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MMW 2ND SHIFTING B: 150 + 240 + 200 C: D: E: → B WINS UNIT 5.1 THE MATHEMATICS OF VOTING ➢ the contribution of mathematics in political science began during the French Revolution in the 18th century through Mary Jean Antoine Nicolas de Caritat Mary Jean Antoine Nicolas de Caritat also known as M arquis de Condorcet a French political philosopher and mathematician who raised important ideas related to voting systems VOTING the action or process of indicating choice, opinion, or will on a question, such as the choosing of a candidate, by or as if by some recognized means, such as a ballot the goal of voting is to have a general agreement called c onsensus → CONSENSUS HAS TO BE MADE VOTING METHODS 1. Plurality Method 2. Plurality with Elimination Method 3. Instant Run-off Voting 4. Borda Count Method 5. Pairwise Comparison Method ➢ = 590 = 340 = 360 = 300 the problem with the plurality method is that it may produce a winner, even if the majority didn’t vote for that winner. this is resolved by the plurality with elimination method PLURALITY WITH ELIMINATION METHOD - - each person votes for his or her favorite candidate (or choice) if no candidate receives a majority, then the candidate with the fewest votes is eliminated and another election is held this process continues until a candidate receives a majority of the votes example: A company is planning a company outing next summer. There are three possible locations for the outing: Amanpulo in Palawan, Pandan Island in Mindoro, and BellaRocca Island in Marinduque. The 1000 employees including managers and department heads have to decide based on costs, amenities, and safety. The results of the election are given in the following table. → using the plurality method: PLURALITY METHOD - the one with the most number of votes or with the most first-preference votes wins example: Consider the preference schedule below, in which the voters in a barangay are voting on five different candidates for Brgy. Captain. The candidates are called A, B, C, D, and E here for simplicity. Summary of votes of 1000 employees Amanpulo wins Assuming the result of the new voting, with Pandan Island being eliminated is given below. Where will the company outing be held? → by plurality with elimination, Amanpulo wins ➢ Preference table of voters in a barangay election the problem with this method, aside from the fact that it requires more time and effort, is that it may violate the monotonicity criterion a voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots → eliminate P; all the candidates below the eliminated candidate will be upped by one rank higher INSTANT RUN-OFF VOTING - - - or Ranked-choice method each voter ranks all of the candidates; that is, each voter selects his or her first choice, second choice, third choice, and so on if no candidate receives a majority, then the candidate with the fewest first choice votes is eliminated and those votes are given to the next preferred candidate if a candidate now has a majority of first-choice votes, that candidate is declared the winner. If no candidate receives a majority, this process continues until a candidate receives a majority example: Consider the voting of the 1000 employees to choose the island for their company outing. They were asked to write their 1st choice, 2nd choice and 3rd choice. The results are shown in the following table preference table. a. BORDA COUNT METHOD - - - named after Jean Charles de Borda, who developed the system in 1770 each voter ranks all of the candidates; that is each voter selects his or her first choice, second choice, third choice, and so on if there are k candidates, each candidate receives ➔ k points for each first-choice vote ➔ k – 1 points for each second-choice vote ➔ k – 2 points for each third-choice vote, and so on the candidate with the most total points is declared the winner example: Consider the voting of the 1000 employees to choose the island for their company outing. They were asked to write their 1st choice, 2nd choice and 3rd choice. The results are shown in the following table preference table. use the plurality method to determine the winner Inspecting the table for first-choice row: A: 168 + 202 = 370 B: 215 + 105 = 320 P: 90 220 = 310 → A wins b. Inspecting the table for first-choice row: A: 168 + 202 + 