Mathematics In The Modern World 1st Semester 2024-2025 PDF

MATHEMATICS IN THE MODERN WORLD 1st Semester┃ A.Y. 2024 - 2025 ┃Midterms┃School of Dental Medicine LESSON 1 FUNCTIONS AND ITS OPERATIONS RELATION IN REAL WORLD RELATION ➔ Money won after buying a lotto...

MATHEMATICS IN THE MODERN WORLD 1st Semester┃ A.Y. 2024 - 2025 ┃Midterms┃School of Dental Medicine LESSON 1 FUNCTIONS AND ITS OPERATIONS RELATION IN REAL WORLD RELATION ➔ Money won after buying a lotto ticket. ➔ A relation may have more than 1 ➔ The high temperature on July 1st in output for any given input. New York City. (Depends on the ➔ The set whose elements are the first year.) coordinates in the ordered pairs is ➔ How many words your friend uses the domain of the relation. when answering, “How are you?” ➔ The set whose elements are the ➔ The number of calories in a fast second coordinates is the range. food hamburger. ➔ Places you can drive to with 1 gallon Examples: left in your gas tank. ➔ A = { (1, 1), (2, 3), (2, 4) } ➔ Domain: { (1, 2) } FUNCTION ➔ Range: { (1, 3, 4) } ➔ It is a relation in which each ➔ B = { (5,2), (7,6), (7,4)} element in the domain is paired ➔ Domain: {2,3,4} with exactly one element in the ➔ Range: {5,7} range KEY CONCEPTS: ➔ A function can have no more than one output for any given input. ➔ The domain is the set of all first coordinates (inputs) of the ordered ONE-IS-TO-ONE CORRESPONDENCE pairs. ➔ The range is the set of all second coordinates (outputs) of the ordered pairs. ONE-IS-TO-MANY CORRESPONDENCE ➔ A relation can have one input associated within multiple outputs. ➔ Example: A = { (1,2), (2,3), (3,4)} Domain: {1,2,3} Range: {2,3,4} B = { (5,2), (7,6), (9,6)} Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Domain: {5,7,9} FUNCTION NOTATION Range: {2,6} ➔ The notation f(x) defines a function named f. This is read as “y is a ➔ Each input has exactly one unique function of x.” The letter x represents output the input value, or independent variable. The letter y is replaced by MANY-IS-TO-ONE CORRESPONDENCE f(x) and represents the output value, or dependent variable. Example: Evaluate f(x) = x + 5 when x = 4 Solution: f(4) = 4 + 5 f(4) = 9 Evaluate f(x) = x³ - 4x² + 3x + 10 for x = -2 Solution: f(-2) = (-2)³ - 4(-2)² + 3(-2) + 10 f(-2) = -8 - 16 - 6 + 10 f(-2) = -30 + 10 ➔ A function can have no more than 1 f(-2) = -20 output for any given input. ➔ Multiple inputs can have the same UNARY OPERATION output, but each input still has only one output. ➔ It involves only one value or accepts one value or operand REAL WORLD EXAMPLES OF FUNCTION ➔ For example: -5 𝛑 cos 40° tan 𝛑/3 Examples: BINARY OPERATORS ➔ The amount of sodas that come out ➔ It can act on two operands “+” and of a vending machine, depending “-” on how much money you insert. ➔ It takes two values and includes the ➔ The amount of carbon left in a fossil operations of addition, subtraction, after so many years. multiplication, division, and ➔ The velocity of an object in freefall exponentiation. after being dropped for so many seconds, excluding air resistance. PROPERTIES OF TWO BINARY OPERATORS ➔ The height of a person at a given time in their life. ➔ The intensity of a light as you slide Closure of The product and the sum its dimmer switch. Binary of any two real numbers Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Operations is also a real number = 6x - 10y Example: x + y = R and x y=R Identity An element of the set of 5+3 = 8 ; 5 3 = 15 Elements of real numbers is an Binary identity element for Commutativit Addition and Functions addition/multiplication if y of Binary multiplication of any two x+e=e+x=x and x e=e Operations real numbers is x=x commutative as seen in This means that the their mathematical identity is the number that symbols you add to any