contents 7.2 applications of percentage in banking and 7.3 time-series graphs 7.4 interpreting and...

Contents

7.2 Applications of Percentage in Banking and

7.3 Time-series Graphs

7.4 Interpreting and Analysing Data Collected

7.1 Arithmetic and Geometric Means

7 Further Applications (3)

Finance

from Surveys

7.5 Determining the Relation between Two Variables

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Further Applications (3)7

Content

.0 if undefined

ismean geometric the

that Note . is ratio

common the, ,

, sequence in the while

;is ratiocommon the

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In means. geometric are

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a

b

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7.1 Arithmetic and Geometric Means

In between two numbers, arithmetic means are numbers inserted to form an arithmetic sequence. Similarly. Geometric means are numbers inserted to form a geometric sequence. However, if we add only one term x between two numbers, say, a and b, the sequence becomes

a, x, b.

2

2

bax

bax

axxb

To form an arithmetic sequence,

abx

abxa

x

x

b

2

To form a geometric sequence,

The value (s) of x can be calculated as follow.

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Content

7.2 Applications of Percentage in Banking and Finance

Suppose a person deposits $P in a bank account for n years and the

interest rate is r% per annum. The total amount he can get after n years

depends on whether the interest is calculated as simple interest or

compound interest.

1. For simple interest,

total amount = P (1 + r% n)

nrP %)1( amount total(a)

nr

P12

12

%1

amount total (b)

if the interest is compounded monthly.

if the interest is compounded yearly;

2. For compound interest,

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Content

7.2 Applications of Percentage in Banking and Finance

1%)1(

%%)1(

n

n

r

rrPQ

In general, if we want to fully repay a loan of $p with interest rate 12r% per annum in n months, then the amount of monthly instalments $Q can be calculated as follow:

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Content

A. Introduction


Definition 7.1:

All the data in a time-series graph are connected by broken lines (or curves).

From the graph, we can observe the following features;

1. Trend --- it is the long-term movement of the data over a period of time.

A time-series graph is a graph that shows a sequence of observations (or a

group of data) over a period of time.

2. Seasonal variations --- it is the short-term fluctuations in recorded

values that are affected by different times of

a year, different days of a week or different

times of a day.

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Content


B. Moving Average of a Series of Data

Definition 7.2:

For example, the closing prices of the stock of the Culture Company in the last ten transaction days are given below.

Date Price ($) Date Price ($)

1/12 2.8 8/12 3.8

2/12 2.6 9/12 4.4

3/12 3.5 10/12 4.0

4/12 3.2 11/12 4.5

5/12 3.6 12/12 4.5

The trend of the prices of a stock can be studied by the moving average method.

A moving average is the average value of a group of data over a fixed number of periods.

Table 7.6

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Content

Fig. 7.10 shows the time-series graph of the closing prices of the stock of the Culture Company.


From the graph, the price moves up and down.

Fig. 7.10

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Content


If we take the moving average of the price over a period of three consecutive days, we can get the following results:

The moving average shows that there is an upward trend in the stock price.

Date Price ($) Moving average ($)

1/12 2.8 -----

2/12 2.6 -----

3/12 3.5 2.97

4/12 3.2 3.1

5/12 3.6 3.43

8/12 3.8 3.53

9/12 4.4 3.93

10/12 4.0 4.07

11/12 4.5 4.3

12/12 4.5 4.33 Table 7.7

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Content

Different statistical diagrams may help to present the data in different ways.

For example, if we consider the types of cars involved in road accidents in a

city in a certain period, we can use different diagrams to present the data.

7.4 Interpreting and Analysing Data Collected from Surveys

From the bar chart, we can get the actual frequency and the order of magnitude. But from the pie chart, we notice that private cars were involved in almost 50% of the accidents, so, we may choose different statistical diagrams to fulfill different purposes.

Fig. 7.15

Fig. 7.16

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Content


From the above two figures, we observe that the higher the level of investment, the more the number of people employed. But the number of people employed decreases from 1995 to 2005 with the same investment.

The following scatter diagrams show the data collected from a survey that was conducted to examine the relationship between the number of people employed and the amount of investment by ten factories in 1995 and 2005.

Fig. 7.17 Fig. 7.18

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Content


Consider the sales (in bottle) of two types of soft drink in two weeks.

From Fig. 7.19, we observe that both brands of soft drink increased their sales by 100 and the slope of the two graphs are the same. This may give us an impression that the relative changes in sales are the same.

Last week This week

Soft drink A 800 900

Soft drink B 100 200

However, if we consider the percentage changes of their sales, we will find that the percentage increase in sales of soft drink B is eight times that of soft drink A.

Fig. 7.19

Thus, the broken line graph cannot show the relative change of the survey

data.

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If we are only interested in the absolutes changes in figures, then using the exact values of the data to plot a statistical graph is acceptable.

Year Price ($) log P

1 100.0 2.00

2 110.0 2.04

3 121.0 2.08

4 133.1 2.12

5 146.4 2.17

Suppose the price of a toy increases by 10% each year and its price is $100 at the first year.

If we consider the inflation rate, then the percentage increase in the prices will be more important than the increase in the actual prices.

Table 7.18

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The first graph (Fig. 7.20) shows the actual prices of the toy over the past five years, while the second graph (Fig. 7.21) shows the increase in the prices of the toy at a constant rate, that is, 10% each year.

Fig. 7.20 Fig. 7.21

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Content


A linear function f (x) is in the form

f (x) = ax + b,

while the equation of a straight line can be expressed as

y = mx + c,

For two variables x and y, we say they obey a linear law if y is a linear

function of x.

We observe that the form of a linear function is very similar to the equation

of a straight line. The graph of a linear function is also a straight line

where m is the slope of the line and c is the y-intercept.

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Content


then If ,310 xy

If the original values of the two variables do not obey a linear law, they may obey a linear law after making some transformation. For an index function or an exponential function, we may take logarithm on both sides of the equation to convert them into a linear function.

Fig. 7.27 shows the linear graph log y against x.

For example,

1)3(log

)3(log10log

)310log(log

x

x

y x

So , we say that log y and x obey a linear law.

Fig. 7.27

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Content


x 1 2 4 5 10

y 20 10 5 4 2

have we Consider ,20

xy

Fig. 7.28whose graph is a curve as shown in Fig. 7.28.

Table 7.28

law. linear aobey and that observe can wegraph, the Fromx

y1

2 4 5 101x

10 5 4 220y

0.10.20.250.51x

1

have wethen against of graph the plot weIf ,1

xy

Fig. 7.29

Table 7.27

contents 7.2 applications of percentage in banking and 7.3 time-series graphs 7.4 interpreting and...

Documents

applications of percentage

timeseries graphs definition

period of time

series of data definition

geometric sequence

arithmetic sequence

moving average method

geometric means