Engineering Data Analysis PDF

MATH019A Engineering Data Analysis ENGR. JAN JUSTINE A. RAZON FACULTY, CPE DEPARTMENT MATH 019A  PRELIM TOPICS 1. OBTAINING DATA 1.1 Methods of Data Collection 1.2 Planning and Conducting Surveys 1.3. Introduction to Design Expe...

MATH019A Engineering Data Analysis ENGR. JAN JUSTINE A. RAZON FACULTY, CPE DEPARTMENT MATH 019A  PRELIM TOPICS 1. OBTAINING DATA 1.1 Methods of Data Collection 1.2 Planning and Conducting Surveys 1.3. Introduction to Design Experiments 2. PROBABILITY 2.1 Relationship among Events MATH 019A 2.2 Rules of Probability  PRELIM TOPICS 3. Discrete Probability Distribution 3.1 Random Variables 3.2 Cumulative Distribution 3.3 Binomial Distribution 3.4 Poisson Distribution 4. Continuous Probability Distribution 4.1 Continuous Random Variables MATH 019A 4.2 Normal and Exponential Distribution  METHODS OF DATA COLLECTION Data collection is the process of gathering and measuring information on variables of interest, in an established systematic fashion that enables one to answer stated MATH research questions, test hypotheses, and evaluate outcomes. 019A  TYPES OF DATA 1. PRIMARY DATA data which are collected a fresh and for the first time and thus happen to be original in character and known as PRIMARY DATA. 2. SECONDARY DATA data which have been collected by someone else and which have already MATH been passed through the statistical 019A process. METHODS OF DATA COLLECTION: PRIMARY DATA 1. Observation 2. Interview 3. Questionnaire 4. Case Study 5. Survey MATH 019A  METHODS OF DATA COLLECTION: PRIMARY DATA OBSERVATION Observation method is a method under which data from the field is collected with the help of observation by the observer or by personally going to the field. ADVANTAGES DISADVANTAGES Subjective bias Time consuming eliminated Current information Limited information Independent to Unforeseen factors MATH respondent’s variable 019A  TYPES OF OBSERVATION STRUCTURED and UNSTRUCTURED 1. Structured Observation when observation is done by characterizing style of recording the observed information, standardized conditions of observation , definition of the units to be observed , selection of pertinent data of observation. Example: An auditor performing inventory analysis in store 2. Unstructured Observation when observation is done without any thought before observation. MATH 019A Example: Observing children playing with new toys.  TYPES OF OBSERVATION PARTICIPANT and NON- PARTICIPANT 1. Participant when the Observer is member of the group which he is observing. Advantages: 1. Observation of natural behavior 2. Closeness with the group 3. Better understanding 2. Non-participant when observer is observing people without giving any information to them. Advantages: 1. Objectivity and neutrality MATH 2. More willingness of the respondent 019A  TYPES OF OBSERVATION CONTROLLED and UNCONTROLLED 1. Controlled when the observation takes place in natural condition. It is done to get spontaneous picture of life and persons. 2. Uncontrolled when observation takes place according to definite pre arranged plans , with experimental procedure then it is controlled observation generally done in laboratory under controlled condition. MATH 019A  METHODS OF DATA COLLECTION: PRIMARY DATA INTERVIEW METHOD INTERVIEW METHOD This method of collecting data involves presentation or oral- verbal stimuli and reply in terms of oral-verbal responses. Interview Method is an oral verbal communication where interviewer asks questions (which are aimed to get information required for study) to respondent. MATH 019A  TYPES OF INTERVIEW Personal interviews : The interviewer asks questions generally in a face to face contact to the other person or persons. Structured interviews : in this case, a set of pre- decided questions are there. Unstructured interviews : in this case, we don’t follow a system of pre-determined questions. Focused interviews : attention is focused on the given experience of the respondent and its possible effects. Clinical interviews : concerned with broad underlying feelings or motivations or with the MATH 019A course of individual’s life experience, rather than with the effects of the specific experience, as in  TYPES OF INTERVIEW Group interviews : a group of 6 to 8 individuals is interviewed. Qualitative and quantitative interviews : divided on the basis of subject matter i.e. whether qualitative or quantitative. Individual interviews : interviewer meets a single person and interviews him. Selection interviews : done for the selection of people for certain jobs. Depth interviews : it deliberately aims to elicit unconscious as well as other types of material relating especially to personality dynamics and motivations. MATH 019A Telephonic interviews : contacting samples on telephone.  