Analytical Methods

Assignment 01

Dinuka Lakshan Denipitiya CSCT 2013383

[ ] CSCT2013382

Acknowledgement

First of all I would like to thank Mr.Isuru Withanage the lecturer of Unit: ANM003 Analytical Methods. Because he guide me to do this assignment properly.

I should thank the management of CSCT campus for organizing HND in Quantity Surveying for us to improve our knowledge and the environment facilitated with resources such as computer lab, library and other required facilities to do our work without any problems.

I would thank Prof. Chitra Wedikkara, with great pleasure that everything she has done for us, for our future.

Then I would like to specially thank for my parents for being by the side of me in my all difficulties, encourage me to achieve my target.

2

[ ] CSCT2013382

Table of Contents

Acknowledgement.......................................................................................................................................2

Introduction.................................................................................................................................................4

Task 01........................................................................................................................................................5

Task 02.......................................................................................................................................................10

Task 03.......................................................................................................................................................13

Task 04.......................................................................................................................................................19

Task 05.......................................................................................................................................................23

Conclusion.................................................................................................................................................27

References.................................................................................................................................................28

3

[ ] CSCT2013382

Introduction

This assignment is about “Analyze and solve problems using statistics and probability”. Statistics is a type of analysis involving the use of quantified models, representations and summaries for

given set of empirical data or real world observation.

As a person it is needed to have some sort of knowledge of statistics. Even if we are sellers, doctors, accountants, journalist we should have a somewhat knowledge about the statistics.

Statistics as a subjects, it can be applied to every fields. It helps to set out the final result of a data collection. Data representation of statics can be divided in to 2 main parts, called

diagrammatic & graphical representation of Data.

4

[ ] CSCT2013382

Task 01

Define “Statistics” and explain why and how it is useful in various fields. (P8)

Definition of statistics

The modern world is based on calculations more than anything else. Calculations of money, calculation of wages, calculation of taxes and all these calculations need to be assimilated, and utilized in making more calculations. Thus, a proper system of data usage involves collecting all the data and organizing it properly. Moreover, not everyone will be able to understand data in the form it is organized, which, in turn, requires proper interpretation of data. In fact, it is just the tip of the iceberg. Over the years, the importance of statistics has risen dramatically. Today, statistics is used in almost all the fields. In the following lines, we have given a note on the significance of statistics in the different fields. Statistics plays a vital role in every fields of human activity. Statistics has important role in determining the existing position of per capita income, unemployment, population growth rate, housing, schooling medical facilities etc…in a country. Now statistics holds a central position in almost every field like Industry, Commerce, Trade, Physics, Chemistry, Economics, Mathematics, Biology, Botany, Psychology, Astronomy etc…, so application of statistics is very wide. Now we discuss some important fields in which statistics is commonly applied

5

[ ] CSCT2013382

Some consider statistics a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data, while others consider it a branch of mathematics concerned with collecting and interpreting data. Because of its empirical roots and its focus on applications, statistics is usually considered a distinct mathematical science rather than a branch of mathematics.

Much of statistics is non-mathematical: ensuring that data collection is undertaken in a way that produces valid conclusions; coding and archiving data so that information is retained and made useful for international comparisons of official statistics; reporting of results and summarized data (tables and graphs) in ways comprehensible to those who must use them; implementing procedures that ensure the privacy of census information.

Statisticians improve data quality by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting the use of data and statistical models. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical consultants can help organizations and companies that don't have in-house expertise relevant to their particular questions.

Statistical methods can summarize or describe a collection of data. This is called descriptive statistics. This is particularly useful in communicating the results of experiments and research. In addition, data patterns may be modeled in a way that accounts for randomness and uncertainty in the observations.

Significance of Statistics

"Significance level" is a misleading term that many researchers do not fully understand. This article may help you understand the concept of statistical significance and the meaning of the numbers produced by The Survey System.

This article is presented in two parts. The first part simplifies the concept of statistical significance as much as possible; so that non-technical readers can use the concept to help make decisions based on their data. The second part provides more technical readers with a fuller discussion of the exact meaning of statistical significance numbers.

In normal English, "significant" means important, while in Statistics "significant" means probably true (not due to chance). A research finding may be true without being important. When statisticians say a result is "highly significant" they mean it is very probably true. They do not (necessarily) mean it is highly important.

