Introduction to Statistics PDF

AAU Introduction to statistics Instructor Anteneh T. M.T 2005 WWW.AAU.EDU.ET  1. INTRODUCTION Definition and classifications of statistics Definition: Statistics: we can define it i...

AAU Introduction to statistics Instructor Anteneh T. M.T 2005 WWW.AAU.EDU.ET  1. INTRODUCTION Definition and classifications of statistics Definition: Statistics: we can define it in two senses a. In the plural sense : statistics are the raw data themselves , like statistics of births, statistics of deaths, statistics of students, statistics of imports and exports, etc. b. In the singular sense statistics is the subject that deals with the collection, organization, presentation, analysis and interpretation of numerical data Classifications: Depending on how data can be used statistics is some times divided in to two main areas or branches. 1. Descriptive Statistics: is concerned with summary calculations, graphs, charts and tables. 2. Inferential Statistics: is a method used to generalize from a sample to a population. For example, the average income of all families (the population) in Ethiopia can be estimated from figures obtained from a few hundred (the sample) families. It is important because statistical data usually arises from sample. Statistical techniques based on probability theory are required. Stages in statistical investigation. There are five stages or steps in any statistical investigation. 1. Collection of data: the process of measuring, gathering, assembling the raw data up on which the statistical investigation is to be based. ƒ Data can be collected in a variety of ways; one of the most common methods is through the use of survey. Survey can also be done in different methods, three of the most common methods are: Telephone survey Mailed questionnaire Personal interview. Exercise: discuss the advantage and disadvantage of the above three methods with respect to each other. 2. organization of data: Summarization of data in some meaningful way, e.g table form 3. Presentation of the data: The process of re-organization, classification, compilation, and summarization of data to present it in a meaningful form. 4. Analysis of data: The process of extracting relevant information from the summarized data, mainly through the use of elementary mathematical operation. 1 5. Inference of data: The interpretation and further observation of the various statistical measures through the analysis of the data by implementing those methods by which conclusions are formed and inferences made. Statistical techniques based on probability theory are required. Definitions of some terms 1. A (statistical) population: is the complete set of possible measurements for which inferences are to be made. The population represents the target of an investigation, and the objective of the investigation is to draw conclusions about the population hence we sometimes call it target population. Examples 9 Population of trees under specified climatic conditions 9 Population of animals fed a certain type of diet 9 Population of farms having a certain type of natural fertility 9 Population of house holds, etc ƒ The population could be finite or infinite (an imaginary collection of units) ƒ There are two ways of investigation: Census and sample survey. 2. Census: a complete enumeration of the population. But in most real problems it can not be realized, hence we take sample. 3. Sample: A sample from a population is the set of measurements that are actually collected in the course of an investigation. It should be selected using some pre-defined sampling technique in such a way that they represent the population very well. Examples: 9 Monthly production data of a certain factory in the past 10 years. 9 Small portion of a finite population. In practice, we don’t conduct census, instead we conduct sample survey 4. Parameter: Characteristic or measure obtained from a population. 5. Statistic: Characteristic or measure obtained from a sample. 6. Sampling: The process or method of sample selection from the population. 7. Sample size: The number of elements or observation to be included in the sample. 8. Variable: It is an item of interest that can take on many different numerical values. 2 Types of Variables or Data 1. Qualitative Variables are nonnumeric variables and can't be measured. Examples: gender, religious affiliation, and state of birth. 2. Quantitative Variables are numerical variables and can be measured. Examples include balance in checking account, number of children in family. Note that quantitative variables are either discrete (which can assume only certain values, and there are usually "gaps" between the values, such as the number of bedrooms in your house) or continuous (which can assume any value within a specific range, such as the air pressure in a tire.) Applications, Uses and Limitations of statistics. Applications of statistics: In almost all fields of human endeavor. Almost all human beings in their daily life are subjected to obtaining numerical facts e.g. abut price. Applicable in some process e.g. invention of certain drugs, extent of environmental pollution. In industries especially in quality control area. Uses of statistics: The main function of statistics is to enlarge our knowledge of complex phenomena. The following are some uses of statistics: 1. It presents facts in a definite and precise form. 2. Data reduction. 3. Measuring the magnitude of variations in data. 4. Furnishes a technique of comparison 5. Estimating unknown population characteristics. 6. Testing and formulating of hypothesis. 7. Studying the relationship between two or more variable. 8. Forecasting future events. Limitations of statistics As a science statistics has its own limitations. The following are some of the limitations: Deals with only quantitative information. Deals with only aggregate of facts and not with individual data items. Statistical data are only approximately and not mathematical correct. Statistics can be easily misused and therefore should be used be experts. 3  Scales of measurement Proper knowledge about the nature and type of data to be dealt with is essential in order to specify and apply the proper statistical method for their analysis and inferences. Measurement scale refers to the property of value assigned to the data based on the properties of order, distance and fixed zero. In mathematical terms measurement is a functional mapping from the set of objects {Oi} to the set of real numbers {M(Oi)}. The goal of measurement systems is to structure the rule for assigning numbers to objects in such a way that the relationship between the objects is preserved in the numbers assigned to the objects. The different kinds of relationships preserved are called properties of the measurement system. Order The property of order exists when an object that has more of the attribute than another object, is given a bigger number by the rule system. This relationship must hold for all objects in the "real world". The property of ORDER exists When for all i, j if Oi > Oj, then M(Oi) > M(Oj). 4 Distance The property of distance is concerned with the relationship of differences between objects. If a measurement system possesses the property of distance it means that the unit of measurement means the same thing throughout the scale of numbers. That is, an inch is an inch, no matters were it falls - immediately ahead or a mile downs the road. More precisely, an equal difference between two numbers reflects an equal difference in the "real world" between the objects that were assigned the numbers. In order to define the property of distance in the mathematical notation, four objects are required: Oi, Oj, Ok, and Ol. The difference between objects is represented by the "-" sign; Oi - Oj refers to the actual "real world" difference between object i and object j, while M(Oi) - M(Oj) refers to differences between numbers. The property of DISTANCE exists, for all i, j, k, l If Oi-Oj ≥ Ok- Ol then M(Oi)-M(Oj) ≥ M(Ok)-M( Ol ). Fixed Zero A measurement system possesses a rational zero (fixed zero) if an object that has none of the attribute in question is assigned the number zero by the system of rules. The object does not need to really exist in the "real world", as it is somewhat difficult to visualize a "man with no height". The requirement for a rational zero is this: if objects with none of the attribute did exist would they be given the value zero. Defining O0 as the object with none of the attribute in question, the definition of a rational zero becomes: The property of FIXED ZERO exists if M(O0) = 0. The property of fixed zero is necessary for ratios between numbers to be meaningful. SCALE TYPES Measurement is the assignment of numbers to objects or events in a systematic fashion. Four levels of measurement scales are commonly distinguished: nominal, ordinal, interval, and ratio and each possessed different properties of measurement systems. 