IB2 HL Questionbank questions on Probability and Statistics

IB2 Mathematics HL, 69 questions on Statistics and Probability
1.
Events A and B are such that P(A) = 0.3 and P(B) = 0.4.
(a)
(b)
2.
(4)
(3)
(Total 7 marks)
In a population of rabbits, 1 % are known to have a particular disease. A test is developed for the
disease that gives a positive result for a rabbit that does have the disease in 99 % of cases. It is also
known that the test gives a positive result for a rabbit that does not have the disease in 0.1 % of
cases. A rabbit is chosen at random from the population.
(a)
(b)
3.
Find the value of P(A ∪ B) when
(i) A and B are mutually exclusive;
(ii) A and B are independent.
Given that P(A ∪ B) = 0.6, find P(A | B).
Find the probability that the rabbit tests positive for the disease.
(2)
Given that the rabbit tests positive for the disease, show that the probability that the rabbit does not
have the disease is less than 10 %.
(3)
(Total 5 marks)
The random variable X has probability density function f where
f(x) =
(a)
(b)
⎧kx( x + 1)(2 − x),
⎨
0,
⎩
0≤ x≤2
otherwise.
Sketch the graph of the function. You are not required to find the coordinates of the
maximum.
(1)
Find the value of k.
(5)
(Total 6 marks)
4.
A batch of 15 DVD players contains 4 that are defective. The DVD players are selected at random,
one by one, and examined. The ones that are checked are not replaced.
(a)
(b)
What is the probability that there are exactly 3 defective DVD players in the first 8 DVD
players examined?
(4)
What is the probability that the 9th DVD player examined is the 4th defective one found?
(3)
(Total 7 marks)
5.
A continuous random variable X has a probability density function given by the function f(x), where
f(x) =
(a)
⎧k ( x + 2) 2
⎪
⎪
k,
⎨
⎪
0,
⎪
⎩
−2≤ x < 0
4
0≤ x≤
3
otherwise.
Find the value of k.
(2)
(b)
Hence find
(i)
the mean of X;
(ii)
the median of X.
(5)
(Total 7 marks)
6.
A student arrives at a school X minutes after 08:00, where X may be assumed to be normally
distributed. On a particular day it is observed that 40 % of the students arrive before 08:30 and 90 %
arrive before 08:55.
(a) Find the mean and standard deviation of X.
(5)
(b)
The school has 1200 students and classes start at 09:00. Estimate the number of students who
will be late on that day.
(c)
Maelis had not arrived by 08:30. Find the probability that she arrived late.
(3)
(2)
At 15:00 it is the end of the school day, and it is assumed that the departure of the students from
school can be modelled by a Poisson distribution. On average, 24 students leave the school every
minute.
(d) Find the probability that at least 700 students leave school before 15:30.
(3)
(e)
7.
8.
There are 200 days in a school year. Given that Y denotes the number of days in the year that
at least 700 students leave before 15:30, find
(i) E(Y);
(ii) P(Y > 150).
(4)
(Total 17 marks)
The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a standard
deviation of 0.2 kg.
(a) Determine the probability that a fish that is caught weighs less than 1.4 kg.
(1)
(b) John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg.
(3)
(c) Determine the probability that a fish that is caught weighs less than 1 kg, given that it weighs
less than 1.4 kg.
(2)
(Total 6 marks)
The number of accidents that occur at a large factory can be modelled by a Poisson distribution with
a mean of 0.5 accidents per month.
(a)
Find the probability that no accidents occur in a given month.
(1)
(b)
Find the probability that no accidents occur in a given 6 month period.
(2)
(c)
Find the length of time, in complete months, for which the probability that at least 1 accident
occurs is greater than 0.99.
(6)
(d)
To encourage safety the factory pays a bonus of $1000 into a fund for workers if no accidents
occur in any given month, a bonus of $500 if 1 or 2 accidents occur and no bonus if more than
2 accidents occur in the month.