METHODS OF DATA COLLECTION: PRIMARY DATA QUESTIONNAIRE METHOD QUESTIONNAIRE METHOD This method of data collection is quite popular, particularly in case of big enquiries. The questionnaire is mailed to respondents who are expected to read and understand the questions and write down the reply in the space meant for the purpose in the questionnaire itself. The respondents have to answer the questions on MATH their own. 019A  METHODS OF DATA COLLECTION: PRIMARY DATA QUESTIONNAIRE METHOD ADVANTAGES DISADVANTAGES Low cost even if the Low rate of return of duly filled geographical area is too large questionnaire. Answers are in respondents Slowest method of data word so free from bias. collection. Adequate time to think for Difficult to know if the expected answers. respondent have filled the form or it is filled by someone else. Non approachable respondents may be conveniently contacted. Large samples can be used so results are more reliable. MATH 019A  METHODS OF DATA COLLECTION: PRIMARY DATA CASE STUDY METHOD CASE STUDY METHOD is essentially an intensive investigation of the particular unit under consideration. ADVANTAGES DISADVANTAGES They are less costly and less They are subject to selection time-consuming; they are bias advantageous when exposure data is expensive or hard to obtain. They are advantageous when They generally do not allow MATH studying dynamic populations calculation of incidence 019A in which follow-up is difficult. (absolute risk).  METHODS OF DATA COLLECTION: PRIMARY DATA SURVEY METHOD SURVEY METHOD is one of the common methods of diagnosing and solving of social problems is that of undertaking surveys. ADVANTAGES DISADVANTAGES Relatively easy to administer Respondents may not feel encouraged to provide accurate, honest answers Can be developed in less time Surveys with closed-ended (compared to other data- questions may have a lower collection methods) validity rate than other question types. MATH Cost-effective, but cost Data errors due to question 019A depends on survey mode non-responses may exist.  SECONDAY DATA: SOURCES OF DATA Publications of Central, state , local government Technical and trade journals Books, Magazines, Newspaper Reports & publications of industry ,bank, stock exchange Reports by research scholars, Universities, economist Public Records MATH 019A FACTORS TO BE CONSIDERED BEFORE USING SECONDARY DATA Reliability of data – Who, when , which methods, at what time etc. Suitability of data – Object ,scope, and nature of original inquiry should be studied, as if the study was with different objective then that data is not suitable for current study Adequacy of data– Level of accuracy, Area differences then data is not adequate for study MATH 019A SELECTION OF PROPER METHOD FOR COLLECTION OF DATA Nature ,Scope and object of inquiry Availability of Funds Time Factor Precision Required MATH 019A  DESIGNING A SURVEY Surveys can take different forms. They can be used to ask only one question or they can ask a series of questions. We can use surveys to test out people’s opinions or to test a hypothesis. When designing a survey, the following steps are useful: 1. Determine the goal of your survey: What question do you want to answer? 2. Identify the sample population: Whom will you interview? 3. Choose an interviewing method: face-to-face MATH interview, phone interview, self-administered paper survey, 019A or internet survey.  DESIGNING A SURVEY 4. Decide what questions you will ask in what order, and how to phrase them. (This is important if there is more than one piece of information you are looking for.) 5. Conduct the interview and collect the information. 6. Analyze the results by making graphs and drawing conclusions. MATH 019A  DESIGNING A SURVEY Example: 1. Martha wants to construct a survey that shows which sports students at her school like to play the most. Step 1: List the goal of the survey Step 2: What population should she interview? Step 3: How should she administer the survey? Step 4: Create a data collection sheet that she can use to record her results MATH 019A  DESIGNING A SURVEY Step 1: GOAL The goal of the survey is to find the answer to the question: “Which sports do students at Martha’s school like to play the most?” Step 2: POPULATION A sample of the population would include a random sample of the student population in Martha’s school. A good strategy would be to randomly select students (using dice or a random number generator) as they walk into an all-school assembly. MATH 019A  DESIGNING A SURVEY Step 3: METHODS Face-to-face interviews are a good choice in this