)6 + 6(0:6)5 (0:4) + 15(0:6)4 (0:4)2 + 20(0:6)3 (0:4)3
X wins 6 X wins 5
X wins 4
X wins 3
Y wins 1
Y wins 2
Y wins 3
¼ 1:56
+
b 0:298
¡5 1 ¢5
+ 15(0:6)2 (0:4)4 + 6(0:6)(0:4)5 + (0:4)6
X wins 2
X wins 1
Y wins 6
Y wins 4
Y wins 5
b
i 20(0:6)3 (0:4)3 ¼ 0:276
ii 6(0:6)(0:4)5 + (0:4)6 ¼ 0:0410
10 0:156
11 a p = 0:3
b 0:850
5
11
20
12
R
a k=
12
145
b ¼ 2:81, ¼ 1:19
c median = 3, mode = 4
Er
R'
Qr
R
Er
R'
13 m ¼ 1:2, ¼ 0:879
R'
Wt
i P(X = 1) =
P(X = 9) =
¡ 10 ¢ ¡ 11 ¢1 ¡ 9 ¢9
1
20
20
9
20
20
¡ 10 ¢ ¡ 11 ¢9 ¡ 9 ¢1
REVIEW SET 25C
,
ii It is more likely that exactly one red will occur 9 times.
a y=
7
R
Et
11
a 42 donations
b 0:334
a 0:849
b 2:56 £ 10¡6
a
i 0:100
ii 0:114
9
¼ 0:205
Square
spinner
4
5
6
¡ 4 ¢3 ¡ 1 ¢2
¡ 4 ¢ ¡ 1 5¢4
128
625
ii
14
9
a 0:302
b
i ¼ 0:0501
8 a = 16
1 15
b $75
a
c
5
¼ 0:0205
a $4
10
¡ 3 ¢ ¡ 2 ¢3
c
+ 10
b
¡ 3 ¢2 ¡ 2 ¢2
b No, she would lose $1 per game in the long run.
a k=
7
ii
d ¼ 0:954
+6
c 2:55
2
1
4
b x = 16
c median = 14, mode = 6
1
a k=
2
a
12
11
b k=
0
0:0625
x
P(X = x)
b ¹=2
c 1
3 ¹ = 480, ¾ ¼ 17:0
5
a
i
2
5
1
2
ii
4
1
10
1
0:25
2
0:375
a 0:259
iii
1
10
6
a ¼ 0:0660
b ¼ 0:563
8
a ¼ 0:0388
b 25 of them
9
a ( 13 + 23 )5
b ¼ 0:313
a
11
a m = 4, m > 0
b 0:337
4
0:0625
c 0:922
b $2:70
7 n = 11
i ¹ = 1:28, ¾ = 1:13
ii E(Y ) = 1:14, Var(Y ) = 0:566
10
3
0:25
b ¼ 0:366
13
¼ 0:238
e4
a 0:0516
b No
b
13
cyan
magenta
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_an\951IB_HL-3ed_an.cdr Monday, 21 May 2012 4:39:50 PM BEN
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
12 ¼ 0:736 as p ¼ 0:1
black
IB HL 3ed
REVIEW SET 26A
1
2
a 2:28%
a
i 2:28%
3
a a = 6:3 grams
4
a
R1
0
b 95:4%
ii 84%
c 68:3%
b 0:341
b b ¼ 32:3 grams
a(x + 1)x(x ¡ 1)(x ¡ 2) dx = 1 gives a =
b mode is
1
2
d median is
30
11
1
2
a Harri’s score is 2 standard deviations below the mean.
b 97:7%
c 17
b mode = p2
d ¹=
6 k¼2
7 a a = 14
5
3
a 0:136
b 0:341
10
8 29:5 m
a 2:28%
b 84:1%
c 81:9%
11
a k = ln 2
b 0:3
c ¹ = 1 ¡ ln 2
9
16
15
REVIEW SET 26B
1 a 68:3%
b 95:4%
c 81:9%
2 a
i 81:9%
ii 84:1%
b 0:477
3 ¹ ¼ 31:2 mm
4 k ¼ 0:885
5
3
a a = ¡ 10
c
d
b
3
y y = - 10 x(x - 3)
1:2
1:5
¼ 1:24
0:24
i
ii
iii
iv
(2, 0.6)
(1.5, 0.675)
x
2
13
20
6
a ¹ = 29:0, ¾ ¼ 10:7
7
8
a 6:68%
b 0:854
a 1438 candidates
b 71:1 marks
9
a 0:260
10
11
d 13:6%
c x ¼ 61:9
i 0:713
b
ii 0:250
c IQR ¼ 20:2 marks
b 29:3 weeks
a ¹ = 61:2, ¾ ¼ 22:6
b ¼ 0:244
a The relative difficulty of each test is not known.
b z-score for English = 1, z-score for Chemistry = 1
) Kerry’s performance relative to the rest of the class is the
same in both tests.
REVIEW SET 26C
1
a mean is 18:8, standard deviation is 2:6
b 13:6 to 24:0
2 a 0:364
b 0:356
c k ¼ 18:2
3 0:207
4 ¹ ¼ 80:0 cm, median and mode are also 80:0 cm.
5 a ¼ 0:330
b ¼ 0:796
6 0:0708
7
8
a 68:3%
Z
1
b 0:0884
4
dx = ¼
1 + x2
a
0
b F (x) =
c ¹=
2
¼
1
¼
f (x), )
which is 6= 1
k=
ln 2, Var(X) =
4
¼
1
¼
¡1¡
³ 2 ln 2 ´2
cyan
magenta
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_an\954IB_HL-3ed_an.cdr Monday, 21 May 2012 5:07:59 PM BEN
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
¼
9 ¾ ¼ 0:501 mL
10 0:403
11 a k = 8 b median = 2 27 c ¹ ¼ 2:75, Var(X) ¼ 2:83
black
IB HL 3ed