Chapter 20 Further Applications (3)

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Further Applications (3) Contents 7.1 Arithmetic and Geometric Means 7.2 Applications of Percentage in Banking and Finance 7.3 Time-series Graphs 7.4 Interpreting and Analysing Data Collected from Surveys 7.5 Determining the Relation between Two Variables

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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. The value (s) of x can be calculated as follow. To form an arithmetic sequence,

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b−x = x−a 2x = a + b a+b x= 2

To form a geometric sequence, b x = x a x 2 = ab x = ± ab

Both + ab and − ab are geometric means. In the sequence a, ab , b, the common ratio is

b

a while in the sequence a,

;

− ab , b, the common ratio is −

b

. Note that a the geometric mean is undefined if ab ≤ 0.

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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)

1.

For compound interest, (a) total amount = P × (1 + r %) n if the interest is compounded yearly;

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12 n

 r%  (b) total amount = P × 1 +  12  

if the interest is compounded monthly.

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7.2 Applications of Percentage in Banking and Finance 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: P × (1 + r %) n × r % Q= (1 + r %) n − 1

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7.3 Time-series Graphs A. Introduction Definition 7.1: A time-series graph is a graph that shows a sequence of observations (or a group of data) over a period of time. All the data in a time-series graph are connected by broken lines (or curves). From the graph, we can observe the following features;

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1.

Trend --- it is the long-term movement of the 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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7.3 Time-series Graphs B. Moving Average of a Series of Data Definition 7.2: A moving average is the average value of a group of data over a fixed number of periods. The trend of the prices of a stock can be studied by the moving average method. For example, the closing prices of the stock of the Culture Company in the last ten transaction days are given below.

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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

Table 7.6

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

Content Fig. 7.10

From the graph, the price moves up and down.

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

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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

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

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7.4 Interpreting and Analysing Data Collected from Surveys 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.

Fig. 7.16 Fig. 7.15 Content

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.

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7.4 Interpreting and Analysing Data Collected from Surveys 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.

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Fig. 7.17

Fig. 7.18

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.

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7.4 Interpreting and Analysing Data Collected from Surveys Consider the sales (in bottle) of two types of soft drink in two weeks. Last week

This week

Soft drink A

800

900

Soft drink B

100

200

Fig. 7.19

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

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7.4 Interpreting and Analysing Data Collected from Surveys 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. If we consider the inflation rate, then the percentage increase in the prices will be more important than the increase in the actual prices. Suppose the price of a toy increases by 10% each year and its price is $100 at the first year.

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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

Table 7.18

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7.4 Interpreting and Analysing Data Collected from Surveys

Fig. 7.20 Content

Fig. 7.21

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.

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7.5 Determining the Relation between Two Variables 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, where m is the slope of the line and c is the y-intercept. 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

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

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7.5 Determining the Relation between Two Variables 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. For example, If y = 10 × 3 x , then log y = log(10 × 3 x )

= log 10 + (log 3) x = (log 3) x + 1 So , we say that log y and x obey a linear law. Content

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

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7.5 Determining the Relation between Two Variables Consider y =

20 , we have x

x

1

2

4

5

10

y

20

10

5

4

2 Table 7.28

whose graph is a curve as shown in Fig. 7.28. If we plot the graph of y against

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Fig. 7.28

1 , then we have x 5 10

x

1

2

4

1 x

1

0.5

0.25

0.2

0.1

y

20

10

5

4

2 Table 7.27

From the graph, we can observe that y and

1 obey a linear law. x

Fig. 7.29

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