case. Interviews will be easy to conduct since the survey consists of only one question which can be quickly answered and recorded, and asking the question face to face will help eliminate non-response bias. Step 4: DATA MATH 019A  DESIGNING A SURVEY Example: 1. Juan wants to construct a survey that shows how many hours per week the average student at his school works. Step 1: List the goal of the survey Step 2: What population should she interview? Step 3: How should she administer the survey? Step 4: Create a data collection sheet that she can use to record her results MATH 019A  DESIGNING A SURVEY Step 1: GOAL The goal of the survey is to find the answer to the question “How many hours per week do you work?” Step 2: POPULATION Juan suspects that older students might work more hours per week than younger students. He decides that a stratified sample of the student population would be appropriate in this case. The strata are grade levels 9th through 12th. He would need to find out what proportion of the students in his school are in each grade level, and then include the same proportions in his sample. MATH 019A  DESIGNING A SURVEY Step 3: METHODS Face-to-face interviews are a good choice in this case since the survey consists of two short questions which can be quickly answered and recorded. Step 4: DATA MATH 019A  THE BASIS OF CONDUCTING AN EXPERIMENT 1. With an experiment, the researcher is trying to learn something new about the world, an explanation of 'why' something happens. 2. The experiment must maintain internal and external validity, or the results will be useless. 3. When designing an experiment, a researcher must follow all of the steps of the scientific method, from making sure that the hypothesis is valid and testable, to using controls and statistical tests MATH 019A PROBABILITY SAMPLE SPACE The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol S. ELEMENT Each outcome in a sample space is called an element or a member of the sample space. Example #1: Consider the experiment of tossing a die. If we are interested in the number that shows on the top face, the sample space would be MATH S = {1,2,3,4,5,6} 019A PROBABILITY | Sample Spaces & Events Example #2: An experiment consists of flipping a coin and then flipping it a second time if a head occurs. If a tail occurs on the first, flip, then a die is tossed once. To list the elements of the sample space providing the most information, we construct the tree diagram S = {HH, HT, T1, T2, T3, T4, T5, T6} MATH 019A PROBABILITY | Sample Spaces & Events EVENT Is any collection of sample points called subset of a sample space Example #3. An experiment that tosses a coin 3 times. S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} A = event that has at least 1 head B = event that has at most 1 head EA = {HHH, HHT, HTH, HTT, THH, THT, TTH} EB = {HTT, THT, TTH, TTT} MATH 019A PROBABILITY | Basic Rules 1. The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement of A by the symbol A’. 2. The intersection of two events A and B, denoted by the symbol A ∩ B, is the event containing all elements that are common to A and B. 3. Two event A and B are mutually exclusive, or disjoint, if A ∩ B = Ø that is, A and B have no elements in common. 4. The union of events A and B, denoted by A ∪B , is the event containing all the elements that belong to A or B or both. S = {1,2,3,4,5,6,7,8,9,10} A = {1,3,5,8,9} B = {1,4,6,8,10} MATH 019A PROBABILITY | Basic Rules Example #4. If M = {x | 3 < x < 9} and N= {y | 5 < y < 12}, then M U N = {z | 3 < z < 12} VENN DIAGRAMS A∩B= B∩C = A∪C= B’ ∩ A = A∩B∩C= (A ∪ B) ∪ C = MATH 019A Counting Sample Points 1 st Rule: If operations can be performed in n ways, and if for each of these ways a second operation can be performed in n2 ways, then two operations can be performed in n1n2 ways Example#1: How many 4-digit even number can be formed from 0, 1, 2, 5, 6, and 9 if each digit can be used only once? 2 nd Rule: The number of permutations of n objects is n! **A permutation is an arrangement of all or part of a set of objects. Example#2: The number of permutations of letters a,b,c,d. MATH 019A Counting Sample Points 3 rd Rule: The number of permutation of n distinct object taken r at a time is Example #3: In one year, three awards (research, teaching, and service) will be given for a class of 25 graduate students in a statistics department. If each student can receive at most one award, how many possible selections are there? Example #4: A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if (a) there are no restrictions; (b) A will serve only if he is president; MATH (c) B and C will serve together or not at all: 019A (d) D and E will not serve