(i)
Calculate the expected amount that the company will pay in bonuses each month.
(ii)
Find the probability that in a given 3 month period the company pays a total of exactly
$2000 in bonuses.
(9)
(Total 18 marks)
9.
Two players, A and B, alternately throw a fair six–sided dice, with A starting, until one of them
obtains a six. Find the probability that A obtains the first six.
(Total 7 marks)
10.
The ten numbers x1, x2, ..., x10 have a mean of 10 and a standard deviation of 3.
Find the value of
10
∑ (x
i
− 12) 2 .
i =1
(Total 6 marks)
11.
A continuous random variable X has probability density function
x<0
⎧ 0,
− ax
⎩ae , x ≥ 0.
f(x) = ⎨
It is known that P(X < 1) = 1 –
1
2
.
1
ln 2 .
2
(a)
Show that a =
(b)
Find the median of X.
(c)
Calculate the probability that X < 3 given that X > 1.
(6)
(5)
(9)
(Total 20 marks)
12.
A continuous random variable X has the probability density function f given by
⎧c( x − x 2 ), 0 ≤ x ≤ 1
0,
otherwise.
⎩
f(x) = ⎨
(a)
(b)
13.
Determine c.
Find E(X).
(3)
(2)
(Total 5 marks)
4
. The coin is tossed 6
7
P( X = 3)
times and X denotes the number of heads observed. Find the value of the ratio
.
P( X = 2)
A biased coin is weighted such that the probability of obtaining a head is
(Total 4 marks)
14.
15.
In a factory producing glasses, the weights of glasses are known to have a mean of 160 grams. It is
also known that the interquartile range of the weights of glasses is 28 grams. Assuming the weights
of glasses to be normally distributed, find the standard deviation of the weights of glasses.
(Total 6 marks)
Casualties arrive at an accident unit with a mean rate of one every 10 minutes.
Assume that the number of arrivals can be modelled by a Poisson distribution.
(a) Find the probability that there are no arrivals in a given half hour period.
(3)
(b) A nurse works for a two hour period. Find the probability that there are fewer than ten
casualties during this period.
(3)
(c)
(d)
Six nurses work consecutive two hour periods between 8am and 8pm.
Find the probability that no more than three nurses have to attend to less than ten casualties
during their working period.
(4)
Calculate the time interval during which there is a 95 % chance of there being at least two
casualties.
(5)
(Total 15 marks)
16.
A discrete random variable X has a probability distribution given in the following table.
(a)
x
0.5
1.5
2.5
3.5
4.5
5.5
P(X = x)
0.15
0.21
p
q
0.13
0.07
If E(X) = 2.61, determine the value of p and of q.
(4)
(b)
Calculate Var (X) to three significant figures.
(2)
(Total 6 marks)
17.
After being sprayed with a weedkiller, the survival time of weeds in a field is normally distributed
with a mean of 15 days.
(a)
If the probability of survival after 21 days is 0.2, find the standard deviation of the survival
time.
(3)
When another field is sprayed, the survival time of weeds is normally distributed with a mean of 18
days.
(b)
If the standard deviation of the survival time is unchanged, find the probability of survival
after 21 days.
(2)
(Total 5 marks)
18.
The random variable X follows a Poisson distribution with mean λ.
(a)
Find λ if P(X = 0) + P(X = 1) = 0.123.
(4)
(b)
With this value of λ, find P(0 < X < 9).
(2)
(Total 6 marks)
19.
After a shop opens at 09:00 the number of customers arriving in any interval of duration t minutes
follows a Poisson distribution with mean
(a)
t
.
10
(i)
Find the probability that exactly five customers arrive before 10:00.
(ii)
Given that exactly five customers arrive before 10:00, find the probability that exactly
two customers arrive before 09:30.
(7)
(b)
Let the second customer arrive at T minutes after 09:00.
