It is not necessary to decode the values of
x– 3.
TIP Forty values of x are coded in the following table.
3
x– 0– 18– 24–32
Frequency 9 13 18
Calculate an estimate of the mean value of x.
Answer ( 3)
3
(9 9) (21 13) (28 18)
40 3
24.45
x x f
= Σ −f
Σ +
= × + × + × +
=
We calculate an estimate for the mean of the coded data using class mid-values of 9, 21 and 28, and then add 3 to obtain our estimate for x.
WORKED EXAMPLE 2.10
For two datasets coded as (x − a) and (y − b), we can use the totals Σx and Σy to find the mean of the combined set of values of x and y.
The exact age of an individual boy is denoted by b, and the exact age of an individual girl is denoted by g.
Exactly 5 years ago, the sum of the ages of 10 boys was 127.0 years, so Σ( −b 5)=127.0. In exactly 5 years’ time, the sum of the ages of 15 girls will be 351.0 years, so Σ( +g 5) =351.0. Find the mean age today of
a the 10 boys b the 15 girls c the 10 boys and 15 girls combined.
Answers
a b=127+ =
10 5 17.7years
b b
b b
( 5) (10 5) 127, so 127 50 177 and 177
10 17.7 years.
Σ − = Σ − × =
Σ = + = = =
b
= − =
g 351
15 5 18.4 years
Σ +(g 5)= Σ +g (15 5)× =351, so
Σ =g 351 75− =276 andg= 276=
15 18.4 years.
c b g 10 15
177 276
25 18.12 years Σ + Σ
+ = + =
Alternatively, we expand the brackets.
We backdate the girls’ future mean age by subtraction.
Alternatively, we expand the brackets.
We update the boys’ past mean age by addition.
WORKED EXAMPLE 2.9
38
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy
Copyright Material - Review Only - Not for Redistribution
EXERCISE 2C
1 For 10 values denoted x, it is given that x=7.4. Find:
a Σx b Σ +(x 2) c Σ −(x 1)
2 Twenty-five values of z are such that Σ −(z 7)=275. Find z.
3 Given q=22 and (Σ −q 4)=3672, find the number of values of q.
4 The lengths of 2500 bolts, xmm, are summarised by (Σ −x 40)=875. Find the mean length of the bolts.
5 Six data values are coded by subtracting 13 from each. Five of the coded values are 9.3, 5.4, 3.9, 7.6 and 2.2, and the mean of the six data values is 17.6.
Find the sixth coded value.
6 The SD card slots on digital cameras are designed to accommodate a card of up to 24 mm in width. Due to low sales figures, a manufacturer suspects that the machine used to cut the cards needs to be recalibrated. The widths, wmm, of 400 of these cards were measured and are coded in the following table, where x=w– 24.
– 24 (mm)
w –0.15<x<–0.1 –0.1<x<0 0<x<0.1 0.1<x<0.2
No. cards ( )f 32 360 6 2
a Suggest a reason why the widths have been coded in this way.
b What percentage of the SD cards are too wide to fit into the slots?
c Use the coded data to estimate the mean width of these 400 cards.
7 Sixteen bank accounts have been accidentally under-credited by the following amounts, denoted by $ . x
917.95 917.98 918.03 917.97 918.01 917.94 918.05 918.07 918.02 917.93 918.01 917.88 918.10 917.85 918.11 917.94 To calculate x manually, Fidel and Ramon code these figures using ( – 917) x and ( – 920), respectively.x
Who has the simpler maths to do? Explain your answer.
8 Throughout her career, an athlete has been timed in 120 of her 400-metre races.
Her times, denoted by t seconds, were recorded on indoor tracks 45 times and are summarised by Σ −(t 60)=83.7, and on outdoor tracks where Σ −(t 65)= −38.7. Calculate her average 400-metre running time and comment on the accuracy of your answer.
9 All the interior angles of n triangular metal plates, denoted by y°, are measured.
a State the number of angles measured and write down the value of y.
b Hence, or otherwise, find the value of Σ −(y 30).