5  Nominal Scales Nominal scales are measurement systems that possess none of the three properties stated above. Level of measurement which classifies data into mutually exclusive, all inclusive categories in which no order or ranking can be imposed on the data. No arithmetic and relational operation can be applied. Examples: o Political party preference (Republican, Democrat, or Other,) o Sex (Male or Female.) o Marital status(married, single, widow, divorce) o Country code o Regional differentiation of Ethiopia. Ordinal Scales Ordinal Scales are measurement systems that possess the property of order, but not the property of distance. The property of fixed zero is not important if the property of distance is not satisfied. Level of measurement which classifies data into categories that can be ranked. Differences between the ranks do not exist. Arithmetic operations are not applicable but relational operations are applicable. Ordering is the sole property of ordinal scale. Examples: o Letter grades (A, B, C, D, F). o Rating scales (Excellent, Very good, Good, Fair, poor). o Military status. Interval Scales Interval scales are measurement systems that possess the properties of Order and distance, but not the property of fixed zero. 6  Level of measurement which classifies data that can be ranked and differences are meaningful. However, there is no meaningful zero, so ratios are meaningless. All arithmetic operations except division are applicable. Relational operations are also possible. Examples: o IQ o Temperature in oF. Ratio Scales Ratio scales are measurement systems that possess all three properties: order, distance, and fixed zero. The added power of a fixed zero allows ratios of numbers to be meaningfully interpreted; i.e. the ratio of Bekele's height to Martha's height is 1.32, whereas this is not possible with interval scales. Level of measurement which classifies data that can be ranked, differences are meaningful, and there is a true zero. True ratios exist between the different units of measure. All arithmetic and relational operations are applicable. Examples: o Weight o Height o Number of students o Age The following present a list of different attributes and rules for assigning numbers to objects. Try to classify the different measurement systems into one of the four types of scales. (Exercise) 1. Your checking account number as a name for your account. 2. Your checking account balance as a measure of the amount of money you have in that account. 3. The order in which you were eliminated in a spelling bee as a measure of your spelling ability. 7  4. Your score on the first statistics test as a measure of your knowledge of statistics. 5. Your score on an individual intelligence test as a measure of your intelligence. 6. The distance around your forehead measured with a tape measure as a measure of your intelligence. 7. A response to the statement "Abortion is a woman's right" where "Strongly Disagree" = 1, "Disagree" = 2, "No Opinion" = 3, "Agree" = 4, and "Strongly Agree" = 5, as a measure of attitude toward abortion. 8. Times for swimmers to complete a 50-meter race 9. Months of the year Meskerm, Tikimit… 10. Socioeconomic status of a family when classified as low, middle and upper classes. 11. Blood type of individuals, A, B, AB and O. 12. Pollen counts provided as numbers between 1 and 10 where 1 implies there is almost no pollen and 10 that it is rampant, but for which the values do not represent an actual counts of grains of pollen. 13. Regions numbers of Ethiopia (1, 2, 3 etc.) 14. The number of students in a college; 15. the net wages of a group of workers; 16. the height of the men in the same town; Introduction to methods of data collection There are two sources of data: 1. Primary Data Data measured or collect by the investigator or the user directly from the source. Two activities involved: planning and measuring. a) Planning: ƒ Identify source and elements of the data. ƒ Decide whether to consider sample or census. ƒ If sampling is preferred, decide on sample size, selection method,… etc ƒ Decide measurement procedure. ƒ Set up the necessary organizational structure. b) Measuring: there are different options. ƒ Focus Group ƒ Telephone Interview ƒ Mail Questionnaires 8  ƒ Door-to-Door Survey ƒ Mall Intercept ƒ New Product Registration ƒ Personal Interview and ƒ Experiments are some of the sources for collecting the primary data. 2. Secondary Data Data gathered or compiled from published and unpublished sources or files. When our source is secondary data check that: ƒ The type and objective of the situations. ƒ The purpose for which the data are collected and compatible with the present problem. ƒ The nature and classification of data is appropriate to our problem. ƒ There are no biases and misreporting in the published data. Note: Data which are primary for one may be secondary for the other. 9  2. METHODS OF DATA PRESNTATION -Having collected and edited the data, the next important step is to organize it. That is to present it in a readily comprehensible condensed form that aids in order to draw inferences from it. It is also necessary that the like be separated from the unlike ones. - The presentation of data is broadly classified in to the following two categories: Tabular presentation Diagrammatic and Graphic presentation. -The process of arranging data in to classes or categories according to similarities technically is called classification. -Classification is a preliminary and it prepares the ground for proper presentation of data. Definitions: Raw data: recorded information in its original collected form, whether it be counts or measurements, is referred to as raw data. Frequency: is the number of values in a specific class of the distribution. Frequency distribution: is the organization of raw data in table form using classes and frequencies. -There are three basic types of frequency distributions ƒ Categorical frequency distribution ƒ Ungrouped frequency distribution ƒ Grouped frequency distribution -There are specific procedures for constructing each type. 1) Categorical frequency Distribution: -Used for data that can be place in specific categories such as nominal, or ordinal. e.g. marital status. 10 Example: a social worker collected the following data on marital status for 25 persons.(M=married, S=single, W=widowed, D=divorced) M S D W D S S M M M W D S M M W D D S S S W W D D Solution: Since the data are categorical, discrete classes can be used. There are four types of marital status M, S, D, and W. These types will be used as class for the distribution. We follow procedure to construct the frequency distribution. Step 1: Make a table as shown. Class Tally Frequency Percent (1) (2) (3) (4) M S D W Step 2: Tally the data and place the result in column (2). Step 3: Count the tally and place the result in column (3). Step 4: Find the percentages of values in each class by using; f %= * 100 Where f= frequency of the class, n=total number of value. n -Percentages are not normally a part of frequency distribution but they can be added since they are used in certain types diagrammatic such as pie charts. Step 5: Find the total for column (3) and (4). Combing all the steps one can construct the following frequency distribution. 11  Class Tally Frequency Percent (1) (2) (3) (4) M //// 5 20 S //// // 7 28 D //// // 7 28 W //// / 6 24 2) Ungrouped frequency Distribution: -Is a table of all the potential raw score values that could possible occur in the data along with the number of times each actually occurred. -Is often constructed for small set or data on discrete variable. Constructing ungrouped frequency distribution: First find the smallest and largest raw score in the collected data. Arrange the data in order of magnitude and count the frequency. To facilitate counting one may include a column of tallies. Example: The following data represent the mark of 20 students. 80 76 90 85 80 70 60 62 70 85 65 60 63 74 75 76 70 70 80 85 Construct a frequency distribution, which is ungrouped. Solution: Step 1: Find the range, Range=Max-Min=90-60=30. Step 2: Make a table as shown Step 3: Tally the data. 12  Step 4: Compute the frequency. Mark Tally Frequency 60 // 2 62 / 1 63 / 1 65 / 1 70 //// 4 74 / 1 75 // 2 76 / 1 80 /// 3 85 /// 3 90 / 1 -Each individual value is presented separately, that is why it is named ungrouped frequency distribution. 