(i) Show that, for t > 0,
t
t ⎞ −
⎛
P(T > t) = ⎜1 + ⎟e 10 .
⎝ 10 ⎠
(ii) Hence find in simplified form the probability density function of T.
(iii) Evaluate E(T).
(You may assume that, for n ∈ + and a > 0, lim t n e − at = 0 .)
t →∞
(12)
(Total 19 marks)
20.
Jenny goes to school by bus every day. When it is not raining, the probability that the bus is late is
3
7
. When it is raining, the probability that the bus is late is
. The probability that it rains on a
20
20
9
particular day is
. On one particular day the bus is late. Find the probability that it is not raining
20
on that day.
(Total 5 marks)
21.
22.
23.
The weight loss, in kilograms, of people using the slimming regime SLIM3M for a period of three
months is modelled by a random variable X. Experimental data showed that 67 % of the individuals
using SLIM3M lost up to five kilograms and 12.4 % lost at least seven kilograms. Assuming that X
follows a normal distribution, find the expected weight loss of a person who follows the SLIM3M
regime for three months.
(Total 5 marks)
The random variable X follows a Poisson distribution with mean m and satisfies
P(X = 1) + P(X = 3) = P(X = 0) + P(X = 2).
(a) Find the value of m correct to four decimal places.
(4)
(b) For this value of m, calculate P(1 ≤ X ≤ 2).
(2)
(Total 6 marks)
Tim throws two identical fair dice simultaneously. Each die has six faces: two faces numbered 1,
two faces numbered 2 and two faces numbered 3. His score is the sum of the two numbers shown
on the dice.
(a)
(i)
Calculate the probability that Tim obtains a score of 6.
(ii)
Calculate the probability that Tim obtains a score of at least 3.
(3)
Tim plays a game with his friend Bill, who also has two dice numbered in the same way. Bill’s
score is the sum of the two numbers shown on his dice.
(b)
(i)
Calculate the probability that Tim and Bill both obtain a score of 6.
(ii)
Calculate the probability that Tim and Bill obtain the same score.
(4)
(c)
Let X denote the largest number shown on the four dice.
16
.
81
(i)
Show that P(X ≤ 2) =
(ii)
Copy and complete the following probability distribution table.
x
1
P(X = x)
1
81
2
3
(iii) Calculate E(X) and E(X2) and hence find Var(X).
(10)
(d)
Given that X = 3, find the probability that the sum of the numbers shown on the four dice is 8.
(4)
(Total 21 marks)
24.
A random variable has a probability density function given by
⎧kx(2 − x), 0 ≤ x ≤ 2
0,
elsewhere.
⎩
f(x) = ⎨
25.
(a)
Show that k =
(b)
Find E(X).
3
.
4
(4)
(2)
(Total 6 marks)
Bob measured the heights of 63 students. After analysis, he conjectured that the height, H, of the
students could be modelled by a normal distribution with mean 166.5 cm and standard deviation 5 cm.
(a)
Based on this assumption, estimate the number of these students whose height is at least 170 cm. (3)
Later Bob noticed that the tape he had used to measure the heights was faulty as it started at the 5 cm
mark and not at the zero mark.
(b)
26.
27.
What are the correct values of the mean and variance of the distribution of the heights of these
students?
(3)
(Total 6 marks)
Mr Lee is planning to go fishing this weekend. Assuming that the number of fish caught per hour
follows a Poisson distribution with mean 0.6, find
(a)
the probability that he catches at least one fish in the first hour;
(2)
(b)
the probability that he catches exactly three fish if he fishes for four hours;
(2)
(c)
the number of complete hours that Mr Lee needs to fish so that the probability of catching more
than two fish exceeds 80 %.
(3)
(Total 7 marks)
In a class of 20 students, 12 study Biology, 15 study History and 2 students study neither Biology
nor History.
(a)
Illustrate this information on a Venn diagram.