real Example: x+y=y and x number and the result will y=y x be the same real number 4+5=5+4 4 5=5 4 Example: 5+0=0+5=5 9=9 20=20 50 1 1 50=50 Therefore the identity Associativity Given any three real element e in the above of Binary numbers you may take definition is zero for Operations any two and perform addition and one for addition or multiplication multiplication as the case maybe and you will end with the Inverse of Addition: Additive inverse same number Binary For any number x, the Example: Function additive inverse is -x, and (x + y) + z = x + (y + z) Example: ○ (4 + 5) + 7 = x + (-x) = 0. 4 + (5 + 7) x + (-x) = -x + x = 0 ○ 9 + 7 = 4 + 12 4+(-4) = -4 + 4 = 0 ○ 16 = 16 Multiplication: Reciprocal (x y) z = x (y z) For any non-zero number ○ (4 5) 7 = 4 (5 7) x, the inverse is 1/x, and x × ○ 20 7 = 4 35 (1/x) = 1. ○ 140 = 140 Example: x 1/x = 1/x x = 1, x ≠ 0 Distributivity Applies when of Binary multiplication is Operations performed on a group of OPERATIONS ON FUNCTIONS two numbers added or Sum of Functions subtracted together Example: z(x ± y) = zx ± zy ➔ If f(x) and g(x) are two functions, 2(3x ± 5y) = 6x ± 10y their sum is given by: = 6x + 10y ➔ (f + g)(x) = f(x) + g(x) Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Example 1. f(x) = x − 2 g(x) = 2x² + 5x − 3 Given: Solution: f(x) = 3x + 5 (f − g)(x) = (x − 2) − (2x² + 5x − 3) g(x) = 5x + 11 (f − g)(x) = −2x² − 4x + 1 Solution: (f + g)(x) = (3x + 5) + (5x + 11) Example 4. (f + g)(x) = 8x + 16 Given: Example 2. f(x) = x² + 6x + 8 g(x) = x² − 16 Given: Solution: f(x) = −5x + 3 (f − g)(x) = (x² + 6x + 8) − (x² − 16) g(x) = 11x − 5 (f − g)(x) = 6x + 24 Solution: (f + g)(x) = (−5x + 3) + (11x − 5) Product of Functions (f + g)(x) = 6x − 2 ➔ The product of two functions is Difference of Functions defined as: (f ⋅ g)(x) = f(x) ⋅ g(x) ➔ The difference between two Example 1. functions f(x) and g(x) is: (f − g)(x) = f(x) − g(x) Given: f(x) = 3x + 5 g(x) = 5x + 11 Example 1. Solution: (f ⋅ g)(x) = (3x + 5)⋅(5x + 11) Given: (f ⋅ g)(x) = 15x² + 58x + 55 f(x) = 3x + 5 g(x) = 5x + 11 Example 2. Solution: (f − g)(x) = (3x + 5) − (5x + 11) Given: (f − g)(x) = −2x−6 f(x) = x − 2 g(x) = 2x² + 5x − 3 Example 2. Solution: (f ⋅ g)(x) = (x − 2) ⋅ (2x² + 5x − 3) Given: (f ⋅ g)(x) = 2x³ + x² − 13x + 6 f(x) = −5x + 3 g(x) = 11x − 5 Example 3. Solution: (f − g)(x) = (−5x + 3) − (11x − 5) Given: (f − g)(x) = −16x + 8 f(x) = x² + 6x + 8 g(x) = x² − 16 Example 3. Solution: (f ⋅ g)(x) = (x² + 6x + 8) ⋅ (x² − 16) Given: (f ⋅ g)(x) = x⁴ + 6x³ − 8x² − 96x − 128 Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Quotient of Functions POLYA’S FOUR STEP PROBLEM SOLVING STRATEGY ➔ The quotient of two functions is given by: , 𝑓 /( 𝑔 )(𝑥) = 𝑓(𝑥) / 𝑔(𝑥) , Step 1. Understand the Problem where 𝑔(𝑥) ≠ 0 ➔ What is the goal? Example 1. ➔ What is being asked? ➔ What is the condition? ➔ What sort of a problem is it? ➔ What is known or unknown? ➔ Is there enough information? ➔ Can you draw a figure to illustrate the problem? ➔ Is there a way to restate the Example 2. problem in your own words? Step 2. Devise a plan ➔ Act it out ➔ Be systematic ➔ Work backwards ➔ Consider special cases Example 3. ➔ Eliminate possibilities ➔ Perform an experiment ➔ Draw a picture/diagram ➔ Make a list or table/chart ➔ Use a variable, such as x ➔ Look for a formula/formulas ➔ Write an equation or model ➔ Look for a pattern/patterns PROBLEM SOLVING STRATEGIES ➔ Use direct or indirect reasoning GEORGE POLYA (1887-1985) ➔ Solve a simple version of the problem ➔ The father of problem solving ➔ Guess and check your answer (trial ➔ A mathematics educator who and error) strongly believed that skill of problem solving can be taught Step 3. Carry out the plan ➔ Be patient ➔ Work carefully ➔ Modify the plan or try a new plan ➔ Keep trying