together? Counting Sample Points 4 th Rule: The number of distinct permutations of n things of which n1 are one of a kind, n2 of a second kind, …, nk of nth kind is Example#5: In a college football training session, the defensive coordinator needs to have 10 players standing in a row. Among these 10 players, there are 1 freshman, 2 sophomores, 4 juniors, and 3 seniors, respectively. How many different ways can they be arranged in a row if only their class level will be distinguished? MATH 019A Counting Sample Points 5 th Rule: The number of combinations of n distinct objects taken r at a time is Example#6: A young boy asks his mother to get five Game-BoyTM cartridges from his collection of 10 arcade and 5 sports games. How many ways are there that his mother will get 3 arcade and 2 sports games, respectively? MATH 019A SW#1(Prelim) 1. If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8} B = {1, 3, 5, 7, 9} C = {2, 3, 4, 5} D = {1, 6, 7}, list all the elements of the sets corresponding to the following events: a. A ∪ C b. A ∩ B c. C’ d. (C’∩ D) ∪ B e. (S ∩ C)’ f. A ∩ C ∩ D’ MATH 019A Draw venn diagram for each item. SW#1(Prelim) 2. The resumes of 2 male applicants for a college teaching position in chemistry are placed in the same file as the resumes of 2 female applicants. Two positions become available and the first, at the rank of assistant professor, is filled by selecting 1 of the 4 applicants at random. The second position, at the rank of instructor, is then filled by selecting at random one of the remaining 3 applicants. Using the notation M2F1, for example, to denote the simple event that the first position is filled by the second male applicant and the second position is then filled by the first female applicant, (a) list the elements of a sample space S; (b) list the elements of S corresponding to event A that the position of assistant professor is filled by a male applicant; (c) list the elements of S corresponding to event B that exactly 1 of the 2 positions was filled by a male applicant; (d) list the elements of S corresponding to event C that neither position was filled by a male applicant; (e) list the elements of S corresponding to the event A ∩ B ; (f) list the elements of S corresponding to the event A ∪ C: MATH (g) construct a Venn diagram to illustrate the intersections and unions of the events A, B, 019A and C. SW#2 (Prelim) 1. (a) In how many ways can 6 people be lined up to get on a bus? (b) If 3 specific persons, among 6, insist on following each other, how many ways are possible? (c) If 2 specific persons, among 6, refuse to follow each other, how many ways are possible? 2. (a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6, if each digit can be used only once? (b) How many of these are odd numbers? (c) How many are greater than 330? 3. If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct, (a) In how many different ways can a student check off one answer to each question? (b) In how many different ways can a student check off one answer to each question and get all the answers wrong? MATH 4. Nine people are going on a skiing trip in 3 cars that hold 2, 4 and 5 passengers, 019A respectively. In how many ways is it possible to transport 9 people to the ski lodge, using all cars? Random Variables and Probability Distribution A random variable is a function that associate a real number with each element in the sample space. X denotes a random variable x denotes its cases Types: 1. Discrete – if a sample space contains finite number of possibilities 2. Continuous – if a sample space contains infinite number of possibilities Examples: 3. Number of automobiles accidents per year in Q.C. 4. Length of time to play 15 holes of golf 5. Amount of milk produced yearly by a particular cow MATH 6. Number of eggs laid each month by a hen 019A 7. Length of grain produced per hectare. Discrete Probability Distribution - A discrete random variable assumes each of its values with a certain possibility. - The set of ordered pairs (x, f(x)) is a probability function, probability mass function or probability distribution. 1. f(x) ≥ 0 2. 3. P(X=x) = f(x) Example: 8 computers Find: the probability distribution of the number of 3 out of 8 defectives defectives Randomly get 2 computers MATH 019A Cumulative Distribution Function The cumulative distribution function F(x) of a discrete random variable x with probability distribution f(x) is F(x) = P(X≤x) = , for -∞2, Y ≤ 1) d. P(X+Y = 4) 019A Joint Probability Distribution 3. A sack of fruit contains 3 oranges, 2 apples, and 3 bananas. A random sample of 4 pieces of fruit is selected. Find the joint probability distribution if X = oranges and Y = apples MATH 019A