Take a look at the table below. The chi (pronounced kie like pie) squares at the bottom of the table show two rows of numbers. The top row numbers of 0.07 and 24.4 are the chi square statistics themselves. The meaning of these statistics may be ignored for the purposes of this article. The second row contains values .795 and .001. These are the significance levels and are explained following the table.

6

[ ] CSCT2013382

Significance levels show you how likely a result is due to chance. The most common level, used to mean something is good enough to be believed, is .95. This means that the finding has a 95% chance of being true. However, this value is also used in a misleading way. No statistical package will show you "95%" or ".95" to indicate this level.

Instead it will show you ".05," meaning that the finding has a five percent (.05) chance of not being true, which is the converse of a 95% chance of being true. To find the significance level, subtract the number shown from one. For example, a value of ".01" means that there is a 99% (1-.01=.99) chance of it being true. In this table, there is probably no difference in purchases of gasoline X by people in the city center and the suburbs, because the probability is .795 (i.e., there is only a 20.5% chance that the difference is true). In contrast the high significance level for type of vehicle (.001 or 99.9%) indicates there is almost certainly a true difference in purchases of Brand X by owners of different vehicles in the population from which the sample was drawn.

The Field Of Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία (oikonomia, "management of a household, administration") from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)". Political economy was the earlier name for the subject, but economists in the late 19th century suggested "economics" as a shorter term for "economic science" that also avoided a narrow political-interest connotation and as similar in form to "mathematics", "ethics", and so forth.

A focus of the subject is how economic agents behave or interact and how economies work. Consistent with this, a primary textbook distinction is between microeconomics and macroeconomics. Microeconomics examines the behavior of basic elements in the economy, including individual agents (such as households and firms or as buyers and sellers) and markets, and their interactions. Macroeconomics analyzes the entire economy and issues affecting it, including unemployment, inflation, economic growth, and monetary and fiscal policy.

Other broad distinctions include those between positive economics (describing "what is") and normative economics (advocating "what ought to be"); between economic theory and applied economics; between rational and behavioral economics; and between mainstream economics (more "orthodox" and dealing with the "rationality-individualism-equilibrium nexus") and heterodox economics (more "radical" and dealing with the "institutions-history-social structure nexus").

Economic analysis may be applied throughout society, as in business, finance, health care, and government, but also to such diverse subjects as crime, education, the family, law, politics, religion, social institutions, war, and science.] At the turn of the 21st century, the expanding domain of economics in the social sciences has been described as economic imperialism

7

[ ] CSCT2013382

The Field of Natural & Social Sciences

For any of the natural sciences, experimentation is very important. Now, for the precise measurement of results, their tabulation, i.e. recording and comparison, is required, which is nothing but a form of statistics. Similarly, in social sciences scenario, it is very important to consider all the data related to the subjects in observation. Since social science spectrums includes a great number of subjects, mass data collection and assimilation is inevitable. This can be a really tedious task, if not done using proper statistical channels.

The Field Of Astronomy

Astronomy is an ancient art and one of the most primitive too. Star gazing may be one of the easiest things to do in the night, but capturing exact distances in the space and predicting celestial events can be next to impossible, if a statistical approach is not used. Astronomers have been using statistical methods, like method of least squares, for a long time for calculation of celestial distances and comparing objects. Astronomy is one of the oldest branches of statistical study; it deals with the measurement of distance, sizes, masses and densities of heavenly bodies by means of observations. During these measurements errors are unavoidable so most probable measurements are founded by using statistical methods.

The Banking Sector

Banking sector is very important for general public. This sector, too, uses statistical data utilization approach for its functioning. The banking sector works on the money exchange system. They collect money from depositors and give money to people who want to borrow. All the while banks earn interest, a part of which they offer to the depositors. The banks calculate through statistical approach, which is they calculate the number of people who will appear each day as depositors or withdrawers and plans their actions similarly. If the calculation goes wrong, chances of the business collapsing and public money being at jeopardy is high. Statistics play an important role in banking. The banks make use of statistics for a number of purposes.