(2)
(b)
Find the probability that a randomly selected student from this class is studying both Biology and
History.
(1)
(c)
Given that a randomly selected student studies Biology, find the probability that this student
also studies History.
(1)
(Total 4 marks)
28.
An influenza virus is spreading through a city. A vaccination is available to protect against the virus. If
a person has had the vaccination, the probability of catching the virus is 0.1; without the vaccination,
the probability is 0.3. The probability of a randomly selected person catching the virus is 0.22.
(a)
Find the percentage of the population that has been vaccinated.
(3)
(b)
A randomly chosen person catches the virus. Find the probability that this person has been
vaccinated.
(2)
(Total 5 marks)
29.
30.
31.
32.
Testing has shown that the volume of drink in a bottle of mineral water filled by Machine A at a
bottling plant is normally distributed with a mean of 998 ml and a standard deviation of 2.5 ml.
(a) Show that the probability that a randomly selected bottle filled by Machine A contains more
than 1000 ml of mineral water is 0.212.
(1)
(b) A random sample of 5 bottles is taken from Machine A. Find the probability that exactly 3 of
them each contain more than 1000 ml of mineral water.
(3)
(c) Find the minimum number of bottles that would need to be sampled to ensure that the
probability of getting at least one bottle filled by Machine A containing more than 1000 ml of
mineral water is greater than 0.99.
(4)
(d) It has been found that for Machine B the probability of a bottle containing less than 996 ml of
mineral water is 0.1151. The probability of a bottle containing more than 1000 ml is 0.3446.
Find the mean and standard deviation for the volume of mineral water contained in bottles
filled by Machine B.
(6)
(e) The company that makes the mineral water receives, on average, m phone calls every 10
minutes. The number of phone calls, X, follows a Poisson distribution such that
P(X = 2) = P(X = 3) + P(X = 4).
(i) Find the value of m.
(ii) Find the probability that the company receives more than two telephone calls in a
randomly selected 10 minute period.
(6)
(Total 20 marks)
The random variable X has a Poisson distribution. Given that P(X > 2) = 0.6, find
(a) the mean of the distribution;
(4)
(b) the mode of the distribution.
(2)
(Total 6 marks)
At a nursing college, 80 % of incoming students are female. College records show that 70 % of the
incoming females graduate and 90 % of the incoming males graduate.
A student who graduates is selected at random. Find the probability that the student is male, giving
your answer as a fraction in its lowest terms.
(Total 5 marks)
The annual weather-related loss of an insurance company is modelled by a random variable X with
probability density function
f(x) =
⎧ 2.5( 200) 2.5
⎪
⎨
x 3.5
⎪
0,
⎩
x ≥ 200
otherwise.
Find the median.
33.
34.
(Total 8 marks)
Tim goes to a popular restaurant that does not take any reservations for tables. It has been
determined that the waiting times for a table are normally distributed with a mean of 18 minutes and
standard deviation of 4 minutes.
(a) Tim says he will leave if he is not seated at a table within 25 minutes of arriving at the
restaurant. Find the probability that Tim will leave without being seated.
(2)
(b) Tim has been waiting for 15 minutes. Find the probability that he will be seated within the
next five minutes.
(4)
(Total 6 marks)
In each round of two different games Ying tosses three fair coins and Mario tosses two fair coins.
(a)
The first game consists of one round. If Ying obtains more heads than Mario, she receives $5 from
Mario. If Mario obtains more heads than Ying, he receives $10 from Ying. If they obtain the same
number of heads, then Mario receives $2 from Ying. Determine Ying’s expected winnings.
(12)
(b)
They now play the second game, where the winner will be the player who obtains the larger
number of heads in a round. If they obtain the same number of heads, they play another round
until there is a winner. Calculate the probability that Ying wins the game.
(8)
(Total 20 marks)
35.