until something works Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ➔ Implement the strategy/strategies Make a table In this strategy, data or from step 2 or an information are ➔ Try another strategy if the first one organized list organized by listing them isn’t working or recording them ➔ Keep a complete and accurate systematically in tables. record of your work The data are then ➔ Be determined and don’t get analyzed to discover discouraged if the plan does not relationships and work immediately patterns, allowing for generalizations or Step 4. Review the solution solutions to emerge regarding the problem at ➔ Look for an easier solution hand. ➔ Does the answer make sense? ➔ Check the results in the original Guess and To use the guess-check problem Check strategy, one follows a ➔ Interpret the solution with the facts series of steps. First, make of the problem a logical guess at the ➔ Recheck any computations answer, which helps the involved in the solution student learn more about ➔ Can the solution be extended to a the problem. After making more general case? the guess, check it ➔ Ensure that all the conditions carefully, ensuring that related to the problem are met. computations are ➔ Determine whether there is another accurate to avoid wasting method of finding the solution. time on unnecessary ➔ Ensure the consistency of the additional guesses. If the solution in the context of the guess is incorrect, use the problem. information obtained to make another, adjusting PROBLEM SOLVING STRATEGIES AND whether the next guess is RECREATION smaller or larger based on estimation skills and Draw a It is a process of logical reasoning. The diagram, translating a problem process continues until picture, or scenario into a drawing or the correct answer is model using of a picture or found. To minimize the model helps learners number of trials, it's visualize the problem important to use logical situation which may reasoning and refine successfully lead to the guesses as you proceed. desired solution Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Act it out! Acting out the Problem is 4 contestants a strategy in which people before. At the physically act out what is 10-minute mark, happening in a word half of the problem. You may use remaining dancers people or objects exactly were eliminated, so as described, or use items there were 8 to represent the people or contestants before. objects. By using this At the 3-minute strategy, individuals can mark, half of the visualize and simulate the original actions described in the contestants were problem, making it easier eliminated, which to understand and solve. means there were 16 contestants in Work The “Work Backward” total at the start of backwards method is ideal for the competition. problems where a series Therefore, there of operations is were 16 dancers at performed on an the beginning. unknown number, and you are given the result. To use this method, start LESSON 2 REASONING with the final result and INDUCTIVE REASONING apply the operations in reverse order until you find ➔ It is a type of reasoning that uses the starting number. specific examples to reach a general conclusion. ★ In the given ➔ The conclusion formed through problem, there was inductive reasoning is called a 1 winner at the end conjecture. of the competition. ➔ A conjecture is an idea that may or At the 20-minute may not be correct. mark, half of the ★ Example: Use inductive reasoning remaining dancers to predict the next number of the were eliminated, pattern below. meaning there ★ 5, 10, 15, 20, 25, _____. were 2 contestants ★ 1, 4, 9, 16, 25, _____. before this. At the 15-minute mark, Example: Use the data in the table and half again were answer the following questions. eliminated, leaving Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD 2. The product of two positive Earthquake Max. Tsunami numbers is greater than either Magnitude height (meters) number. 