The Field Of Business

In business structures, there may be nothing more important than statistical data and vice versa. Every business collects data about its consumers and their needs. This data is then assimilated and interpreted in order to form the base of their future plans and goals. Only on the basis of accurate data collection and assimilation can accurate projections for the future of the business be derived. Everything, right from expansion plans, disinvestment moves and new business and product proposals depend on such statistical calculations. Any small mistake here can spell doom for the business. Statistics play an important role in business.

8

[ ] CSCT2013382

Mathematics

Statistical plays a central role in almost all natural and social sciences. The methods of natural sciences are most reliable but conclusions draw from them are only probable, because they are based on incomplete evidence. Statistical helps in describing these measurements more precisely. Statistics is branch of applied mathematics.

State Management

Statistics is essential for a country. Different policies of the government are based on statistics. Statistical data are now widely used in taking all administrative decisions. Suppose if the government wants to revise the pay scales of employees in view of an increase in the living cost, statistical methods will be used to determine the rise in the cost of living.

9

[ ] CSCT2013382

Task 02

Select at least 2 problems/situations relevant to your field and explain how you use statistical methods to solve, analyze these problems/situations. (P8)

Statistics released by the Health and Safety Executive for 2005/06 show the rate of fatal and major injuries in the construction industry is continuing to fall. The Health and Safety Statistics 2005/06 report shows:

1. The rate of reported fatal injuries (to all workers, including the self-employed) in construction between 1999/00 and 2005/06 has reduced by around 36%;

2. The rate of reported major injuries to employees in construction between 1999/00 and 2005/06 has reduced by around 22%;

10

[ ] CSCT2013382

Construction injury rates since 1999/2000

There has also been good progress towards the all industry Revitalising targets of 2000, although the industry is still falling short of its own ambitious targets set at the 2001 Construction Summit.

For the first time the proportion of major injuries due to falls from height has been overtaken by those resulting from slips, trips and falls on the same level. Injuries due to slips and trips and manual handling have risen steadily over the past few years while those from falls from height and being struck by an object have been reduced substantially.

The Health and Safety Statistics 2005/06 report shows:

There were 981 major injuries in construction caused by slips, trips or falls on the same level;

There were 917 major injuries in construction caused by falls from height;

There were 577 major injuries in construction while handling, lifting or carrying;

11

[ ] CSCT2013382

There were 572 major injuries in construction caused by struck by moving, including flying/falling

object.

Detail of causes for fatalities can be found at

Commenting on the statistics, HSE's Chief Inspector of Construction, Stephen Williams said: "My congratulations go out to the industry on the progress it has made so far. I am particularly pleased to see the encouraging signs that it has taken ownership of its health and safety performance and worked hard to achieve the lowest incidence rates ever for fatal, major injuries and over three day injuries."

"There has been very good progress in reducing injuries due to falls and being hit by objects. This, coupled with the decrease in fatal injuries announced earlier this year (down to 59 from last year's total of 69) shows that the hard work by many in the industry is paying off. However, slips and trips and manual handling injuries are increasing. It may be that they receive less attention because they rarely result in fatalities; nevertheless more needs to be done to prevent these types of injury. In particular, sites need to be kept tidy, which is hard work and requires persistence.

"We need to focus more on these areas, while maintaining the progress on falls from height. If the industry had made the same progress in these other areas it would have been close to meeting its own challenging targets."

12

[ ] CSCT2013382

Task 03

(A) State and explain one diagrammatic and one graphical representation used to illustrate/ interpret the data. Which is more suitable to interpret a discrete set of data?

Diagrammatic representation of data

Data can be represented diagrammatically by pictograms, column graphs, horizontal bar charts, pie charts, line graphs etc. In bar chart, numerical data is represented by bars, which may be vertical (column graph) or horizontal (in horizontal bar graph). These bars must be of uniform width and equally spaced. The labels on horizontal and vertical axes should be clear. If subcomponents of items are given, then the bars may be subdivided. If two or more different values are given for each item, and comparison between these values is required, then multiple bar charts are better. Here bars for an item are placed side-by-side.

Bar chart

In a bar chart, the height of the bar shows the frequency of the result. As the height of bar represents frequency, label the vertical axis 'Frequency'.

13

[ ] CSCT2013382

Graphical representation of data

Visualization techniques are ways of creating and manipulating graphical representations of data. We use these representations in order to gain better insight and understanding of the problem we are studying - pictures can convey an overall message much better than a list of numbers. In this section we describe some graphical presentations of data.