The random variable T has the probability density function
f (t) =
π
⎛ πt ⎞
cos ⎜ ⎟, − 1 ≤ t ≤ 1.
4
⎝ 2 ⎠
Find
(a) P(T = 0);
(b) the interquartile range.
36.
The probability distribution of a discrete random variable X is defined by
(a)
(b)
37.
38.
P(X = x) = cx(5 − x), x = 1, 2, 3, 4.
Find the value of c.
Find E(X).
(3)
(3)
(Total 6 marks)
Let A and B be events such that P(A) = 0.6, P(A È B) = 0.8 and P(A | B) = 0.6.
Find P(B).
(Total 6 marks)
A company produces computer microchips, which have a life expectancy that follows a normal
distribution with a mean of 90 months and a standard deviation of 3.7 months.
(a)
(b)
(c)
39.
(2)
(5)
(Total 7 marks)
If a microchip is guaranteed for 84 months find the probability that it will fail before the
guarantee ends.
(2)
The probability that a microchip does not fail before the end of the guarantee is required to be
99%. For how many months should it be guaranteed?
(2)
A rival company produces microchips where the probablity that they will fail after 84 months
is 0.88. Given that the life expectancy also follows a normal distribution with standard
deviation 3.7 months, find the mean.
(2)
(Total 6 marks)
Only two international airlines fly daily into an airport. UN Air has 70 flights a day and IS Air has
65 flights a day. Passengers flying with UN Air have an 18% probability of losing their luggage and
passengers flying with IS Air have a 23% probability of losing their luggage. You overhear
someone in the airport complain about her luggage being lost.
Find the probability that she travelled with IS Air.
(Total 6 marks)
40.
The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at any
given time are independent of one another and can be modelled using a Poisson Distribution with
mean 0.2 per day.
(a)
Determine the probability that there will be exactly four breakdowns during the month of June
(June has 30 days).
(3)
(b)
Determine the probability that there are more than three breakdowns during the month of June.
(2)
(c)
Determine the probability that there are no breakdowns during the first five days of June.
(2)
(d)
Find the probability that the first breakdown in June occurs on June 3rd.
(3)
(e)
It costs 1850 euros to service the lifts when they have breakdowns. Find the expected cost of
servicing lifts for the month of June.
(1)
(f)
Determine the probability that there will be no breakdowns in exactly 4 out of the first 5 days
in June.
(2)
(Total 13 marks)
41.
A continuous random variable X has probability density function
⎧ 2
f (x) = ⎨12 x (1 − x ), for 0 ≤ x ≤1,
⎩
0,
otherwise.
Find the probability that X lies between the mean and the mode.
(Total 6 marks)
42.
Over a one month period, Ava and Sven play a total of n games of tennis.
The probability that Ava wins any game is 0.4. The result of each game played is independent of
any other game played.
Let X denote the number of games won by Ava over a one month period.
(a)
Find an expression for P(X = 2) in terms of n.
(3)
(b)
If the probability that Ava wins two games is 0.121 correct to three decimal places, find the
value of n.
(3)
(Total 6 marks)
43.
The distance travelled by students to attend Gauss College is modelled by a normal distribution
with mean 6 km and standard deviation 1.5 km.
(a) (i) Find the probability that the distance travelled to Gauss College by a randomly selected
student is between 4.8 km and 7.5 km.
(ii) 15% of students travel less than d km to attend Gauss College. Find the value of d.
(7)
At Euler College, the distance travelled by students to attend their school is modelled by a normal
distribution with mean m km and standard deviation s km.
(b) If 10% of students travel more than 8 km and 5% of students travel less than 2 km, find the
value of m and of s.
(6)
The number of telephone calls, T, received by Euler College each minute can be modelled by a
Poisson distribution with a mean of 3.5.
(c) (i) Find the probability that at least three telephone calls are received by Euler College in
each of two successive one-minute intervals.
(ii) Find the probability that Euler College receives 15 telephone calls during a randomly
selected five-minute interval.