3) Grouped frequency Distribution: -When the range of the data is large, the data must be grouped in to classes that are more than one unit in width. Definitions: Grouped Frequency Distribution: a frequency distribution when several numbers are grouped in one class. Class limits: Separates one class in a grouped frequency distribution from another. The limits could actually appear in the data and have gaps between the upper limits of one class and lower limit of the next. Units of measurement (U): the distance between two possible consecutive measures. It is usually taken as 1, 0.1, 0.01, 0.001, -----. Class boundaries: Separates one class in a grouped frequency distribution from another. The boundaries have one more decimal places than the row data and therefore do not appear in the data. There is no gap between the upper boundary of one class and lower boundary of the next class. The lower class boundary is found by subtracting U/2 from the corresponding lower class limit and the upper class boundary is found by adding U/2 to the corresponding upper class limit. 13  Class width: the difference between the upper and lower class boundaries of any class. It is also the difference between the lower limits of any two consecutive classes or the difference between any two consecutive class marks. Class mark (Mid points): it is the average of the lower and upper class limits or the average of upper and lower class boundary. Cumulative frequency: is the number of observations less than/more than or equal to a specific value. Cumulative frequency above: it is the total frequency of all values greater than or equal to the lower class boundary of a given class. Cumulative frequency blow: it is the total frequency of all values less than or equal to the upper class boundary of a given class. Cumulative Frequency Distribution (CFD): it is the tabular arrangement of class interval together with their corresponding cumulative frequencies. It can be more than or less than type, depending on the type of cumulative frequency used. Relative frequency (rf): it is the frequency divided by the total frequency. Relative cumulative frequency (rcf): it is the cumulative frequency divided by the total frequency. Guidelines for classes 1. There should be between 5 and 20 classes. 2. The classes must be mutually exclusive. This means that no data value can fall into two different classes 3. The classes must be all inclusive or exhaustive. This means that all data values must be included. 4. The classes must be continuous. There are no gaps in a frequency distribution. 5. The classes must be equal in width. The exception here is the first or last class. It is possible to have an "below..." or "... and above" class. This is often used with ages. Steps for constructing Grouped frequency Distribution 1. Find the largest and smallest values 2. Compute the Range(R) = Maximum - Minimum 3. Select the number of classes desired, usually between 5 and 20 or use Sturges rule k = 1 + 3.32 log n where k is number of classes desired and n is total number of observation. 4. Find the class width by dividing the range by the number of classes and R rounding up, not off. w =. k 14  5. Pick a suitable starting point less than or equal to the minimum value. The starting point is called the lower limit of the first class. Continue to add the class width to this lower limit to get the rest of the lower limits. 6. To find the upper limit of the first class, subtract U from the lower limit of the second class. Then continue to add the class width to this upper limit to find the rest of the upper limits. 7. Find the boundaries by subtracting U/2 units from the lower limits and adding U/2 units from the upper limits. The boundaries are also half-way between the upper limit of one class and the lower limit of the next class. !may not be necessary to find the boundaries. 8. Tally the data. 9. Find the frequencies. 10. Find the cumulative frequencies. Depending on what you're trying to accomplish, it may not be necessary to find the cumulative frequencies. 11. If necessary, find the relative frequencies and/or relative cumulative frequencies Example*: Construct a frequency distribution for the following data. 11 29 6 33 14 31 22 27 19 20 18 17 22 38 23 21 26 34 39 27 Solutions: Step 1: Find the highest and the lowest value H=39, L=6 Step 2: Find the range; R=H-L=39-6=33 Step 3: Select the number of classes desired using Sturges formula; k = 1 + 3.32 log n =1+3.32log (20) =5.32=6(rounding up) Step 4: Find the class width; w=R/k=33/6=5.5=6 (rounding up) Step 5: Select the starting point, let it be the minimum observation. ƒ 6, 12, 18, 24, 30, 36 are the lower class limits. 15 Step 6: Find the upper class limit; e.g. the first upper class=12-U=12-1=11 ƒ 11, 17, 23, 29, 35, 41 are the upper class limits. So combining step 5 and step 6, one can construct the following classes. Class limits 6 – 11 12 – 17 18 – 23 24 – 29 30 – 35 36 – 41 Step 7: Find the class boundaries; E.g. for class 1 Lower class boundary=6-U/2=5.5 Upper class boundary =11+U/2=11.5 Then continue adding w on both boundaries to obtain the rest boundaries. By doing so one can obtain the following classes. Class boundary 5.5 – 11.5 11.5 – 17.5 17.5 – 23.5 23.5 – 29.5 29.5 – 35.5 35.5 – 41.5 Step 8: tally the data. Step 9: Write the numeric values for the tallies in the frequency column. Step 10: Find cumulative frequency. Step 11: Find relative frequency or/and relative cumulative frequency. The complete frequency distribution follows: 16  Class Class Class Tally Freq. Cf (less Cf (more rf. rcf (less limit boundary Mark than than than type type) type) 6 – 11 5.5 – 11.5 8.5 // 2 2 20 0.10 0.10 12 – 17 11.5 – 17.5 14.5 // 2 4 18 0.10 0.20 18 – 23 17.5 – 23.5 20.5 ////// 7 11 16 0.35 0.55 24 – 29 23.5 – 29.5 26.5 //// 4 15 9 0.20 0.75 30 – 35 29.5 – 35.5 32.5 /// 3 18 5 0.15 0.90 36 – 41 35.5 – 41.5 38.5 // 2 20 2 0.10 1.00 Diagrammatic and Graphic presentation of data. -These are techniques for presenting data in visual displays using geometric and pictures. Importance: They have greater attraction. They facilitate comparison. They are easily understandable. -Diagrams are appropriate for presenting discrete data. -The three most commonly used diagrammatic presentation for discrete as well as qualitative data are: Pie charts Pictogram Bar charts Pie chart A pie chart is a circle that is divided in to sections or wedges according to the percentage of frequencies in each category of the distribution. The angle of the sector is obtained using: 17  Value of the part Angle of sector = * 100 the whole quantity Example: Draw a suitable diagram to represent the following population in a town. Men Women Girls Boys 2500 2000 4000 1500 Solutions: Step 1: Find the percentage. Step 2: Find the number of degrees for each class. Step 3: Using a protractor and compass, graph each section and write its name corresponding percentage. Class Frequency Percent Degree Men 2500 25 90 Women 2000 20 72 Girls 4000 40 144 Boys 1500 15 54 CLASS Boys Men Girls Women 18  Pictogram -In these diagram, we represent data by means of some picture symbols. We decide abut a suitable picture to represent a definite number of units in which the variable is measured. Example: draw a pictogram to represent the following population of a town. Year 1989 1990 1991 1992 Population 2000 3000 5000 7000 Bar Charts: - A set of bars (thick lines or narrow rectangles) representing some magnitude over time space. - They are useful for comparing aggregate over time space. - Bars can be drawn either vertically or horizontally. - There are different types of bar charts. The most common being : Simple bar chart Deviation o0r two way bar chart Broken bar chart Component or sub divided bar chart. Multiple bar charts. Simple Bar Chart -Are used to display data on one variable. -They are thick lines (narrow rectangles) having the same breadth. The magnitude of a quantity is represented by the height /length of the bar. Example: The following data represent sale by product, 1957- 1959 of a given company for three products A, B, C. Product Sales($) Sales($) Sales($) In 1957 In 1958 In 1959 A 12 14 18 B 24 21 18 C 24 35 54 19  Solutions: Sales by product in 1957 30 25 Sales in $ 20 15 10 5 0 A B C product Component Bar chart -When there is a desire to show how a total (or aggregate) is divided in to its component parts, we use component bar chart. -The bars represent total value of a variable with each total broken in to its component parts and different colours or designs are used for identifications Example: Draw a component bar chart to represent the sales by product from 1957 to 1959. Solutions: SALES BY PRODUCT 1957-1959 100 80 Sales in $ Product C 60 Product B 40 Product A 20 0 1957 1958 1959 Year of production Multiple Bar charts - These are used to display data on more than one variable. - They are used for comparing different variables at the same time. Example: Draw a component bar chart to represent the sales by product from 1957 to 1959. 