90 = 460 votes B: 215 + 105 + 220 = 540 votes →B wins use the instant runoff to determine the winner Inspecting the table for first-choice row: A: 168 + 202 = 370 B: 215 + 105 = 320 P: 90 220 = 310 → but 370 is not yet the majority a. use Borda Count Method in coming up with a winner Sixth column: B: 220 A: 168 + 202 + 90 = 460 B: 215 + 105 + 220 =540 → B gets 1 point → A wins PAIRWISE COMPARISON METHOD - - - also known as the Copeland’s method named after Arthur Herbert Copeland, who proposed the method in 1951 each voter ranks all of the candidates; that is each voter selects his or her first choice, second choice, third choice, and so on for each possible pairing of candidates (head to head challenge), the candidate with the most votes receives 1 point; if there is a tie, each candidate receives ½ point the candidate who receives the most points is declared as the winner example: Let us consider again the decision of a company to determine the island destination to have their summer outing. ● Because there are k=3 candidates, there must be 3 pairwise comparisons; that is 3! c23 = (3−2)!2! =3 ● Specifically, we investigate A versus B, A versus P, and B versus P A versus B: First column: A: 168 Second column: A: 202 Third column: B: 215 Fourth column: B: 105 Fifth column: A: 90 A versus P: First column: Second column: Third column: Fourth column: Fifth column: Sixth column: A: 168 A: 202 A: 215 P: 105 P: 90 P: 220 A: 168 + 202 + 215 = 585 P: 105 + 90 + 220 = 415 → A gets 1 point B versus P: First column: Second column: Third column: Fourth column: Fifth column: Sixth column: B: 168 P: 202 B: 215 B: 105 P: 90 P: 220 B: 168 + 215 + 105 = 488 P: 105 + 90 + 220 = 512 → P gets 1 point ➢ - - B > A, A > C, but C > B Condorcet Paradox collective preferences can be cyclic, even if the preferences of individual voters are not cyclic transitive property does not apply UNIT 5.2 ● APPORTIONMENT - the act of dividing items between different groups according to some plan, especially to make proportionate distribution in a f air manner STATES the parties having stake in the apportionment may be cities, regions, towns or municipalities SEATS the number of indivisible objects which are assigned in different states may also be the number of firemen, number of teachers, or number of nurses assigned to different cities HAMILTON’S METHOD named after Alexander Hamilton (1792) first method used to apportion the seats in the US Congress adopted to apportion the US House of Representatives every ten years between 1852 and 1900 Hamilton’s method tends to favor larger states also called the largest remainder method Steps in Hamilton’s Method 1. Determine how many people each representative should represent by calculating the standard divisor. d= 2. POPULATION a positive integer n which refers to the number of citizens or residents in a locale the basis of the seats to the states 3. 4. APPORTIONMENT the act of dividing people among places METHODS OF APPORTIONMENT ● ● ● ● ● Hamilton’s Method Jefferson’s Method Adam’s Method Webster’s Method Hill-Huntington’s Method total population no. of representatives Divide each state’s population by the divisor to determine how many representatives it should have, aka standard quota. q= STANDARD DIVISOR (d) ratio of the total population to the total number of seats to be allocated STANDARD QUOTA (q) ratio of a state’s population to the standard divisor Lowndes’ Method state′s population d Cut off all the decimal parts of all the quotas. Add up the remaining whole numbers. Get the lower quota of each state, which is the rounded down value of the standard quota (add all the l ower quota). If the sum of the lower quota is less than the total number of representatives, the remaining number of seats are assigned to the state whose decimal value of the standard quota i s the highest. example/s: ● Suppose the Phil. Constitution was amended and the 250 members of the House of Representatives will be apportioned among regions. Using the data in the 2015 National Population Census per Region as shown in the table below, determine the apportionment using Hamilton’s method. 