(8)
(Total 21 marks)
44.
Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When she
tosses it, the score is defined as the number on the uppermost face. One day, she decides to toss the
die repeatedly until all the possible scores have occurred at least once.
(a)
Having thrown the die once, she lets X2 denote the number of additional throws required to
obtain a different number from the one obtained on the first throw. State the distribution of X2
and hence find E(X2).
(3)
(b)
(c)
She then lets X3 denote the number of additional throws required to obtain a different number
from the two numbers already obtained. State the distribution of X3 and hence find E(X3).
(2)
By continuing the process, show that the expected number of tosses needed to obtain all six
possible scores is 14.7.
(5)
(Total 10 marks)
45.
A factory makes wine glasses. The manager claims that on average 2% of the glasses are imperfect.
A random sample of 200 glasses is taken and 8 of these are found to be imperfect.
Test the manager’s claim at a 1% level of significance using a one-tailed test.
46.
The number of telephone calls received by a helpline over 80 one-minute periods are summarized in
the table below.
(a)
(b)
47.
(Total 7 marks)
Number of calls
0
1
2
3
4
5
6
Frequency
9
12
22
10
11
8
8
Find the exact value of the mean of this distribution.
(2)
Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson
distribution.
(12)
(Total 14 marks)
The heights, x metres, of the 241 new entrants to a men’s college were measured and the following
statistics calculated.
∑ x = 412.11, ∑ x
2
= 705.5721
(a)
Calculate unbiased estimates of the population mean and the population variance.
(b)
The Head of Mathematics decided to use a χ2 test to determine whether or not these heights
could be modelled by a normal distribution. He therefore divided the data into classes as
follows.
Interval
Frequency
(i)
(3)
x < 1.60
1.60 < x < 1.65
1.65 < x < 1.70
1.70 < x < 1.75
1.75 < x < 1.80
x ³ 1.80
5
34
70
72
48
12
State suitable hypotheses.
Calculate the value of the χ2 statistic and state your conclusion using a 10% level of
significance.
(12)
(Total 15 marks)
The independent random variables X and Y have Poisson distributions and Z = X + Y. The
means of X and Y are l and m respectively. By using the identity
(ii)
48.
(a)
n
P (Z = n ) =
∑ P (X = k ) P (Y = n − k )
k =0
show that Z has a Poisson distribution with mean (l + m).
(6)
(b)
Given that U1, U2, U3, … are independent Poisson random variables each having mean m,
use mathematical induction together with the result in (a) to show that
n
∑U
r
has a Poisson
r =1
distribution with mean nm.
49.
(6)
(Total 12 marks)
John removes the labels from three cans of tomato soup and two cans of chicken soup in order to
enter a competition, and puts the cans away. He then discovers that the cans are identical, so that he
cannot distinguish between cans of tomato soup and chicken soup. Some weeks later he decides to
have a can of chicken soup for lunch. He opens the cans at random until he opens a can of chicken
soup. Let Y denote the number of cans he opens.
Find
(a) the possible values of Y,
(1)
(b) the probability of each of these values of Y,
(4)
(c) the expected value of Y.
(2)
(Total 7 marks)
50.
A continuous random variable X has a probability density function given by
⎧ ( x + 1) 3
, for 1 ≤ x ≤ 3,
f(x) = ⎪⎨ 60
⎪
otherwise.
⎩ 0,
Find
(a) P(1.5 ≤ X ≤ 2.5);
(b) E(X);
(c) the median of X.
51.
(a)
(b)
Ahmed is typing Section A of a mathematics examination paper. The number of mistakes that
he makes, X, can be modelled by a Poisson distribution with mean 3.2. Find the probability
that Ahmed makes exactly four mistakes.
(1)
His colleague, Levi, is typing Section B of this paper. The number of mistakes that he makes,
Y, can be modelled by a Poisson distribution with mean m.