20  Solutions: Sales by product 1957-1959 60 50 Sales in $ 40 Product A 30 Product B 20 Product C 10 0 1957 1958 1959 Year of production Graphical Presentation of data - The histogram, frequency polygon and cumulative frequency graph or ogive are most commonly applied graphical representation for continuous data. Procedures for constructing statistical graphs: Draw and label the X and Y axes. Choose a suitable scale for the frequencies or cumulative frequencies and label it on the Y axes. Represent the class boundaries for the histogram or ogive or the mid points for the frequency polygon on the X axes. Plot the points. Draw the bars or lines to connect the points. Histogram A graph which displays the data by using vertical bars of various heights to represent frequencies. Class boundaries are placed along the horizontal axes. Class marks and class limits are some times used as quantity on the X axes. Example: Construct a histogram to represent the previous data (example *). Frequency Polygon: - A line graph. The frequency is placed along the vertical axis and classes mid points are placed along the horizontal axis. It is customer to the next higher and lower class interval with corresponding frequency of zero, this is to make it a complete polygon. Example: Draw a frequency polygon for the above data (example *). 21  Solutions: 8 6 4 Value Frequency 2 0 2.5 8.5 14.5 20.5 26.5 32.5 38.5 44.5 Class Mid points Ogive (cumulative frequency polygon) - A graph showing the cumulative frequency (less than or more than type) plotted against upper or lower class boundaries respectively. That is class boundaries are plotted along the horizontal axis and the corresponding cumulative frequencies are plotted along the vertical axis. The points are joined by a free hand curve. Example: Draw an ogive curve(less than type) for the above data. (Example *) 22  Introduction to Statistics Lecture notes 3. MEASURES OF CENTERAL TENDENCY. Introduction and objectives of measuring central tendency. When we want to make comparison between groups of numbers it is good to have a single value that is considered to be a good representative of each group. This single value is called the average of the group. -Averages are also called measures of central tendency. -An average which is representative is called typical average and an average which is not representative and has only a theoretical value is called a descriptive average. -A typical average should posses the following: It should be rigidly defined. It should be based on all observation under investigation. It should be as little as affected by extreme observations. It should be capable of further algebraic treatment. It should be as little as affected by fluctuations of sampling. It should be ease to calculate and simple to understand. Objectives: 1. To comprehend the data easily. 2. To facilitate comparison. 3. To make further statistical analysis. The Summation Notation: - Let X1, X2 ,X3 …XN be a number of measurements where N is the total number of observation and Xi is ith observation. - Very often in statistics an algebraic expression of the form X1+X2+X3+...+XN is used in a formula to compute a statistic. It is tedious to write an expression like this very often, so mathematicians have developed a shorthand notation to represent a sum of scores, called the summation notation. N - The symbol ∑X i =1 i is a mathematical shorthand for X1+X2+X3+...+XN 1.  Introduction to Statistics Lecture notes The expression is read, "the sum of X sub i from i equals 1 to N." It means "add up all the numbers." Example: Suppose the following were scores made on the first homework assignment for five students in the class: 5, 7, 7, 6, and 8. In this example set of five numbers, where N=5, the summation could be written: The "i=1" in the bottom of the summation notation tells where to begin the sequence of summation. If the expression were written with "i=3", the summation would start with the third number in the set. For example: In the example set of numbers, this would give the following result: The "N" in the upper part of the summation notation tells where to end the sequence of summation. If there were only three scores then the summation and example would be: Sometimes if the summation notation is used in an expression and the expression must be written a number of times, as in a proof, then a shorthand notation for the shorthand notation is employed. When the summation sign "" is used without additional notation, then "i=1" and "N" are assumed. 2  Introduction to Statistics Lecture notes For example: PROPERTIES OF SUMMATION n 1. ∑ k = nk i =1 where k is any constant n n 2. ∑ kX i = k ∑ X i where k is any constant i =1 i =1 n n 3. ∑ (a + bX i ) = na + b∑ X i where a and b are any constant i =1 i =1 n n n 4. ∑(X i =1 i + Yi ) = ∑ X i + ∑ Yi i =1 i =1 The sum of the product of the two variables could be written: Example: considering the following data determine X Y 5 6 7 7 7 8 6 7 8 8 5 a) ∑X i =1 i 5 b) ∑Y i =1 i 5 c) ∑10 i =1 3  Introduction to Statistics Lecture notes 5 d) ∑(X i =1 i + Yi ) 5 e) ∑(X i =1 i − Yi ) 5 f) ∑X Y i =1 i i 5 ∑X 2 g) i i =1 5 5 h) (∑ X i )(∑ Yi ) i =1 i =1 Solutions: 5 a) ∑X i =1 i = 5 + 7 + 7 + 6 + 8 = 33 5 b) ∑Y i =1 i = 6 + 7 + 8 + 7 + 8 = 36 5 c) ∑10 = 5 *10 = 50 i =1 5 d) ∑(X i =1 i + Yi ) = (5 + 6) + (7 + 7) + (7 + 8) + (6 + 7) + (8 + 8) = 69 = 33 + 36 5 e) ∑(X i =1 i − Yi ) = (5 − 6) + (7 − 7) + (7 − 8) + (6 − 7) + (8 − 8) = −3 = 33 − 36 5 f) ∑X Y i =1 i i = 5 * 6 + 7 * 7 + 7 * 8 + 6 * 7 + 8 * 8 = 241 5 ∑X = 5 2 + 7 2 + 7 2 + 6 2 + 8 2 = 223 2 g) i i =1 5 5 h) (∑ X i )(∑ Yi ) = 33 * 36 = 1188 i =1 i =1 4  Introduction to Statistics Lecture notes Types of measures of central tendency -There are several different measures of central tendency; each has its advantage and disadvantage. The Mean (Arithmetic, Geometric and Harmonic) The Mode The Median Quantiles(Quartiles, deciles and percentiles) -The choice of these averages depends up on which best fit the property under discussion. The Arithmetic Mean: - Is defined as the sum of the magnitude of the items divided by the number of items - The mean of X1, X2 ,X3 …Xn is denoted by A.M ,m or X and is given by: X 1 + X 2 +... + X n X = n n ∑X i ⇒X = i =1 n - If X1 occurs f1 times If X2occurs f2 times.. - If Xn occurs fn times Then the mean will be k ∑fX i i k X= i =1 k , where k is the number of classes and ∑f i =n ∑f i =1 i i =1 5  Introduction to Statistics Lecture notes Example: Obtain the mean of the following number 2, 7, 8, 2, 7, 3, 7 Solutions: Xi fi Xifi 2 2 4 3 1 3 7 3 21 8 1 8 Total 7 36 4 ∑fX i i 36 X = i =1 4 = = 5.15 ∑f 7 i i =1 Arithmetic Mean for Grouped Data If data are given in the shape of a continuous frequency distribution, then the mean is obtained as follows: k ∑fX i i th X = i =1 k ,Where Xi =the class mark of the i class ∑f i =1 i fi = the frequency of the ith class Example: calculate the mean for the following age distribution. Class frequency 6- 10 35 11- 15 23 16- 20 15 21- 25 12 26- 30 9 31- 35 6 6  Introduction to Statistics Lecture notes Solutions: First find the class marks Find the product of frequency and class marks Find mean using the formula. Class fi Xi Xifi 6- 10 35 8 280 11- 15 23 13 299 16- 20 15 18 270 21- 25 12 23 276 26- 30 9 28 252 31- 35 6 33 198 Total 100 1575 6 ∑f X i i 1575 X = i =1 6 = = 15.75 ∑f 100 i i =1 Exercises: 1. Marks of 75 students are summarized in the following frequency distribution: Marks No. of students 40-44 7 45-49 10 50-54 22 55-59 f4 60-64 f5 65-69 6 70-74 3 If 20% of the students have marks between 55 and 59 i. Find the missing frequencies f4 and f5. ii. Find the mean. 7  Introduction to Statistics Lecture notes Special properties of Arithmetic mean 1. The sum of the deviations of a set of items from their mean is always n zero. i.e. ∑ ( X i − X ) = 0. i =1 2. The sum of the squared deviations of a set of items from their mean is n n the minimum. i.e. ∑ ( Xi − X ) 2 < ∑ ( X i − A) 2 , A ≠ X i =1 i =1 3. If X 1 is the mean of n1 observations If X 2 is the mean of n2 observations.. If X k is the mean of n k observations Then the mean of all the observation in all groups often called the combined mean is given by: k X 1 n1 + X 2 n2 +.... + X k nk ∑ X i ni Xc = = k i =1 n1 + n2 +...nk ∑ ni i =1 Example: In a class there are 30 females and 70 males. If females averaged 60 in an examination and boys averaged 72, find the mean for the entire class. Solutions: Females Males X 1 = 60 X 2 = 72 n1 = 30 n2 = 70 2 X 1 n1 + X 2 n 2 ∑ X i ni Xc = = 2 i =1 n1 + n 2 ∑ ni i =1 30(60) + 70(72) 6840 ⇒ Xc = = = 68.40 30 + 70 100 8  Introduction to Statistics Lecture notes 4. If a wrong figure has been used when calculating the mean the correct mean can be obtained with out repeating the whole process using: (CorrectValue − WrongValue ) CorrectMean = WrongMean + n Where n is total number of observations. Example: An average weight of 10 students was calculated to be 65.Latter it was discovered that one weight was misread as 40 instead of 80 k.g. Calculate the correct average weight. Solutions: (CorrectValue − WrongValue ) CorrectMean = WrongMean + n (80 − 40) CorrectMean = 65 + = 65 + 4 = 69k.g. 10 5. The effect of transforming