4. If that the total from Step 3 was less than the total number of representatives, assign the remaining representatives, one each, to the states whose decimal parts of the quota were largest, until the desired total is reached. ● Suppose 100 doctors have to be assigned to different parts of Metro Manila during New Year’s eve due to firecracker reported injuries that may happen. Allocations will depend on the number of reported injuries of the previous years. Below is the report of WHO of reported firecracker injuries from 2010 to 2014. Solution: 1. Compute for d total population d = no. of representatives d= 100,981,437 250 d = 403, 925.75 2. Compute for the standard quota of each state state′s population q= d ex: q = 12,877,253 403,925.75 = 31.880 Solution: 1. Compute for d 2. 3. Cut of all the decimal parts of all the quotas. Add up the remaining whole numbers. 3. 4. Compute for the standard quota of each state Lower quota Final quota ➢ The problem with this method is that increasing the number of seats may cause the party to lose a seat → called Alabama Paradox (happened in the state of Alabama in 1980) JEFFERSON’S METHOD proposed by Thomas Jefferson (1743-1826) first used in US Congress in 1792 until 1840. Jefferson’s method tends to favor larger states first steps are the same with Hamilton’s method example/s: ● The House of Representatives of a certain country needs 41 new members to divide among three states A, B, and C. The population data of the 3 states are as follows: Solution: 1. Compute for d total population d = no. of representatives d= 2. 897,934 41 d = 21, 900.83 Compute for the standard quota of each state q = state′s population d Steps in Jefferson’s Method 1. Determine how many people each representative should represent by calculating the standard divisor. d= 2. total population no. of representatives Divide each state’s population by the divisor to determine how many representatives it should have, aka standard quota. q= 3. 4. 3. Cut of all the decimal parts of all the quotas. Add up the remaining whole numbers. 4. Reduce the standard divisor (21,900.83) until the sum of lower quota is 41. state′s population d Cut off all the decimal parts of all the quotas. Add up the remaining whole numbers. Get the lower quota of each state, which is the rounded down value of the standard quota (add all the lower quota). If the total from Step 3 was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. Continue doing this until the total in Step 3 is equal to the total number of representatives. ➔ The divisor we end up using is called the modified divisor or adjusted divisor. Trying modified divisor as 21,000 ● Use the Jefferson method to determine how many doctors have to be assigned at different parts of Metro Manila. ➔ Using Hamilton’s method, with standard divisor of 28.10 Reduce the divisor. Trying divisor as 27. ➔ ADAM’S METHOD Proposed by John Quincy Adams (1767-1848) Adams proposed using a modified divisor greater than the standard divisor, and he rounded his modified quotas quota up to the upper quota. It tends to favor smaller states. Steps in Adam’s Method ➢ Opposite of Jefferson’s method ➢ Instead of rounding down to get the standard quota, r ound up. ➢ If the sum of the upper quotas does not equal the total number of seats to be apportioned, choose a modified divisor greater than the standard divisor and calculate the modified quotas and upper modified quotas. example/s: ● The House of Representatives of a certain country needs 41 new members to divide among three states A, B, and C. The population data of the 3 states are as follows: MODULO ARITHMETIC - - - - - Related to that of the remainder in division Operation of finding the remainder is sometimes referred to as modulo operation Ex: clock → always on 1 to 12 Dividing an integer Z by 5 will have the remainders {0, 1, 2, 3, 4) 25/5 = 5 + 0/5 → remainder 24/5 = 4 + ⅘ → remainder Thus we define Zn as the set of integers from 0, 1, 2, …, n-1 modulo n, i.e. Zn = {0, 1, 2, …,n-1} Z3 = {0, 1, 2} modulo 3 Z5 = {0, 1, 2, 3, 4, 5} modulo 5 Z8 = {0, 1, 2, 3, 4, 5, 6, 7, 8} modulo 8 Zn has exactly n non-negative integers In Zn, modulo is simply the remainder r when an integer a ∑ Z is simply divided by n a/n has a remainder r < n PERFORM THE FOLLOWING OPERATIONS (REMAINDER) 1. In Z8 what is 4+9 Answer is 5 5 is the remainder when 13 is divided by 8 4+9 = 13 13/8 = 5 → remainder 2. In Z8, what is 15 + 21 15 + 21 = 36 36/8 = 32 +4/8 = 4 3. In Z8 what is 106+102 106 + 102 = 208 Divisible by 8 → 0 is the remainder 4. In Z8, what is 22-3 22-3 = 19 19/8 = 3 