(i) If E(Y2) = 5.5, find the value of m.
(ii)
52.
(2)
(2)
(3)
(Total 7 marks)
Find the probability that Levi makes exactly three mistakes.
(5)
(c)
Given that X and Y are independent, find the probability that Ahmed makes exactly four
mistakes and Levi makes exactly three mistakes.
(2)
(Total 8 marks)
(a)
A box of biscuits is considered to be underweight if it weighs less than 228 grams. It is known
that the weights of these boxes of biscuits are normally distributed with a mean of 231 grams
and a standard deviation of 1.5 grams. What is the probability that a box is underweight?
(b)
(c)
(2)
The manufacturer decides that the probability of a box being underweight should be reduced
to 0.002.
(i)
Bill’s suggestion is to increase the mean and leave the standard deviation unchanged.
Find the value of the new mean.
(ii)
Sarah’s suggestion is to reduce the standard deviation and leave the mean unchanged.
Find the value of the new standard deviation.
(6)
After the probability of a box being underweight has been reduced to 0.002, a group of
customers buys 100 boxes of biscuits. Find the probability that at least two of the boxes are
underweight.
(3)
(Total 11 marks)
53.
Flowering plants are randomly distributed around a field according to a Poisson distribution with
mean µ. Students find that they are twice as likely to find exactly ten flowering plants as to find
exactly nine flowering plants in a square metre of field. Calculate the expected number of flowering
plants in a square metre of field.
(Total 5 marks)
54.
If P(A) =
1
1
5
, P(B) = , and P(A ∪ B) =
, what is P(A′ / B′)?
6
3
12
(Total 6 marks)
55.
A discrete random variable X has its probability distribution given by
P(X = x) = k(x + 1), where x is 0, 1, 2, 3, 4.
(a)
Show that k =
(b)
Find E(X).
1
.
15
(3)
(3)
(Total 6 marks)
56.
The probability density function of the random variable X is given by
⎧ k
, for 0 ≤ x ≤ 1
⎪
f(x) = ⎨ 4 − x 2
⎪
0,
otherwise.
⎩
(a)
Find the value of the constant k.
(5)
(b)
Show that E(X) =
6(2 − 3 )
.
π
(7)
(c)
Determine whether the median of X is less than
1
1
or greater than .
2
2
(8)
(Total 20 marks)
57.
Bag A contains 2 red and 3 green balls.
(a)
Two balls are chosen at random from the bag without replacement. Find
the probability that 2 red balls are chosen.
(2)
Bag B contains 4 red and n green balls.
(b)
Two balls are chosen without replacement from this bag. If the
probability that two red balls are chosen is
2
, show that n = 6.
15
(4)
A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A,
otherwise two balls are chosen from bag B.
(c)
Calculate the probability that two red balls are chosen.
(3)
(d)
Given that two red balls are chosen, find the probability that a 1 or a 6 was obtained on the
die.
(4)
(Total 13 marks)
58.
The speeds of cars at a certain point on a straight road are normally distributed with mean µ and
standard deviation σ. 15 % of the cars travelled at speeds greater than 90 km h–1 and 12 % of them
at speeds less than 40 km h–1. Find µ and σ.
(Total 6 marks)
59.
There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at
random to form a committee. Calculate the probability that the committee contains
(a)
two girls and two boys;
(3)
(b)
students all of the same gender.
(3)
(Total 6 marks)
60.
The random variable X has a Poisson distribution with mean 4. Calculate
(a)
P(3 ≤ X ≤ 5);
(2)
(b)
P(X ≥ 3);
(2)
(c)
P(3 ≤ X ≤ 5|X ≥ 3).
(2)
(Total 6 marks)
61.
The cumulative frequency graph below represents the weight in grams of 80 apples picked from a
particular tree.
(a)
Estimate the
(i)
median weight of the apples;
(ii)
30th percentile of the weight of the apples.