original series on the mean. a) If a constant k is added/ subtracted to/from every observation then the new mean will be the old mean± k respectively. b) If every observations are multiplied by a constant k then the new mean will be k*old mean Example: 1. The mean of n Tetracycline Capsules X1, X2, …,Xn are known to be 12 gm. New set of capsules of another drug are obtained by the linear transformation Yi = 2Xi – 0.5 ( i = 1, 2, …, n ) then what will be the mean of the new set of capsules Solutions: NewMean = 2 * OldMean − 0.5 = 2 *12 − 0.5 = 23.5 2. The mean of a set of numbers is 500. a. If 10 is added to each of the numbers in the set, then what will be the mean of the new set? b. If each of the numbers in the set are multiplied by -5, then what will be the mean of the new set? 9  Introduction to Statistics Lecture notes Solutions: a).NewMean = OldMean + 10 = 500 + 10 = 510 b).NewMean = −5 * OldMean = −5 * 500 = −2500 Weighted Mean - When a proper importance is desired to be given to different data a weighted mean is appropriate. - Weights are assigned to each item in proportion to its relative importance. - Let X1, X2, …Xn be the value of items of a series and W1, W2, …Wn their corresponding weights , then the weighted mean denoted X w is defined as: n ∑XW i i Xw = i =1 n ∑W i −1 i Example: A student obtained the following percentage in an examination: English 60, Biology 75, Mathematics 63, Physics 59, and chemistry 55.Find the students weighted arithmetic mean if weights 1, 2, 1, 3, 3 respectively are allotted to the subjects. Solutions: 5 ∑XW i i 60 *1 + 75 * 2 + 63 *1 + 59 * 3 + 55 * 3 615 Xw = i =1 = = = 61.5 1+ 2 +1+ 3 + 3 5 ∑W i −1 i 10 Merits and Demerits of Arithmetic Mean Merits: It is rigidly defined. It is based on all observation. It is suitable for further mathematical treatment. It is stable average, i.e. it is not affected by fluctuations of sampling to some extent. It is easy to calculate and simple to understand. Demerits: It is affected by extreme observations. It can not be used in the case of open end classes. 10  Introduction to Statistics Lecture notes It can not be determined by the method of inspection. It can not be used when dealing with qualitative characteristics, such as intelligence, honesty, beauty. It can be a number which does not exist in a serious. Some times it leads to wrong conclusion if the details of the data from which it is obtained are not available. It gives high weight to high extreme values and less weight to low extreme values. The Mode - Mode is a value which occurs most frequently in a set of values - The mode may not exist and even if it does exist, it may not be unique. - In case of discrete distribution the value having the maximum frequency is the model value. Examples: 1. Find the mode of 5, 3, 5, 8, 9 Mode =5 2. Find the mode of 8, 9, 9, 7, 8, 2, and 5. It is a bimodal Data: 8 and 9 3. Find the mode of 4, 12, 3, 6, and 7. No mode for this data. - The mode of a set of numbers X1, X2, …Xn is usually denoted by X̂. Mode for Grouped data. If data are given in the shape of continuous frequency distribution, the mode is defined as: ⎛ Δ1 ⎞ Xˆ = Lmo + w⎜⎜ ⎟⎟ Δ ⎝ 1 + Δ 2 ⎠ 11  Introduction to Statistics Lecture notes Where: Xˆ = the mod e of the distribution w = the size of the mod al class Δ1 = f mo − f1 Δ 2 = f mo − f 2 f mo = frequency of the mod al class f1 = frequency of the class preceeding the mod al class f 2 = frequency of the class following the mod al class Note: The modal class is a class with the highest frequency. Example: Following is the distribution of the size of certain farms selected at random from a district. Calculate the mode of the distribution. Size of farms No. of farms 5- 15 ______________________________ 8 15- 25______________________________ 12 25- 35______________________________ 17 35- 45______________________________ 29 45- 55______________________________ 31 55- 65______________________________ 5 65- 75______________________________ 3 Solutions: 45 − 55 is the mod al class, sin ce it is a class with the highest frequency. Lmo = 45 w = 10 Δ1 = f mo − f1 = 2 Δ 2 = f mo − f 2 = 26 f mo = 31 f1 = 29 f2 = 5 ⎛ 2 ⎞ ⇒ Xˆ = 45 + 10⎜ ⎟ ⎝ 2 + 26 ⎠ = 45.71 12  Introduction to Statistics Lecture notes Merits and Demerits of Mode Merits: It is not affected by extreme observations. Easy to calculate and simple to understand. It can be calculated for distribution with open end class Demerits: It is not rigidly defined. It is not based on all observations It is not suitable for further mathematical treatment. It is not stable average, i.e. it is affected by fluctuations of sampling to some extent. Often its value is not unique. Note: being the point of maximum density, mode is especially useful in finding the most popular size in studies relating to marketing, trade, business, and industry. It is the appropriate average to be used to find the ideal size. The Median - In a distribution, median is the value of the variable which divides it in to two equal halves. - In an ordered series of data median is an observation lying exactly in the middle of the series. It is the middle most value in the sense that the number of values less than the median is equal to the number of values greater than it. -If X1, X2, …Xn be the observations, then the numbers arranged in ascending order will be X, X, …X[n], where X[i] is ith smallest value. ⇒ X< X< … 0 then the distributi on is positively skewed. If α 3 = 0 then the distributi on is symmetric. If α 3 < 0 then the distributi on is negatively skewed. Remark: o In a positively skewed distribution, smaller observations are more frequent than larger observations. i.e. the majority of the observations have a value below an average. o In a negatively skewed distribution, smaller observations are less frequent than larger observations. i.e. the majority of the observations have a value above an average. Examples: 1. Suppose the mean, the mode, and the standard deviation of a certain distribution are 32, 30.5 and 10 respectively. What is the shape of the curve representing the distribution? Solutions: Use the Pearsonian coefficient of skewness Mean − Mode 32 − 30.5 α3 = = = 0.15 S tan dard deviation 10 α 3 > 0 ⇒ The distribution is positively skewed. 2. In a frequency distribution, the coefficient of skewness based on the quartiles is given to be 0.5. If the sum of the upper and lower quartile is 28 and the median is 11, find the values of the upper and lower quartiles. 41 Introduction to Statistics Lecture notes Solutions: ~ Given: α 3 = 0.5, X = Q2 = 11 Required: Q1 ,Q3 Q1 + Q3 = 28...........................(*) (Q3 − Q2 ) − (Q2 − Q1 ) Q3 + Q1 − 2Q2 α3 = = = 0.5 Q3 − Q1 Q3 − Q1 Substituting the given values , one can obtain the following Q3 − Q1 = 12...................................(**) Solving (*) and (**) at the same time we obtain the following values Q1 = 8 and Q3 = 20 3. Some characteristics of annually family income distribution (in Birr) in two regions is as follows: Region Mean Median Standard Deviation A 6250 5100 960 B 6980 5500 940 a) Calculate coefficient of skewness for each region b) For which region is, the income distribution more skewed. Give your interpretation for this Region c) For which region is the income more consistent? Solutions: (exercise) 4. For a moderately skewed frequency distribution, the mean is 10 and the median is 8.5. If the coefficient of variation is 20%, find the Pearsonian coefficient of skewness and the probable mode of the distribution. (exercise) 5. The sum of fifteen observations, whose mode is 8, was found to be 150 with coefficient of variation of 20% (a) Calculate the pearsonian coefficient of skewness and give appropriate conclusion. (b) Are smaller values more or less frequent than bigger values for this distribution? (c) If a constant k was added on each observation, what will be the new pearsonian coefficient of skewness? Show your steps. What do you conclude from this? 42  Introduction to Statistics Lecture notes Kurtosis Kurtosis is the degree of peakdness of a distribution, usually taken relative to a normal distribution. A distribution having relatively high peak is called leptokurtic. If a curve representing a distribution is flat topped, it is called platykurtic. The normal distribution which is not very high peaked or flat topped is called mesokurtic. Measures of kurtosis The moment coefficient of kurtosis: Denoted by α 4 and given by M4 M4 α4 = = 4 M2 2 σ Where : M 4 is the fourth moment about the mean. M 2 is the sec ond moment about the mean. σ is the population s tan dard deviation. The peakdness depends on the value of α 4. If α 4 > 3 then the curve is leptokurtic. If α 4 = 3 then the curve is mesokurtic. If α 4 < 3 then the curve is platykurtic. Examples: 1. If the first four central moments of a distribution are: M 1 = 0, M 2 = 16, M 3 = −60, M 4 = 162 a) Compute a measure of skewness b) Compute a measure of kurtosis and give your interpretation. Solutions: M3 − 60 α3 = = = −0.94 < 0 a) M2 32 163 2 ⇒ The distribution is negatively skewed. M 4 162 α4 = 2 = 2 = 0.6 < 3 b) M2 16 ⇒ The curve is platykurtic. 