5. In Z8, what is 3-20 3 - 20 = -17 -17/8 = -16 - 1 / 8 = 7 8 was added to -1 6. In Z8, what is 11 x 5 48 + 7 = 55 Answer is 7 THE MODULO TABLE Zn is closed under the binary operations of addition and multiplication of integers modulo n Any 2 numbers in the set, the answer are also in the set Examples for addition a. b. In Z3 = {0, 1, 2} → 3 integers only starting from 0 to 2 (limitations) 2 + 1 = 0 remainder 2 + 2 = 1 remainder Example for multiplication 2 x 2 = 4 → remainder 1 THE MODULO CONGRUENCE 2 integers a and b are said to be congruent modulo n, where n is a natural number if a-b/n is an integer Example a = 13 b=4 n=9 - a is congruent to b modulo n Additive Inverse In Zn , two numbers a and b are additive inverse of each other if a + b ≡ 0 mod n - TRUE OR FALSE 1. 29 ≡ 8 mod 3 3 │29-8 3 │21 TRUE 2. 15 ≡ 4 mod 6 6 │15-4 6 │11 FALSE 3. 37 ≡ 9 mod 7 7 │37-9 7 │28 TRUE 4. 115 ≡ 10 mod 3 3 │115-10 3 │105 TRUE 5. 84 ≡ 3 mod 9 9 │84-3 9 │81 TRUE EQUATIONS INVOLVING MODULO CONGRUENCE 1. Example: Find the additive inverse of 7 in mod 16 Answer: 9 because 7 + (9) = 0 mod 16 Find the additive inverse of 2 in mod 12 Answer: 10 Find the additive inverse of 5 in Z7 5+ (2) ≡ 0 mod 7 Answer: 2 Multiplicative Inverse In Zn , two numbers a and b are multiplicative inverse of each other if ab ≡ 1 mod n. Example: Find the multiplicative inverse of 5 in Z7 Answer: 3 because (5)(3) = 1 mod 7 Find the multiplicative inverse of 3 in Z10 Answer: 7 because (3)(7) = 1 mod 10 Find the multiplicative inverse of 4 in mod 6 Answer: none No multiple of 6 can be divided by 4 to have a remainder of 1 Every number has an additive inverse but not necessarily a multiplicative inverse. 2. APPLICATIONS OF MODULO ARITHMETIC Zeller’s congruence Julius Christian Johannes Zeller (June 24 1822 - May 31, 1899) X - any real number N - whole number lang EX. Floor Function of ½ = 0.5 → floor func is 0 - 3/2= 1.5 → floor func is 1 PI → 3 - Thus when we divide a certain number by another number, the floor function is the quotient excluding the remainder Kumbaga parating round down Cautions: ● [x] is the rounded down value of x. ● Jan and Feb are the 13th and 14th months of the previous year. ● January 20, 2011 is like the 13th month of 2010 - EXAMPLE - Using Gregorian Calendar On what day is University of Santo Tomas founded? → April 28, 1611 On what day is January 15, 2019 Modulo Function Two integers a and b are said to be congruent modulo n, where n is natural number a - b is divisible by n. In case we write: DETERMINE THE DAY YOU WERE BORN VIDEO ON ZELLER'S CONGRUENCE Floor Function - For any real number x, - No need for the decimal point for the floor function of x → only care about the quotient not the remainder - The number n is called the modulus. The statement a=b mod n is called the congruence If b mod n only means the remainder when b is divided by n. Sometimes written as Rem (b, n) = b mod n 5 mod 2 = 1 Floor Function = quotient only Modulo Function = r emainder only Julian Calendar 365.25 days in a year (leap year is every 4 years Proposed by Julius Cesar Gregorian Calendar calendar we use today 365.2422 in a year Leap year occurs if the year is divisible by 4 except for years divisible by 100 (i.e. 1700, 1800). But if the year is divisible by 400 (like year 2000) its a leap year The calendar repeats itself every 400 years Difference of gregorian calendar and julian calendar is just 0.0078 → but has a very big effect if we use diff ones Introduced by Pope Gregory XII (June 7 1502- April 10 1585) Accepted in 1752 onwards Example: If today is June 29, 2020 in Gregorian Calendar, in Julian Calendar it is stil June 16, 2020 → 13 days The discrepancy of both both calendars, 0.0078 → is actually more than 10 mins Kaya Ganyan cause Leap Year John Harton Conway December 26, 1937 - April 11, 2020 Doomsday Method INTRODUCTORY CONCEPTS IN STATISTICS STATISTICS - comes from the Latin word “statisticum collegium”, meaning “council of state”; Italian word “statista”, meaning “statesman” or “politician” or “political state” - a discipline concerned with the analysis of data and decision-making based upon data - involves collecting, organizing, summarizing, interpreting and presenting data - why is it being learned: ● in medicine, it is used to determine the efficacy of a drug ● in business and economics, it is used to forecast sales, stocks, etc. ● in