(2)
(b)
Estimate the number of apples that weigh more than 110 grams.
(2)
(Total 4 marks)
62.
The heights in metres of a random sample of 80 boys in a certain age group were measured and the
following cumulative frequency graph obtained.
(a)
(i)
Estimate the median of these data.
(ii)
Estimate the interquartile range for these data.
(3)
(b)
(i)
Produce a frequency table for these data, using a class width of 0.05 metres.
(ii)
Calculate unbiased estimates of the mean and variance of the heights of the population
of boys in this age group.
(5)
(c)
A boy is selected at random from these 80 boys.
(i)
Find the probability that his height is less than or equal to 1.15 metres.
(ii)
Given that his height is less than or equal to 1.15 metres, find the probability that his
height is less than or equal to 1.12 metres.
(5)
(Total 13 marks)
63.
The company Fresh Water produces one-litre bottles of mineral water. The company wants to
determine the amount of magnesium, in milligrams, in these bottles.
A random sample of ten bottles is analysed and the results are as follows:
6.7, 7.2, 6.7, 6.8, 6.9, 7.0, 6.8, 6.6, 7.1, 7.3.
Find unbiased estimates of the mean and variance of the amount of magnesium in the one-litre
bottles.
(Total 4 marks
64.
(a)
Consider the set of numbers a, 2a, 3a, ..., na where a and n are positive integers.
a (n + 1)
.
2
(i)
Show that the expression for the mean of this set is
(ii)
Let a = 4. Find the minimum value of n for which the sum of these numbers exceeds its
mean by more than 100.
(6)
(b)
Consider now the set of numbers x1, ... , xm, y1, ... , y1, ... , yn where xi = 0 for i = 1, ... , m
and yi = 1 for i = 1, ... , n.
(i)
Show that the mean M of this set is given by
S by
(ii)
n
and the standard deviation
m+n
mn
.
m+n
Given that M = S, find the value of the median.
(11)
(Total 17 marks)
65.
The test scores of a group of students are shown on the cumulative frequency graph below.
(a)
Estimate the median test score.
(1)
(b)
The top 10 % of students receive a grade A and the next best 20 % of students receive a grade
B. Estimate
(i)
the minimum score required to obtain a grade A;
(ii)
the minimum score required to obtain a grade B.
(4)
(Total 5 marks)
66.
Consider the data set {k − 2, k, k +1, k + 4}, where kÎ .
(a)
Find the mean of this data set in terms of k.
(3)
Each number in the above data set is now decreased by 3.
(b)
Find the mean of this new data set in terms of k.
(2)
(Total 5 marks)
67.
A recruitment company tests the aptitude of 100 applicants applying for jobs in engineering. Each
applicant does a puzzle and the time taken, t, is recorded.
The cumulative frequency curve for these data is shown below.
Using the cumulative frequency curve,
(a)
write down the value of the median;
(1)
(b)
determine the interquartile range;
(2)
(c)
complete the frequency table below.
Time to complete puzzle in
seconds
Number of applicants
20 < t ≤ 30
30 < t ≤ 35
35 < t ≤ 40
40 < t ≤ 45
45 < t ≤ 50
50 < t ≤ 60
60 < t ≤ 80
(2)
(Total 5 marks)
68.
A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks
is given below.
(a)
Write down the number of students who scored 40 marks or less on the test.
(2)
(b)
The middle 50 % of test results lie between marks a and b, where a < b.
Find a and b.
(4)
(Total 6 marks)
69.
The continuous random variable X has probability density function
f(x) =
1
x(1 + x2) for 0 ≤ x ≤ 2,
6
f(x) = 0
(a)
otherwise.
Sketch the graph of f for 0 ≤ x ≤ 2.
(2)
(b)
Write down the mode of X.
(1)
(c)
Find the mean of X.
(4)
(d)
Find the median of X.
(5)
(Total 12 marks)