Maths Quest Maths A Year 12 for Queensland
WorkSHEET 10.1
Chapter 10 The normal distribution and games of chance
WorkSHEET 10.1
Statistics
1
Name: ___________________________
/50
5
1
In a class test where the mean was 62% and
the standard deviation 12.6%, Jan received
74%. Calculate her result as a z-score.
x  x
z
s
74  62

12
12

12
1
2
On the same test, Bob received 50%. What is
Bob’s result as a z-score?
x  x
z
s
50  62

12
 12

12
 1
5
3
The heights of eight professional basketball
players are (in centimetres):
188, 189, 192, 193, 194, 194, 195, 195.
(a)
6
(a)

x  192.5
s  2 .5
Calculate the mean and standard
deviation of their heights (to one decimal
place).
(b)
(b)
4
John is 190 cm tall. Express his height as
a z-score compared with the basketball
players’ heights (to one decimal place).
Bill’s height, when converted to a z-score
compared with the basketball players in
question 3, is +1. What is Bill’s height?
Enter the data into a calculator which
provides statistical calculations.
x  x
s
190  192.5

2.5
 2.5

2.5
 1
z
Bill’s height is one standard deviation above
the mean height of the basketball players.
So, Bill’s height = 192.5 cm + 2.5 cm
= 195 cm
4
Maths Quest Maths A Year 12 for Queensland
5
Chapter 10 The normal distribution and games of chance
WorkSHEET 10.1
Ryan received 82% in his History exam where
the mean was 70% and the standard deviation
10%. In his Biology exam he received 75%
when the mean was 50% and the standard
deviation 12%. He felt he had performed
better in History than in Biology. Is this the
case? Explain.
History
x  x
s
82  70

10
12

10
 1 .2
z
Biology
2
5
x  x
s
75  50

12
25

12
 2 .1
z
Ryan is 1.2 standard deviations above the class
mean in History and 2.1 standard deviations
above the class mean in Biology. He has
therefore performed better in Biology than
History.
6
In international swimming the mean time for
the men's 100 m freestyle is 50.46 sec with a
standard deviation of 0.6 sec. For the 200 m
freestyle, the mean time is 110.4 sec with a
standard deviation of 1.4 sec.
Jason’s best time for the 100 m is 48.76 sec
and for the 200 m is 108.43 sec.
7
100 m
x  x
s
48.76  50.46

0 .6
 2.83
z
200 m
x  x
s
108.43  110.4

1.4
 1.41
z
If he can only enter one of these events in the
competition, which one should he enter?
Jason’s z-score for the 100 m is lower,
indicating that his time is further below the
mean for this event than for the 200 m event.
So, he should enter the 100 m event.
What percentage of the scores lie:
(a) within one standard deviation of the
mean?
(a)
68% of the scores lie within one standard
deviation either side of the mean.
(b)
95% of the scores lie within two standard
deviations either side of the mean.
(c)
99.7% of the scores lie within three
standard deviations either side of the
mean.
(b)
within two standard deviations of the
mean?
(c)
within three standard deviations of the
mean?
5
6
Maths Quest Maths A Year 12 for Queensland
8
Chapter 10 The normal distribution and games of chance
WorkSHEET 10.1
On a common IQ test, the distribution is
normal with a mean of 100 and a standard
deviation of 12. What percentage of the scores
lie between 76 and 124?
Lower z - value
x  x
s
76  100

12
 24

12
 2
z
Upper z - value
3
5
x  x
s
124  100

12
24

12
2
z
The score of 76 is two standard deviations
below the mean and the score of 124 is two
standard deviations above the mean. So, 95%
of the scores lie in the range 76 to 124.
9
In the above IQ test in question 8, what range
of scores would account for 99.7% of the
people?
99.7% of the scores lie within three standard
deviations either side of the mean.
4
3s  3  12
 36
So a score 36 below 100 to a score 36 above
100 would account for 99.7% of the people.
So the range 64 to 136 would include 99.7% of
people.
10
On an end of semester maths test the mean
result was 62% and the standard deviation was
12%.
What percentage of the results would lie above
86%?
x  x
z
s
86  62

12
24

12
2
So a score of 86% lies two standard deviations
above the mean. 95% of the scores lie within
two standard deviations either side of the mean.
This means that 5% of the scores lie outside
this range. Half of these scores lie below –2
and half above +2.
 Percentage of results above 86% = 2.5%
5