everyday life, it is used to see weather condition forecasts ● in marketing, it is used to study consumer behavior ● in psychology, it is used to study human behavior ● in sports, it is used to summarize the performance of athletes - two categories of statistics: ● Descriptive Statistics ● Inferential Statistics DESCRIPTIVE STATISTICS - involves the method of organizing, summarizing and presenting the data in an informative form - computing for the average or presenting data through graphs - totality of methods and treatments employed in the collection, description and analysis of numerical data - its purpose is to tell something about a particular group of observation - limited to data set INFERENTIAL STATISTICS - also called statistical inference or inductive statistics - from the data, we go beyond the presentation; from the analysis and presentation of the available data, we predict or come up with an intelligent guess on what’s next upon careful investigation - it is the logical process from sample analysis to a generalization or a conclusion about a population - involves using information from a sample to draw conclusions about the population ➔ Population - all the members of the subject of interest → result: p arameter ➔ Sample - selected members of the subject of interest; not all possible respondents under a certain study are always available for a certain procedure of statistical treatment → result: s tatistics ➔ statistics is an estimate of the parameter ➔ Variables - the one being measured; a characteristic of objects, people or events that can take of different values, can vary in quantity ➢ Qualitative variables ➢ Quantitative variables ➔ Constants - held fixed; a characteristic of objects or groups, people or events that does not vary Sources of Data ● Primary Data comes from one’s personal encounter; data collected personally ● Secondary Data comes from other resources, may come from journals, books, etc. Given the following scenarios, identify: ● population and sample ● parameter and statistics ● variable and constant Scenario 1: ● When all University of Sleep freshmen students were asked, it was found that, on the average, they spend 3.7 hours of sleep per day during exam week. ● But from a 30 randomly selected University of Sleep freshmen students, it was found to be 3.6 hours per day. ➢ population: all University of Sleep freshmen students ➢ samples: 30 randomly selected University of Sleep freshmen students ➢ ➢ ➢ ➢ parameter: 3.7 hours statistics: 3.6 hours variable: number of hours of sleep constant: year level of University of Sleep students Scenario 2: ● From 100 randomly selected residents of northern Alkovia, it was found that 13% of them had Dengue fever in 2019. But according to the Institute of Health’s Epidemiology Center (IHEC) 11.9% of all residents of northern Alkovia had Dengue fever in 2019. ➢ population: all residents of northern Alkovia ➢ samples: 100 randomly selected residents of northern Alkovia ➢ parameter: 11.9% ➢ statistics: 13% ➢ variable: occurrence of Dengue fever ➢ constant: disease (Dengue), year Scenario 3: ● 5% of Asian men suffer from red-green color blindness. From 250 randomly selected men in Southeast Asia, it was found that 3% suffer from this type of color blindness. ➢ population: all Asian men ➢ samples: 250 randomly selected men in Southeast Asia ➢ parameter: 5% ➢ statistics: 3% ➢ variable: occurrence of color blindness ➢ constant: type of color blindness, race DATA PRESENTATION - results of data may be presented through: ● TEXTUAL results are expressed in declarative form this type of presentation is advised when you are trying to give an overview of a single group, like reporting their age, sex, and others examples: → The mean age of the respondents is 28.7 years. → Twenty (40%) of the customers are male. ● - TABULAR results are displayed in rows and columns this method of presentation is advised when giving a summary or repetition of categories example: Male (n=20) Female (n=30) 21.1 20.8 No. of smokers 3 (15.0%) 1 (3.3%) Monthly income 21.2 21.7 Age (years) * values expressed as mean, or counts (%) ● - GRAPHICAL results are presented in diagrams Types of Graphs: Bar graphs used to compare average, counts, or percentage of different groups Line graphs used to observe trends and to compare trends or observe gaps between groups