43  Introduction to Statistics Lecture notes 2. The median and the mode of a mesokurtic distribution are 32 and 34 respectively. The 4th moment about the mean is 243. Compute the Pearsonian coefficient of skewness and identify the type of skewness. Assume (n-1 = n). 3. If the standard deviation of a symmetric distribution is 10, what should be the value of the fourth moment so that the distribution is mesokurtic? Solutions (exercise). 44  5. ELEMENTARY PROBABILITY Introduction Probability theory is the foundation upon which the logic of inference is built. It helps us to cope up with uncertainty. In general, probability is the chance of an outcome of an experiment. It is the measure of how likely an outcome is to occur. Definitions of some probability terms 1. Experiment: Any process of observation or measurement or any process which generates well defined outcome. 2. Probability Experiment: It is an experiment that can be repeated any number of times under similar conditions and it is possible to enumerate the total number of outcomes with out predicting an individual out come. It is also called random experiment. Example: If a fair die is rolled once it is possible to list all the possible outcomes i.e.1, 2, 3, 4, 5, 6 but it is not possible to predict which outcome will occur. 3. Outcome :The result of a single trial of a random experiment 4. Sample Space: Set of all possible outcomes of a probability experiment 5. Event: It is a subset of sample space. It is a statement about one or more outcomes of a random experiment.They are denoted by capital letters. Example: Considering the above experiment let A be the event of odd numbers, B be the event of even numbers, and C be the event of number 8. ⇒ A = {1,3,5} B = {2,4,6} C = { } or empty space or impossible event Remark: If S (sample space) has n members then there are exactly 2n subsets or events. 6. Equally Likely Events: Events which have the same chance of occurring. 7. Complement of an Event: the complement of an event A means non- ' occurrence of A and is denoted by A , or A , or A contains those points of the c sample space which don’t belong to A. 8. Elementary Event: an event having only a single element or sample point. 9. Mutually Exclusive Events: Two events which cannot happen at the same time. 1 10. Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other occurring. 11. Dependent Events: Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed. Example:.What is the sample space for the following experiment a) Toss a die one time. b) Toss a coin two times. c) A light bulb is manufactured. It is tested for its life length by time. Solution a) S={1,2,3,4,5,6} b) S={(HH),(HT),(TH),(TT)} c) S={t /t≥0} ¾ Sample space can be ƒ Countable ( finite or infinite) ƒ Uncountable. Counting Rules In order to calculate probabilities, we have to know The number of elements of an event The number of elements of the sample space. That is in order to judge what is probable, we have to know what is possible. In order to determine the number of outcomes, one can use several rules of counting. - The addition rule - The multiplication rule - Permutation rule - Combination rule To list the outcomes of the sequence of events, a useful device called tree diagram is used. 2 Example: A student goes to the nearest snack to have a breakfast. He can take tea, coffee, or milk with bread, cake and sandwitch. How many possibilities does he have? Solutions: Tea Bread Cake Sandwich Coeffee Bread Cake Sandwitch Milk Bread Cake Sandwitch Î There are nine possibilities. 3 The Multiplication Rule: If a choice consists of k steps of which the first can be made in n1 ways, the second can be made in n2 ways…, the kth can be made in nk ways, then the whole choice can be made in ( n1 * n2 *........ * nk ) ways. Example: The digits 0, 1, 2, 3, and 4 are to be used in 4 digit identification card. How many different cards are possible if a) Repetitions are permitted. b) Repetitions are not permitted. Solutions a) 1st digit 2nd digit 3rd digit 4th digit 5 5 5 5 There are four steps 1. Selecting the 1st digit, this can be made in 5 ways. 2. Selecting the 2nd digit, this can be made in 5 ways. 3. Selecting the 3rd digit, this can be made in 5 ways. 4. Selecting the 4th digit, this can be made in 5 ways. ⇒ 5 * 5 * 5 * 5 = 625 different cards are possible. b) 1st digit 2nd digit 3rd digit 4th digit 5 4 3 2 There are four steps 5. Selecting the 1st digit, this can be made in 5 ways. 6. Selecting the 2nd digit, this can be made in 4 ways. 7. Selecting the 3rd digit, this can be made in 3 ways. 8. Selecting the 4th digit, this can be made in 2 ways. ⇒ 5 * 4 * 3 * 2 = 120 different cards are possible. 4 Permutation An arrangement of n objects in a specified order is called permutation of the objects. Permutation Rules: 1. The number of permutations of n distinct objects taken all together is n! Where n!= n * (n − 1) * (n − 2) *..... * 3 * 2 *1 2. The arrangement of n objects in a specified order using r objects at a time is called the permutation of n objects taken r objects at a time. It is written as n Pr and the formula is n! n Pr = (n − r )! 3. The number of permutations of n objects in which k1 are alike k2 are alike ---- etc is n! n Pr = k1!*k 2 *... * k n Example: 1. Suppose we have a letters A,B, C, D a) How many permutations are there taking all the four? b) How many permutations are there two letters at a time? 2. How many different permutations can be made from the letters in the word “CORRECTION”? Solutions: 1. a) Here n = 4, there are four disnict object ⇒ There are 4!= 24 permutations. b) Here n = 4, r = 2 4! 24 ⇒ There are 4 P2 = = = 12 permutations. (4 − 2)! 2 5  2. Here n = 10 Of which 2 are C , 2 are O, 2 are R ,1E ,1T ,1I ,1N ⇒ K1 = 2, k 2 = 2, k3 = 2, k 4 = k5 = k6 = k7 = 1 U sin g the 3rd rule of permutation , there are 10! = 453600 permutations. 2!*2!*2!*1!*1!*1!*1! Exercises: 1. Six different statistics books, seven different physics books, and 3 different Economics books are arranged on a shelf. How many different arrangements are possible if; i. The books in each particular subject must all stand together ii. Only the statistics books must stand together 2. If the permutation of the word WHITE is selected at random, how many of the permutations i. Begins with a consonant? ii. Ends with a vowel? iii. Has a consonant and vowels alternating? Combination A selection of objects with out regard to order is called combination. Example: Given the letters A, B, C, and D list the permutation and combination for selecting two letters. Solutions: Permutation Combination AB BA CA DA AB BC AC BC CB DB AC BD AD BD CD DC AD DC Note that in permutation AB is different from BA. But in combination AB is the same as BA. 6 Combination Rule The number of combinations of r objects selected from n objects is denoted by ⎛n⎞ C n r or ⎜ ⎜ r ⎟⎟ and is given by the formula: ⎝ ⎠ ⎛n⎞ n! ⎜⎜ ⎟⎟ = ⎝ r ⎠ (n − r )!*r! Examples: 1. In how many ways a committee of 5 people be chosen out of 9 people? Solutions: n=9 , r =5 ⎛n⎞ n! 9! ⎜⎜ ⎟⎟ = = = 126 ways ⎝ ⎠ r ( n − r )!*r! 4!* 5! 2. Among 15 clocks there are two defectives.In how many ways can an inspector chose three of the clocks for inspection so that: a) There is no restriction. b) None of the defective clock is included. c) Only one of the defective clocks is included. d) Two of the defective clock is included. Solutions: n = 15 of which 2 are defective and 13 are non − defective. r =3 a) If there is no restriction select three clocks from 15 clocks and this can be done in : n = 15 , r = 3 ⎛n⎞ n! 15! ⎜⎜ ⎟⎟ = = = 455 ways ⎝ ⎠ r ( n − r )!*r! 12!*3! 7  b) None of the defective clocks is included. This is equivalent to zero defective and three non defective, which can be done in: ⎛ 2 ⎞ ⎛13 ⎞ ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = 286 ways. ⎝0⎠ ⎝ 3 ⎠ c) Only one of the defective clocks is included. This is equivalent to one defective and two non defective, which can be done in: ⎛ 2 ⎞ ⎛13 ⎞ ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = 156 ways. ⎝1⎠ ⎝ 2 ⎠ d) Two of the defective clock is included. This is equivalent to two defective and one non defective, which can be done in: ⎛ 2 ⎞ ⎛13 ⎞ ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = 13 ways. ⎝ 2⎠ ⎝ 3 ⎠ Exercises: 1. Out of 5 Mathematician and 7 Statistician a committee consisting of 2 Mathematician and 3 Statistician is to be formed. In how many ways this can be done if a) There is no restriction b) One particular Statistician should be included c) Two particular Mathematicians can not be included on the committee. 2. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, in how many ways this can be don if a) There is no restriction. b) The dictionary is selected? c) 2 novels and 1 book of poems are selected? 