Visualization techniques are ways of creating and manipulating graphical representations of data. We use these representations in order to gain better insight and understanding of the problem we are studying - pictures can convey an overall message much better than a list of numbers.

In this section we describe some graphical presentations of data.

Line or Dot Plots

Line plots are graphical representations of numerical data. A line plot is a number line with x’s placed above septic numbers to show their frequency.

By the frequency of a number we mean the number of occurrence of that number. Line plots are used to represent one group of data with fewer than 50 values.

Example 30.1

Suppose thirty people live in an apartment building. These are the following ages:

58 30 37 36 34 49 35 40 47 47

39 54 47 48 54 50 35 40 38 47

48 34 40 46 49 47 35 48 47 46

14

[ ] CSCT2013382

Simple Bar Diagram

Presentation of statistical data by means of rectangular bars with height proportional to magnitude of the items is called bar diagram. Usually, vertical bars are drawn, but, horizontal bars can as will be of use.

Simple bar diagram is drawn when items have to be compared with regard to a single characteristic. Here, the items are represented by rectangular bars of equal width and height proportional to their magnitude. The bars are drawn on a common base line, with equal distance between consecutive bars. The bars may be shaded.

Histogram

In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson. A histogram consists of tabular frequencies, shown as adjacent rectangles, erected over discrete intervals, with an area equal to the frequency of the observations in the interval.

15

[ ] CSCT2013382

The most suitable representation to interpret a discrete set of data is the graphical method. It is because it gives all the necessary information and the details so that at once instance the viewer can spot the correct details out of it.

Example

For the following frequency distribution of height of students in a tourism section draw the histogram.

Height 140-145 145-150 150-155 155-160 160-165 165-170 170-175 Total

No of Student

4 10 18 20 19 6 3 80

16

[ ] CSCT2013382

(B) The responses of the 25 people who were asked about the type of house/building they lived in.

Where F – Flat, T – Terraced house, SD – Semidetached house, D – Detached house B – Bungalow.

F T SD D B6 5 7 4 3

Suitable diagrammatic representation to illustrate the accommodation

Histogram

17

[ ] CSCT2013382

Pie chart

123456

18

[ ] CSCT2013382

Task 04

During a construction of a building the builder has questioned the consistency and quality if concrete block. As a first check one of the technicians for the builder has collected measurements in the masses of 150 blocks as follows;

Mass (kg) No of blocks (f) x fx

8.0 - 8.4 2 8.2 16.4

8.5 -8.9 7 8.7 60.9

9.0 - 9.4 15 9.2 138.0

9.5 - 9.9 20 9.7 194.0

10.0 - 10.4 27 10.2 275.4

10.5 - 10.9 29 10.7 310.3

11.0 - 11.4 25 11.2 280.0

11.5 - 11.9 12 11.7 140.0

12.0 -12.4 10 12.2 122.0

12.5 - 12.9 3 12.7 38.1

150 1575.5

19

[ ] CSCT2013382

20

[ ] CSCT2013382

ii.

(a)

Mean (M)= ∑fx/∑f

= 1575.5/150

=10.503

(b)

The median,

A + B= 29 1

2 + 7 + 15 + 20 + 27 + A = B + 25 + 12 + 10 + 3

71 + A = B + 52

A-B = -19 2

1 + 2

2A = 10

A = 5

The width of A = 5/29 * class interval

= 5/29* 0.5 = 0.0862

Median = 10.5 + 0.0862

= 10.5862

(c)

The range = 12.9 - 7.95 = 4.95(5 kg)

21

[ ] CSCT2013382

(d)

The variance =

[(8.2-10.5)2 + (8.7-10.5)2 + (9.2-10.5)2 +(9.7-10.5)2 +(10.2-10.5)2 +(10.7-10.5)2 +(11.2-

10.5)2 +( 11.7-10.5)2 + ( 12.2-10.5)2 +( 12.7-10.5)2 ] / 10

= 20.59/10

=2.059.

iii.

According to my idea to builders that should take high weights blocks take to foundation. If more than 10.5kg it’s suitable to use for foundation. If it’s less than 10.5kg it’s suitable to use for walls.