across time; best used when the horizontal axis is an element of time like data expressed in daily, weekly, monthly, or yearly time intervals Pie graphs used to express parts of a whole; recommended to have a maximum of 5 to 7 sectors only Scatterplots used to describe relationship of quantitative variables Statistical maps used to present information with respect to geographical location Pictogram series of repeated resembling icons to visualize simple data Population Pyramid also known as the age-gender pyramid; shows the the two distribution of various age groups in a population according to gender Boxplots shows the five-number summary (quartiles) of a given set of data Violin plots similar to box plot, with the addition of putting densities on each side - also called the o utcome variable LEVELS OF DATA MEASUREMENT - In statistics, it is important that you can classify the variables correctly. Statistical treatments depend on the classification of variables. ➔ 1. example/s: Are you a smoker? ❏ Yes ❏ No = nominal How frequently do you smoke? ❏ Always (>19 sticks/day) ❏ Oftentimes (10-19 sticks/day) ❏ Sometimes (1-9 sticks/day) ❏ Rarely (at most 1 stick/week) ❏ Never = ordinal How many sticks of cigarette do you smoke in a day? Specify: _______ sticks = ratio VARIABLES ● Quantitative variables ● Qualitative variables ● Independent variable ● Dependent variable Quantitative variables - expressed in numerical form - reveals the pattern of a certain data - continuous or discrete variable ➔ Continuous variable every fraction, decimal is considered (ex: height, temperature) ➔ Discrete variable whole numbers are the only ones to be considered (ex: no. of people) - may be categorized into two levels: ➔ Interval ➔ Ratio Qualitative variables - also known as the categorical variables - non-numeric data; expressed in textual form - provides the answer to the question “why?” - may be further classified into two levels: ➔ Nominal ➔ Ordinal Independent variable - variable controlled by the experimenter or the researcher and expected to have an effect on the behavior of the subject - also called the explanatory variable Dependent variable - some measure of the behavior of subject and expected to be influenced by the independent variable 2. 3. NOMINAL LEVEL OF MEASUREMENT categorical variables in which responses have no order all responses are in the same level example/s: race, color, sex, disease occurrence, subscription Keyword: Category It is used to differentiate classes or category for purely classification or identification purposes SSS number Ex. Survey in Tapsilogan about the taste sa isang menu → pakaplog Commonly in survey questionnaires there is a demographic profile Mutually exclusive and exhaustive → for every category there should only one answer Nominal data is qualitative variable ORDINAL LEVEL OF MEASUREMENT categorical variables in which responses have order responses may be ranked or arranged ascendingly or descendingly example/s: position in organization, BMI interpretation, smoking frequency, Likert scale Keyword: RANK, qualitative Somewhat stronger than the nominal level, because in an original data, aside from categorization, one data can be identified as greater or lesser than the other data However there is no numerical specification of the difference between two data. We only know that one data is lesser or higher than the other data Example: a teacher asks a question to the students “what is nominal data” Student ans: the one you discussed a while ago (GOOD) Another question was asked, what is ordinal data? - student answered: the one that we are discussing now (VERY GOOD) Teacher Asked again “what is statistics” student answered It is a branch of mathematics that examines and investigates ways and processes to - analyze the data gathered… (EXCELLENT) We only know the ranking however we don't know the difference between the two (ex the diff of good and very good → numerically) INTERVAL LEVEL OF MEASUREMENT has r elative zero → zero response does not indicate absence → example/s: temperature in Celsius or Fahrenheit (The temperature in A (40°C) is 30°C more as compared to B (10°C)) responses are compared through their difference Higher than nominal and ordinal Used to classify order and differentiate between classes or categories in terms of degrees of differences Either discrete