10 A dataset of 20 values is denoted by x where Σ − =(x 1) 58. Another dataset of 30 values is denoted by y where Σ −(y 2)=36. Find the mean of the 50 values of x and y. P
PS
P
You will study the mean of linear combinations of random variables in the Probability
& Statistics 2 Coursebook, Chapter 3.
FAST FORWARD
39
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy
Copyright Material - Review Only - Not for Redistribution
) Σ(ax b− can be rewritten as a xΣ −nb.
TIP 11 Students investigated the prices in dollars ($) of 1 litre bottles of a certain drink
at 24 shops in a town and at 16 shops in surrounding villages. Denoting the town prices by t and the village prices by v, the students’ data are summarised by the totals Σ −(t 1.1)=1.44 and (Σ −v 1.2)=0.56.
Find the mean price of 1 litre of this drink at all the shops at which the students collected their data.
The total area of cloth produced at a textile factory is denoted by Σx and is measured in square metres. Find an expression in x for the area of cloth produced in square centimetres.
Answer
= 1m 100 cm
= =
1m2 100 cm2 2 10 000 cm2
Total area, in square centimetres, is Σ10 000x or 10 000Σx.
We convert the measurements of x from m2 to cm2.
WORKED EXAMPLE 2.11
For the 20 values of x summarised by Σ(2x−3)=104, find x. Answer
104= 20 5.2 x=5.2+ =3
2 4.1
WORKED EXAMPLE 2.12
A set of data can be coded by multiplication as well as by addition of a constant.
Suppose the monthly take-home salaries of four teachers are $3600, $4200, $3700 and
$4500, which have mean x=$4000.
What happens to the mean if all the teachers receive a 10% increase but must pay an extra
$50 in tax each month?
To find their new take-home salaries, we multiply the current salaries by 1.1 and then subtract 50.
The new take-home salaries are $3910, $4570, $4020 and $4900.
The mean is 3910+4570+4020+4900 =
4 $4350.
The original data, x, has been coded by multiplication and by addition as 1.1 – 50.x The mean of the coded data is 4350, which is equal to (1.1 4000) – 50, where × 4000=x. Data coded as ax– has a mean of b ax−b.
To find x from a total such as Σ(ax−b), we can find the mean of the coded data, then undo
‘– ’ and undo ‘b ×a’, in that order. That is:
x=(4350 50) 1.1 or + ÷ 1 × + = 1.1 (4350 50) 4000.
We first find the mean of the coded values.
Knowing that 2x− =3 5.2, we undo the ‘ – 3’ and then undo the ‘ 2× ’, in that order, to find x.
For ungrouped data, x a
ax b
n b
1 ( )
= Σ − + .
For grouped data, x a
ax b f
f b
1 ( )
Σ .
= Σ − +
These formulae can be summarised by writing
x a1 ax b b
[mean( ) ].
= × − +
KEY POINT 2.4
40
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy
Copyright Material - Review Only - Not for Redistribution
EXERCISE 2D
1 The masses, xkg, of 12 objects are such that x=0.475. Find the value of Σ1000x and state what it represents.
2 The total mass of gold extracted from a mine is denoted by Σx, which is measured in grams. Find an expression in x for the total mass in:
a carats, given that 1 carat is equivalent to 200 milligrams b kilograms.
3 The area of land used for growing wheat in a region is denoted by Σw hectares. Find an expression in w for the total area in square kilometres, given that 1 hectare is equivalent to 10 000 m2.
4 Speeds, measured in metres per second, are denoted by x. Find the constant k such that kx denotes the speeds in kilometres per hour.
5 The wind speeds, x miles per hour (mph), were measured at a coastal location at midday on 40 consecutive days and are presented in the following table.
Speed ( mph)x 15<x<17 17<x<20 20<x<24 24<x<25
No. days ( )f 9 13 14 4
Abel wishes to calculate an estimate of the mean wind speed in kilometres per hour (km/h). He knows that a distance of 5 miles is approximately equal to 8 km.
a Explain how Abel can calculate his estimate without converting the given boundary values from miles per hour to kilometres per hour.
b Use the wind speeds in mph to estimate the mean wind speed in km/h.