8 Approaches to measuring Probability There are four different conceptual approaches to the study of probability theory. These are: The classical approach. The frequentist approach. The axiomatic approach. The subjective approach. The classical approach This approach is used when: - All outcomes are equally likely. - Total number of outcome is finite, say N. Definition: If a random experiment with N equally likely outcomes is conducted and out of these NA outcomes are favourable to the event A, then the probability that event A occur denoted P(A) is defined as: N A No. of outcomes favourable to A n( A) P( A) = = = N Total number of outcomes n( S ) Examples: 1. A fair die is tossed once. What is the probability of getting a) Number 4? b) An odd number? c) An even number? d) Number 8? Solutions: First identify the sample space, say S S = {1, 2, 3, 4, 5, 6} ⇒ N = n( S ) = 6 a) Let A be the event of number 4 A = {4} ⇒ N A = n( A) = 1 n( A) P( A) = =1 6 n( S ) 9  b) Let A be the event of odd numbers A = {1,3,5} ⇒ N A = n( A) = 3 n( A) P( A) = = 3 6 = 0.5 n( S ) c) Let A be the event of even numbers A = {2,4,6} ⇒ N A = n( A) = 3 n( A) P( A) = = 3 6 = 0.5 n( S ) d) Let A be the event of number 8 A= Ø ⇒ N A = n( A) = 0 n( A) P( A) = =0 6=0 n( S ) 2. A box of 80 candles consists of 30 defective and 50 non defective candles. If 10 of this candles are selected at random, what is the probability a) All will be defective. b) 6 will be non defective c) All will be non defective Solutions: ⎛ 80 ⎞ Total selection = ⎜⎜ ⎟⎟ = N = n( S ) ⎝ 10 ⎠ a) Let A be the event that all will be defective. 10  ⎛ 30 ⎞ ⎛ 50 ⎞ Total way in which A occur = ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = N A = n( A) ⎝ 10 ⎠ ⎝ 0 ⎠ ⎛ 30 ⎞ ⎛ 50 ⎞ ⎜ ⎟*⎜ ⎟ n( A) ⎜⎝ 10 ⎟⎠ ⎜⎝ 0 ⎟⎠ ⇒ P( A) = = = 0.00001825 n( S ) ⎛ ⎞ 80 ⎜⎜ ⎟⎟ ⎝ 10 ⎠ b) Let A be the event that 6 will be non defective. ⎛ 30 ⎞ ⎛ 50 ⎞ Total way in which A occur = ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = N A = n( A) ⎝4⎠ ⎝6⎠ ⎛ 30 ⎞ ⎛ 50 ⎞ ⎜ ⎟*⎜ ⎟ n( A) ⎜⎝ 4 ⎟⎠ ⎜⎝ 6 ⎟⎠ ⇒ P( A) = = = 0.265 n( S ) ⎛ 80 ⎞ ⎜⎜ ⎟⎟ ⎝ 10 ⎠ c) Let A be the event that all will be non defective. ⎛ 30 ⎞ ⎛ 50 ⎞ Total way in which A occur = ⎜⎜ ⎟⎟ * ⎜⎜ ⎟⎟ = N A = n( A) ⎝ 0 ⎠ ⎝ 10 ⎠ ⎛ 30 ⎞ ⎛ 50 ⎞ ⎜ ⎟*⎜ ⎟ n( A) ⎜⎝ 0 ⎟⎠ ⎜⎝ 10 ⎟⎠ ⇒ P( A) = = = 0.00624 n( S ) ⎛ ⎞ 80 ⎜⎜ ⎟⎟ ⎝ 10 ⎠ Exercises: 1. What is the probability that a waitress will refuse to serve alcoholic beverages to only three minors if she randomly checks the I.D’s of five students from among ten students of which four are not of legal age? 2. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that a) The dictionary is selected? b) 2 novels and 1 book of poems are selected? 11  Short coming of the classical approach: This approach is not applicable when: - The total number of outcomes is infinite. - Outcomes are not equally likely. The Frequentist Approach This is based on the relative frequencies of outcomes belonging to an event. Definition: The probability of an event A is the proportion of outcomes favourable to A in the long run when the experiment is repeated under same condition. NA P ( A) = lim N →∞ N Example: If records show that 60 out of 100,000 bulbs produced are defective. What is the probability of a newly produced bulb to be defective? Solution: Let A be the event that the newly produced bulb is defective. NA 60 P ( A) = lim = = 0.0006 N →∞ N 100,000 Axiomatic Approach: Let E be a random experiment and S be a sample space associated with E. With each event A a real number called the probability of A satisfies the following properties called axioms of probability or postulates of probability. 1. P( A) ≥ 0 2. P( S ) = 1, S is the sure event. 3. If A and B are mutually exclusive events, the probability that one or the other occur equals the sum of the two probabilities. i. e. P( A ∪ B) = P( A) + P( B) 4. P( A ) = 1 − P( A) ' 5. 0 ≤ P( A) ≤ 1 6. P(ø) =0, ø is the impossible event. Remark: Venn-diagrams can be used to solve probability problems. AUB AnB A 12 In general p( A ∪ B) = p( A) + p( B) − p( A ∩ B) Conditional probability and Independency Conditional Events: If the occurrence of one event has an effect on the next occurrence of the other event then the two events are conditional or dependant events. Example: Suppose we have two red and three white balls in a bag 1. Draw a ball with replacement 2 Let A= the event that the first draw is redÎ p ( A) = 5 2 B= the event that the second draw is red Î p ( B ) = 5 A and B are independent. 2. Draw a ball with out replacement 2 Let A= the event that the first draw is redÎ p ( A) = 5 B= the event that the second draw is red Î p( B) = ? This is conditional. Let B= the event that the second draw is red given that the first draw is red Î p ( B ) = 1 4 Conditional probability of an event The conditional probability of an event A given that B has already occurred, denoted p ( A B ) is p( A ∩ B) p( A B) = , p( B) ≠ 0 p( B) Remark: (1) p( A B) = 1 − p ( A B) ' (2) p( B A) = 1 − p ( B A) ' 13 Examples 1. For a student enrolling at freshman at certain university the probability is 0.25 that he/she will get scholarship and 0.75 that he/she will graduate. If the probability is 0.2 that he/she will get scholarship and will also graduate. What is the probability that a student who get a scholarship graduate? Solution: Let A= the event that a student will get a scholarship B= the event that a student will graduate given p ( A) = 0.25, p ( B ) = 0.75, p( A ∩ B ) = 0.20 Re quired p (B A) p ( A ∩ B ) 0.20 p ( B A) = = = 0.80 p ( A) 0.25 2. If the probability that a research project will be well planned is 0.60 and the probability that it will be well planned and well executed is 0.54, what is the probability that it will be well executed given that it is well planned? Solution; Let A= the event that a research project will be well Planned B= the event that a research project will be well Executed given p ( A) = 0.60, p( A ∩ B ) = 0.54 Re quired p(B A) p( A ∩ B ) 0.54 p ( B A) = = = 0.90 p ( A) 0.60 3. A lot consists of 20 defective and 80 non-defective items from which two items are chosen without replacement. Events A & B are defined as A = {the first item chosen is defective}, B = {the second item chosen is defective} a. What is the probability that both items are defective? b. What is the probability that the second item is defective? Solution……… Exercise 14 Note; for any two events A and B the following relation holds. ( ) ( ) p(B ) = p(B A). p( A) + p B A'. p A' Probability of Independent Events Two events A and B are independent if and only if p ( A ∩ B ) = p ( A). p (B ) Here p ( A B ) = p ( A), P (B A) = p ( B ) Example; A box contains four black and six white balls. What is the probability of getting two black balls in drawing one after the other under the following conditions? a. The first ball drawn is not replaced b. The first ball drawn is replaced Solution; Let A= first drawn ball is black B= second drawn is black Required p( A ∩ B ) a. p( A ∩ B ) = p(B A). p( A) = (4 10 )(3 9 ) = 2 15 b. p ( A ∩ B ) = p( A). p (B ) = (4 10 )(4 10 ) = 4 25 15  6. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Definition: A random variable is a numerical description of the outcomes of the experiment or a numerical valued function defined on sample space, usually denoted by capital letters. Example: If X is a random variable, then it is a function from the elements of the sample space to the set of real numbers. i.e. X is a function X: S → R ÎA random variable takes a possible outcome and assigns a number to it. Example: Flip a coin three times, let X be the number of heads in three tosses. ⇒ S = {(HHH ), (HHT ), (HTH ), (HTT ), (THH ), (THT ), (TTH ), (TTT )} ⇒ X (HHH ) = 3, X (HHT ) = X (HTH ) = X (THH ) = 2, X (HTT ) = X (THT ) = X (TTH ) = 1 X (TTT ) = 0 X = {0, 1, 2, 3} ÎX assumes a specific number of values with some probabilities. Random variables are of two types: 1. Discrete random variable: are variables which can assume only a specific number of values. They have values that can be counted Examples: Toss a coin n times and count the number of heads. Number of children in a family. Number of car accidents per week. Number of defective items in a given company. Number of bacteria per two cubic centimeter of water. 16  2. Continuous random variable: are variables that can assume all values between any two give values. Examples: Height of students at certain college. Mark of a student. Life time of light bulbs. Length of time required to complete a given training. Definition: a probability distribution consists of a value a random variable can assume and the corresponding probabilities of the values. Example: Consider the experiment of tossing a coin three times. Let X be the number of heads. Construct the probability distribution of X. Solution: ƒ First identify the possible value that X can assume. ƒ Calculate the probability of each possible distinct value of X and express X in the form of frequency distribution. X =x 0 1 2 3 P( X = x ) 18 38 38 18 Probability distribution is denoted by P for discrete and by f for continuous random variable. Properties of Probability Distribution: 1. P( x) ≥ 0, if X is discrete. f ( x) ≥ 0, if X is continuous. 