or continuous, In other words we are dealing with numerical values Calculation Example: Temp reading (C and F = numerically diff), calendar, scores of students, clock RATIO LEVEL OF MEASUREMENT has a bsolute zero → zero response indicates absence → example/s: temperature in Kelvin, no. of books, liters of alcohol, height can be expressed as factor of one response over the other → example/s: amount of money (Php 100 is 5 times the value of Php 20) responses are compared through their quotient Pinakamataas na level of measurement Only one diff with interval which is absolute zero or it has a true zero point Example: Weight, height → you can not have a measurement that is below zero, unlike interval data pwede like -10 C, However if Kelvin, ratio na siya, cause in Kelvin no zero → zero reading is considered a null point Almost all that you can use in ratio can be used in interval Sometimes if the data is numeric it is called scale data - SAMPLING TECHNIQUE RANDOM / PROBABILITY Simple Systematic Stratified Cluster NON RANDOM / NON PROBABILITY Purposive/ Judgement Convenience Quota Snowball Voluntary RANDOM SAMPLING 1. Simple - process of selecting a certain number of sample size in a population via random numbers or lottery Example: Lotto (in 6/42) - the machine will pick six numbers out of 42 6 samples in 42 population Classroom Setting Names in the jar for recitation 2. Systematic - process of selecting an nth element in the population until the desired number is obtained Example: Need ng 5 students out of 40 students in the class POPULATION / SAMPLE 40 / 5 = 8 From 1 - 8 you randomly select a number (lets say nabunot ay 2 → so lahat ng number 2 selected) What if may remainder → then you can roundoff nalang Stratified Sampling - is the process if subdividing the population into subgroups or strata and drawing members at random from each subgroup or stratum In accordance with the proportion of each stratum Strata- g roup with commonality Example: Surveying 200 reviewees out of 50 reviewees in review center Out of 50k the breakdown of reviewees are Sample Size 3. 50% of 200 16% of 200 And so on Progra m # of reviewee % Sample Size LET 25 000 50% 100 CSE 8 000 16% 32 NLE 7 000 14 % 28 CLE 10 000 20% 40 Total 50 000 100% 200 Cluster- s omewhat related with stratified sampling, in a sense where both are subdividing the population Process of selecting clusters from a population which is very large or widely spread out over a wide geographical area Subgroups = “cluster” Paghati ng cluster in accordance with geographical area Ex: Philippines- widely dispersed so subdivided into regions Classroom - by row *In reality usually combination of techniques ginagamit 4. NON RANDOM SAMPLING Is called nonrandom if the participant do not have equal chance in becoming part of the study Biased Sampling procedure or technique whose sample are selected in a deliberate manner with little or no attention to randomization 1. Purposive/ Judgemental sampling or subjective sampling - process of selecting based on judgement to select as a sample which the researcher believed, based on prior information, will provide the data they need. → subjective Ex: interview Convenience- t he researcher chooses members merely based on proximity, and does not consider whether they represent the entire population or not → willing & available Ex: research- where you ask your friends, parents and stuff 3. Quota- applied when an investigator survey or collect information from an assign number or quota of individual from one of several sample units fulfilling a certain prescribed criteria or belonging to one stratum Ex: you will conduct a survey to people with diff gender → to make it easier you get a sample that is based on a certain criteria Example: honor students lang interview 4. Snowball- t echnique in which one or more members of a population are located and used to lead the researcher to other members of the population Aka referral method, referral sampling, chain referral sampling Ask referrelas from others with the same interest → like a snowball na lumalaki habang gumugulong 5. Voluntary- technique when samples are composed of respondents who are self select into the survey or study. Most of the time the respondents have a strong interest in the study Ex: national election 2. Cochran sample sizehttp://www.raosoft.com/samplesize.html