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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Advanced Level

STATISTICS 6046/1

PAPER 1
SPECIMEN PAPER 3 hours

Additional materials:
Answer paper
Graph paper
List of Formulae
Electronic calculator

TIME 3 hours

INSTRUCTIONS TO CANDIDATES

Write your name, centre number and candidate number in the spaces provided on the answer
paper/ answer booklet.

Answer all questions.

If a numerical answer cannot be given exactly, and the accuracy required is not specified in
the question, then in the case of an angle it should be given to the nearest degree, and in other
cases it should be given correct to 2 significant figures.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 120.
The use of an electronic calculator is expected, where appropriate.

Y ou are reminded of the need for clear presentation in your answers.

This specimen paper consists of 6 printed pages and 2 blank pages.
Copyright: Zimbabwe School Examinations Council, Specimen paper.

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In a group of 40 students all of whom are studying Statistics or Pure Mathematics
or both, 20 are studying Statistics and 30 are studying Pure Mathematics.

(a) Illustrate the information on a Venn diagram. [1]
(b) Find the probability that a student chosen at random is
@ studying Statistics and Pure Mathematics,

di studying Pure Mathematics but not Statistics.
[4]

Four letters are chosen at random the word DARLING. Find the probability
that

(a) exactly 2 consonants are chosen. [2]
(b) at least 3 consonants are chosen. [3]

The life span, in years of a randomly chosen car battery is normally
distributed with mean 2 and standard deviation 0.4.

(a) Show that a randomly chosen car battery has a life span less than
a year is 0.0062. Correct to 4 decimal places. [2]

(b) =A carbattery dealer, buys 500 randomly chosen car batteries.

Using a suitable approximation. Find the probability that at most
three batteries have a life span less than a year. [4]

Cars arrive at a service station at an average rate of 2 per 5 minutes interval.
Assuming that the cars follow a Poisson distribution find the probability that
@ no cars arrive during a 5 minutes interval,

di at least 3 cars arrive in the next 15 minutes.

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(a) Give 2 examples of situations that can be modelled by an
exponential distribution. [2]

(b) A dart player aims at the bull’s eye. The distance X cm from the

bull’s eye at which the arrow strikes the dart board has a
probability density function defined by

+ e7i0 x>0
f@)= {i ,
0, Otherwise

An arrow scores 8 points if X <2, 5 points if 2 <X <5 and 1 point
if 5 <X< 15 and no points otherwise

@ Construct the probability distribution table for the scores.

di Find the expected score when one arrow is shot at the

bull’s eye.
[6]

A shop’s quarterly electricity bill, $, over a period of 3 years were as
follows:

Year Qi Q2 Q3 Qa

1 112 137 161 154

2 133 147 188 161

3 163 184 209 201
@ Plot these data on a time series graph. [3]
(ii) Calculate 5 point moving averages for the data. [3]
(iii) Plot the 5 point moving averages on the graph in (i). [2]

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(b) A random observation is taken from a binomial distribution

X~ Bin (12, p), and used to test the null hypothesis p = 0,7

against the alternative p > 0,7.

The critical region is chosen to be x> 11.

Find the @ significance of the test.

(ii) probability of making a Type I error

(iii) probability of making a Type II error if
p =0,75.

8 Of the 300 graduands from a college, 120 failed to get employment.
(a) Calculate:

(1) The percentage of graduands who failed to get
employment.

(ii) A 97% confidence interval for the proportion of
graduands who failed to get employment. [4]

(b) Find the sample size that would have been taken in order to
estimate the percentage to within +3% with 97% confidence. [4]

9 The table below shows the weekly wages ($x) of 100 employees.

Wages $x | 45-55 | 55-65 | 65-75 | 75-85 | 85-95 | 95-105 | 105-115 | 115-125 | 125-135

Frequency 1 1 2 6 21 29 24 12 4

(a) @ Construct a cumulative frequency table.

di Draw a cumulative frequency curve.

[6]
(b) Use your graph to estimate:
@ The number of workers who earn a wage greater than $80.
(ii) x, if 20% of the employees earn more than $x. [3]

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11

12

5

A continuous random variable X has a probability density function given by

—kx -2<x<0

fQ)=

0

where k is a constant.

kx OSx<2
otherwise

(a) Sketch the graph of f(x) and hence find the value of k

(b) = Calculate Var(X)

() A random sample of 200 observations of X is taken, find

the probability that the sample mean exceeds 0.2.

A school has two photocopiers X and Y . the number of times per

week that X breaks down has a Poisson distribution with mean

0.3, while independently the number of times that Y breaks down
in a week follows a Poisson distribution with mean 0.2.

Find the probability that in the next 4 weeks.

@ X will not breakdown at all.

di There will be a total of 3 breaks down.

(iii) Each photocopier will breakdown exactly twice.

Ten boys compete in throwing

a cricket bal

the height of each boy (x cm) measured to

and the distance (Y m) to whic:

1, and the table shows
the nearest centimetre
h he can throw the ball.

Boys|A_ |B

C

D

E

F

G H

x 122

124

133

138

144

156

158 | 161

164

168

y Al

38

52

56

29

54

59 | 61

63

67

yx = 1468;

yy? = 28 382;

Calculate

yx? = 218 070;

xy = 77 689

Xv = 520

(1) The regressions line y on x and x ony.

di Coefficient of determination and comment on its significance.

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A college claims that the performance in Mathematics for their non formal learners
depends on the time of day during which they had their lessons. A group of 160
learners gave the following results.

Time of day Performance

Pass Fail
Morning 40 30
Afternoon 44 10
Evening 20 16

Test the claim at the 5% level of significance.

14 The data below were collected about the diameters (cm) of 80 plebbles.

Diameter Number of pebbles
<18 3
1.8-2.2 17
2.2-2.6 33
2.6-3 22
=3 5

Test at 1% of significance whether the diameters follow a normal
distribution with p = 2.44 and 6 =0.4.

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