17  2. ∑ P( X = x ) = 1 , if X is discrete. x ∫ f ( x)dx = 1 , if is continuous. x Note: 1. If X is a continuous random variable then b P(a < X < b) = ∫ f ( x)dx a 2. Probability of a fixed value of a continuous random variable is zero. ⇒ P ( a < X < b) = P ( a ≤ X < b) = P ( a < X ≤ b) = P ( a ≤ X ≤ b) 3. If X is discrete random variable the b −1 P ( a < X < b) = ∑ P ( x ) x = a +1 b −1 P ( a ≤ X < b) = ∑ p ( x ) x=a b P ( a < X ≤ b) = ∑ P ( x ) x = a +1 b P ( a ≤ X ≤ b) = ∑ P ( x ) x=a 4. Probability means area for continuous random variable. 18 Introduction to expectation Definition: 1. Let a discrete random variable X assume the values X1, X2, ….,Xn with the probabilities P(X1), P(X2), ….,P(Xn) respectively. Then the expected value of X ,denoted as E(X) is defined as: E ( X ) = X 1P ( X 1 ) + X 2 P ( X 2 ) +.... + X n P( X n ) n = ∑ X i P( X i ) i =1 2. Let X be a continuous random variable assuming the values in the b interval (a, b) such that ∫ f ( x)dx = 1 ,then a b E ( X ) = ∫ x f ( x)dx a Examples: 1. What is the expected value of a random variable X obtained by tossing a coin three times where is the number of heads Solution: First construct the probability distribution of X X =x 0 1 2 3 P( X = x ) 18 38 38 18 ⇒ E ( X ) = X 1 P ( X 1 ) + X 2 P ( X 2 ) +.... + X n P( X n ) = 0 *1 8 + 1 * 3 8 +..... + 2 *1 8 = 1.5 19  2. Suppose a charity organization is mailing printed return-address stickers to over one million homes in the Ethiopia. Each recipient is asked to donate either $1, $2, $5, $10, $15, or $20. Based on past experience, the amount a person donates is believed to follow the following probability distribution: X =x $1 $2 $5 $10 $15 $20 P( X = x ) 0.1 0.2 0.3 0.2 0.15 0.05 What is expected that an average donor to contribute? Solution: X =x $1 $2 $5 $10 $15 $20 Total P( X = x ) 0.1 0.2 0.3 0.2 0.15 0.05 1 xP( X = x) 0.1 0.4 1.5 2 2.25 1 7.25 6 ⇒ E ( X ) = ∑ xi P( X = xi ) = $7.25 i =1 Mean and Variance of a random variable Let X be given random variable. 1. The expected value of X is its mean ⇒ Mean of X = E (X ) 2. The variance of X is given by: Variance of X = var( X ) = E ( X 2 ) − [ E ( X )]2 20 Where: n 2 E ( X 2 ) = ∑ xi P ( X = xi ) , if X is discrete i =1 = ∫ x 2 f ( x)dx , if X is continuous. x Examples: 1. Find the mean and the variance of a random variable X in example 2 above. Solutions: X =x $1 $2 $5 $10 $15 $20 Total P( X = x ) 0.1 0.2 0.3 0.2 0.15 0.05 1 xP( X = x) 0.1 0.4 1.5 2 2.25 1 7.25 x 2 P( X = x) 0.1 0.8 7.5 20 33.75 20 82.15 ⇒ E ( X ) = 7.25 Var ( X ) = E ( X 2 ) − [ E ( X )]2 = 82.15 − 7.252 = 29.59 2. Two dice are rolled. Let X be a random variable denoting the sum of the numbers on the two dice. i) Give the probability distribution of X ii) Compute the expected value of X and its variance Solution (exercise) 21 There are some general rules for mathematical expectation. Let X and Y are random variables and k be a constant. RULE 1 E (k ) = k RULE 2 Var(k) = 0 RULE 3 E (kX ) = kE ( X ) RULE 4 Var (kX ) = k 2Var ( X ) RULE 5 E ( X + Y ) = E ( X ) + E (Y ) 22  Common Discrete Probability Distributions 1. Binomial Distribution A binomial experiment is a probability experiment that satisfies the following four requirements called assumptions of a binomial distribution. 1. The experiment consists of n identical trials. 2. Each trial has only one of the two possible mutually exclusive outcomes, success or a failure. 3. The probability of each outcome does not change from trial to trial, and 4. The trials are independent, thus we must sample with replacement. Examples of binomial experiments Tossing a coin 20 times to see how many tails occur. Asking 200 people if they watch BBC news. Registering a newly produced product as defective or non defective. Asking 100 people if they favour the ruling party. Rolling a die to see if a 5 appears. Definition: The outcomes of the binomial experiment and the corresponding probabilities of these outcomes are called Binomial Distribution. Let P = the probability of success q = 1 − p = the probability of failure on any given trial Then the probability of getting x successes in n trials becomes: ⎛n⎞ P ( X = x) = ⎜⎜ ⎟⎟ p x q n − x , x = 0,1,2,...., n ⎝ x⎠ And this is some times written as: X ~ Bin(n, p) When using the binomial formula to solve problems, we have to identify three things: The number of trials ( n ) The probability of a success on any one trial ( p ) and The number of successes desired ( X ). 23 Examples: 1. What is the probability of getting three heads by tossing a fair coin four times? Solution: Let X be the number of heads in tossing a fair coin four times X ~ Bin(n = 4, p = 0.50) ⎛ n⎞ ⇒ P ( X = x) = ⎜⎜ ⎟⎟ p x q n − x , x = 0,1,2,3,4 ⎝ x⎠ ⎛ 4⎞ = ⎜⎜ ⎟⎟0.5 x 0.54 − x ⎝ x⎠ ⎛ 4⎞ = ⎜⎜ ⎟⎟0.54 ⎝ x⎠ ⎛ 4⎞ ⇒ P ( X = 3) = ⎜⎜ ⎟⎟0.54 = 0.25 ⎝ 3⎠ 2. Suppose that an examination consists of six true and false questions, and assume that a student has no knowledge of the subject matter. The probability that the student will guess the correct answer to the first question is 30%. Likewise, the probability of guessing each of the remaining questions correctly is also 30%. a) What is the probability of getting more than three correct answers? b) What is the probability of getting at least two correct answers? c) What is the probability of getting at most three correct answers? d) What is the probability of getting less than five correct answers? 24 Solution Let X = the number of correct answers that the student gets. X ~ Bin(n = 6, p = 0.30) a) P( X > 3) = ? ⎛n⎞ ⇒ P ( X = x) = ⎜⎜ ⎟⎟ p x q n − x , x = 0,1,2,..6 ⎝ x⎠ ⎛6⎞ = ⎜⎜ ⎟⎟0.3 x 0.7 6 − x ⎝ x⎠ ⇒ P ( X > 3) = P ( X = 4) + P ( X = 5) + P( X = 6) = 0.060 + 0.010 + 0.001 = 0.071 Thus, we may conclude that if 30% of the exam questions are answered by guessing, the probability is 0.071 (or 7.1%) that more than four of the questions are answered correctly by the student. b) P( X ≥ 2) = ? P( X ≥ 2) = P( X = 2) + P( X = 3) + P( X = 4) + P( X = 5) + P( X = 6) = 0.324 + 0.185 + 0.060 + 0.010 + 0.001 = 0.58 c) P( X ≤ 3) = ? P( X ≤ 3) = P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) = 0.118 + 0.303 + 0.324 + 0.185 = 0.93 d) P( X < 5) = ? P( X < 5) = 1 − P( X ≥ 5) = 1 − {P( X = 5) + P( X = 6)} = 1 − (0.010 + 0.001) = 0.989 25  Exercises: 1. Suppose that 4% of all TVs made by A&B Company in 2000 are defective. If eight of these TVs are randomly selected from across the country and tested, what is the probability that exactly three of them are defective? Assume that each TV is made independently of the others. 2. An allergist claims that 45% of the patients she tests are allergic to some type of weed. What is the probability that a) Exactly 3 of her next 4 patients are allergic to weeds? b) None of her next 4 patients are allergic to weeds? 3. Explain why the following experiments are not Binomial Rolling a die until a 6 appears. Asking 20 people how old they are. Drawing 5 cards from a deck for a poker hand. Remark: If X is a binomial random variable with parameters n and p then E ( X ) = np , Var ( X ) = npq 2. Poisson Distribution - A random variable X is said to have a Poisson distribution if its probability distribution is given by: λx e − λ P( X = x) = , x = 0,1,2,...... x! Where λ = the average number. - The Poisson distribution depends only on the average number of occurrences per unit time of space. - The Poisson distribution is used as a distribution of rare events, such as: Number of misprints. Natural disasters like earth quake. Accidents. Hereditary. Arrivals - The process that gives rise to such events are called Poisson process. 26  Examples: 1. If 1.6 accidents can be expected an intersection on any given day, what is the probability that there will be 3 accidents on any given day? Solution; Let X =the number of accidents, λ = 1.6 1.6 x e −1.6 X = poisson(1.6 ) ⇒ p ( X = x ) = x! 1.63 e −1.6 p ( X = 3) = = 0.1380 3! 2. On the average, five smokers pass a certain street corners every ten minutes, what is the probability that during a given 10minutes the number of smokers passing will be a. 6 or fewer b. 7 or more c. Exactly 8……. (Exercise) If X is a Poisson random variable with parameters λ then E (X ) = λ , Var (X ) = λ Note: The Poisson probability distribution provides a close approximation to the binomial probability distribution when n is large and p is quite small or quite large with λ = np. (np ) x e − ( np ) P( X = x) = , x = 0,1,2,...... x! Where λ = np = the average number. Usually we use this approximation if np ≤ 5. In other words, if n > 20 and np ≤ 5 [or n