6 Given that 15 values of x are such that Σ(3x−2)=528, find x and find the value of b such that x b
Σ(0.5 − )=138.
7 For 20 values of y, it is given that Σ(ax−b)=400 and Σ(bx−a)=545. Given also that x=6.25, find the value of a and of b.
8 The midpoint of the line segment between A and B is at (5.2,–1.2).
Find the coordinates of the midpoint after the following transformations have been applied to A and to B.
a T: Translation by the vector 7 4
− . P
Alternatively, we can expand the brackets in Σ(2x−3), which allows us to find the value of Σx.
x x
x
Σ − =
Σ − × = Σ = (2 3) 104 2 (20 3) 104 82 x=82 =
20 4.1
We will see how to use coded totals such as Σ(ax b− ) and
ax b
( )2
Σ − to find measures of variation in Chapter 3, Section 3.3.
FAST FORWARD
41
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy
Copyright Material - Review Only - Not for Redistribution
2.3 The median
You will recall that the median splits a set of data into two parts with an equal number of values in each part: a bottom half and a top half. In a set of n ordered values, the median is at the value half-way between the 1st and the nth.
Consider a DIY store that opens for 12 hours on Monday and for 15 hours on Saturday.
The numbers of customers served during each hour on Monday and on Saturday last week are shown in the following back-to-back stem-and-leaf diagram.
represents 20 customers on Monday and 22 customers on Saturday
2 3 4
Monday (12) Saturday (15) 2 3 4 6 5 5 6 8 9 9 0 1 3 7 9 4 3 1 1 0
1
Key: 0 2 2 8 6 3 1 0 0
To find the median number of customers served on each of these days, we need to find their positions in the ordered rows of the back-to-back stem-and-leaf diagram.
For Saturday, there are n=15 values arranged in ascending order from top to bottom and from left to right. The median is at the n 1
2 th 15 1
2 8th
( )
+
= + = value.
In the first row, we have the 1st to 4th values, and in the second row we have the 5th to 10th values, so the 8th value is 38.
The median number of customers on Saturday was 38.
For Monday, there are n=12 values arranged in ascending order from top to bottom and from right to left. The median is at the n 1
2 th 12 1
2 th 6.5th
( )
+
= + = value, so we locate the median mid-way between the 6th and 7th values.
In the first row, we have the 1st to 6th values and the 6th is 28.
The first value in the second row is the 7th value, which is 30.
The median number of customers on Monday was 28 30+ =
2 29.
When data appear in an ordered frequency table of individual values, we can use
cumulative frequencies to investigate the positions of the values, knowing that the median is at the n 1
2+ th
value.
b E: Enlargement through the origin with scale factor 5.
c Transformations T and E are carried out one after the other. Investigate whether the location of the mid- point of AB is independent of the order in which the transformations are carried out.
9 Five investors are repaid, each with their initial investment increased by %p plus a fixed ‘thank you’ bonus of $q. The woman who invested $20 000 is repaid double her investment and the man who invested $7500 is repaid triple his investment. Find the total amount that the five people invested, given that the mean amount repaid to them was $33000.
Do you think the method of repayment is fair? Give a reason for your answer.
10 One of the units used to measure pressure is pounds per square inch (psi). The mean pressure in the four tyres of a particular vehicle is denoted by xpsi. Given that 1 pound is approximately equal to 0.4536 kg and that 1 metre is approximately equal to 39.37 inches, express the sum of the pressures in the four tyres of this vehicle in grams per cm2.
M
PS
We can find the 8 th value by counting down and left to right from 22 or by counting up and right to left from 49.
TIP
Take care when locating values at the left side of a back- to-back stem-and-leaf diagram; they ascend from right to left, and descend from left to right, as we move along each row.
TIP
n is equal to the total frequency Σf.
TIP
For n ordered values, the median is at the
n 1 2
+
th value.
For even values of n, the median is the mean of the two middle values.
KEY POINT 2.5
42
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy
Copyright Material - Review Only - Not for Redistribution