22

[ ] CSCT2013382

Task 05A construction firm has 30 designs of houses for the clients, which can be considered under three groups Luxury, Semi-Luxury and Moderate. The details of these designs according to

Roof (tile / sheets)

Floor area ( > 1500 sq.ft / ≤ 1500 sq.ft )

Finished walls (smooth / rough) are given in table 2.

Description of variables

X1 – Type of the house: 1 – Luxury

2 – Semi Luxury

3 – Moderate

X2 – Roof : 1 – Tile

2 – Sheets

X3 – Floor Area : 1 - > 1500 sq.ft

2 - ≤ 1500 sq.ft

X4 – Finished Walls : 1 – Smooth

23

[ ] CSCT2013382

Plan X1 X2 X3 X4

1 1 2 1 12 1 2 2 13 2 2 2 14 2 2 2 25 2 1 1 16 3 1 1 27 2 2 1 28 1 1 1 29 1 2 1 1

10 2 1 1 111 3 1 1 112 3 2 1 213 1 1 1 114 2 1 2 215 3 1 2 116 1 1 2 217 2 2 1 118 2 2 1 119 3 2 2 120 3 1 2 221 2 2 2 222 1 1 1 123 3 2 1 124 2 1 1 125 1 1 2 226 2 2 1 127 2 1 1 128 2 1 2 229 1 1 2 130 2 2 2 1

24

[ ] CSCT2013382

i. (a). A luxury house- 9/30 (b). A semi luxury house- 14/30 (c). Moderate house- 7/30 (d). A luxury house or a house with floor area > 1500 sq.ft- 21/30 (e). A house with a tile roof given that it is a semi luxury house- 6/30

ii.

This combination is not there in the designs. Therefore the probability of selecting a moderate house with a sheet roof, rough finish and with the floor area <1500sq.ft is 0

iii.

(a)

1. Event 1: luxury houses with a floor area which is < or equal to 1500 and rough walls

Event 2: a house with a sheet roof, with rough wall and a floor area less than 1500 square feet

{x1=1 , x2=2 , x3=2 , x4=2 }

2. Event 1: luxury houses with rough walls, more than 1500 in area.

Event 2: a house with a sheet roof, rough walls and more than 1500square feet in area.

{x1=1, x2=2 , x3=1 , x4=2 }

3. Event 1: a moderate house, with sheet roof and with an area of less than 1500 square feet

Event 2: a moderate house with sheet roof and with rough walls.

{x1=3 , x2=2 , x3=2 , x4=2 }

25

[ ] CSCT2013382

a. Three mutually exclusive events,

Sheet roof, rough wall ,floor area less than 1500 square feet. Mutually Exclusive Events is a situation where the occurrence of one event is not influenced or caused by another event.

b. Two independent events Described here

Luxury house with rough wall and floor area more than 1500 square feet. Independent events are events where the outcome of one event does not affect the outcome of the other events.

iv.

Ceiling type1. Wooden ceiling2. Asbestos sheets

Floor type3. Ceramic4. Marble5. Granite6. Teraso

Include doors and windows

7. Yes

8. No

26

[ ] CSCT2013382

Conclusion

As we have finished the assignment we got to know that we can use statistics in everywhere. Here in construction, civil engineering and quantity surveying there’s always statistics and

probability been used and of course mathematics is very widely used.

Data or a collection of data can be represented in diagrammatically and also graphically to interpret the behavior in a meaningful way. Most widely used here is the use of the histogram

and also frequency polygon.

As the conclusion, again I would like to say statistics are important to all the people who live in the world. Because, we need to know somewhat of statistics to survive the life from getting stuck. Especially as a professional in construction industry, sometimes we have to work with

data sets. Therefore as a professional, we should have a fluent knowledge in statistics.

Probability is also a very frequently used mathematical operation for various calculations. It cannot be used where humans are involved such as in a cricket match because we cannot

predict the winning team, where as we can predict the most number of occurrences of masses of bricks to choose a correct mass range of bricks for construction

27

[ ] CSCT2013382

References

http://www.emathzone.com/tutorials/basic-statistics/importance-of- statisticsin-different-fields.html

http://quarknet.fnal.gov/toolkits/new/histograms.html

28