7.5 5 ➔ Answers: 1. Consider the two numbers: a = -2 7.6 9 and b = 1 The sum: -2 + 1 = -1 However, -1 is not greater than 7.7 13 either -2 or 1. Thus, this serves as a counterexample where the sum of 7.8 17 two numbers is not greater than both. Therefore, the conjecture is 8.0 21 false. 8.1 25 2. Consider the numbers a = 0.5 and b = 0.5 The product: 0.5 × 0.5 = 0.25 0.25 is not greater than either 0.5. Question: Thus, this serves as a ➔ If the earthquake magnitude is 8.5, counterexample where the how high (in meters) can the product of two positive numbers is tsunami be? not greater than both. Therefore, ➔ Can a tsunami occur when the the conjecture is false. earthquake magnitude is less than DEDUCTIVE REASONING 7? Explain your answer. ➔ It is a type of reasoning that uses COUNTEREXAMPLE general procedures and principles ➔ Note: Not all conjectures turn out to to reach a conclusion. be true. ➔ It is the process of reaching a You can prove that a conjecture is general conclusion by applying false by finding one general assumptions, procedures, counterexample. or principles. ➔ A counterexample to a conjecture ➔ Example: Use deductive reasoning is an example for which the to make a conjecture. conjecture is incorrect. ➔ Consider the following procedure: ➔ A counterexample is a special kind ★ Pick a number. of example that disproves a ★ Multiply the number by 10. statement or proposition. ★ Add 8 to the product. ➔ Example: Find one ★ Divide the sum by 2, and subtract counterexample to show that each 4 from the quotient. conjecture is false. SOME GENERALIZED PRINCIPLES 1. The sum of two numbers is greater than either number. ➔ PEMDAS ➔ Property of Real Numbers Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ➔ Property of Inequality 2x = 2 Closure Property ➔ Postulates ➔ Theorems 2 x= 2 Division Property of 2 2 Equality Properties of Equality (Some Generalized Principles) x=1 Closure Property Addition Property of If a = b, then a + c = ➔ Solve for the values of x using and Equality b+c supply the properties shown in the equation. Subtraction Property If a = b, then a - c = b of Equality -c Statements Reasons Multiplication If a = b, then ac = bc 4 (3x - 8) + 5 = x -5 Given Property of Equality 12x - 32 + 5 = x -5 Subtraction Property Division Property of If a = b, and c ≠ 0, of Equality Equality then 𝑎 𝑐 = 𝑏 𝑐 11x - 27 = -5 Subtraction Property Reflexive Property of a=a of Equality Equality 11x - 27 + 27 = -5 + 27 Subtraction Property Symmetric Property If a = b, then b = a of Equality of Equality 11x = 22 Closure Property Transitive Property If a = b and b = c, of Equality then a = c 11 x= 22 Division Property of 11 11 Equality Substitution Property If a = b, then b can of Equality be substituted for a x=2 Closure Property in any expression LOGIC PUZZLE EXAMPLES OF DEDUCTIVE REASONING ➔ Logic puzzles can be solved by ➔ Solve for the values of x using and using deductive reasoning and a supply the properties shown in the chart that enables us to display the equation. given information in a visual manner. Statements Reasons ➔ Example: Each one – Ann, Enya, Alvin, and Johnny have different 2x + 6 = 8 Given favorite colors among red, blue, green, and orange. No person’s 2x + 6-6 = 8-6 Subtraction Property of Equality name contains the same number of letters as his/her favorite color. Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Alvin and the boy who likes blue Alvin x x x ○ live in different parts of town. Red is the favorite color of one of the girls. Johnny x ○ x x What is each person’s favorite color? ➔ To find each person’s favorite color, INDUCTIVE we need to look for some clues: ➔ Reaching conclusions based on a 1. No person’s name contains the series of observations. same number of letters as his/her ➔ Conjecture may or may not be favorite color. valid or uncertain. ➔ Example: Emma enjoyed reading Red Blue Green Orange the novel Finders Keepers by Ann Stephen King, so she will enjoy x reading her next novel Enya x DEDUCTIVE Alvin x ➔ Reaching conclusions based on Johnny x previously known facts. ➔ Conjectures are correct and valid or certain. 2. Alvin and the boy who likes blue live in ➔ Example: All pentagons have different parts of town. exactly five sides. Figure A is a pentagon. Therefore, Figure A has Red Blue Green Orange exactly five sides. Ann ➔ Example: Every English setter likes to x x hunt. Duke is an English setter. So, Enya x x Duke like to hunt. Alvin x x FORMS OF DEDUCTIVE REASONING Johnny x ○ x x ➔ Hypothetical Syllogism ➔ Categorical Syllogism 3. Red is the favorite color of one of the Modus Ponens girls. ➔ Is a fundamental rule of inference Red Blue Green Orange logic. It states that if a statement P implies another statement Q, and P Ann x x ○ x is true, then Q must also be true. Enya ○ x x x Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Modus Tollens Modus Tollens ➔ Is another fundamental rule of ➔ It is a hypothetical syllogism with inference in logic, closely related to the form: Modus Ponens. It’s a bit like the ➔ Premise 1: If ppp then qqq opposite of Modus Ponens (conditional statement) ➔ Premise 2: and ¬q\neg q¬q (This is HYPOTHETICAL SYLLOGISM a negation statement) ➔ Conclusion: Therefore, ¬p\neg p¬p ➔ It is a type of deductive reasoning is false, where ppp and qqq are consisting of a conditional major distinct statements. premise, an unconditional minor ➔ Premise 1: If it is raining (P), then the premise, and an unconditional ground is wet (Q) conclusion. ➔ Premise 2: The ground is not wet (Q is false) TYPES OF HYPOTHETICAL SYLLOGISM ➔ Conclusion: Therefore it is not raining (P is false) Modus Ponens ➔ Example, If the traffic is bad, Jim will be late to the movie. Jim was not ➔ It is a hypothetical syllogism with late for the movie. Therefore, the the form: traffic was not bad. ➔ Premise 1: If p then qq (Conditional statement) CATEGORICAL SYLLOGISM ➔ Premise 2: and ppp (true) - This is called antecedent ➔ It is a form of deductive reasoning ➔ Premise 3: Therefore, qqq (True). wherein a categorical conclusion is This is called the consequent. based on two categorical ➔ Premise 1: If it is raining (P), then the premises. ground is wet (Q) ➔ There are four types of ➔ Premise 2: It is raining (P) propositions that are used in the ➔ Conclusion: Therefore the ground is syllogism: wet (Q) Positive Universal: “All A are B” Ex. ➔ For Example, If Allison vacations in All dogs are mammals. Paris, she will have to win a Negative Universal: “No A are B” Ex. scholarship. Allison is vacationing in No dogs are fish. Paris. Therefore, Allison won a Positive Existential: “Some A are B” scholarship. Ex. Some dogs are brown. ➔ Key Points, Modus Ponens is valid Negative Existential: “Some A are argument form, meaning that if the not B” Ex. Some dogs are not brown. premises are true, the conclusion - There are three types of must be also true propositions that will be used to create an argument in the following standard form as defined by: Villanueva, Juliana G. ┃DMED1  LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD Major premise (universal processed, and by which its quantifier) - a general statement accessibility, reliability, and about a category of things. timeliness is ensured to satisfy the Minor premise (existential needs of the data users. quantifier) - a statement about a specific member or subset of that STATISTICS category. Conclusion (universal or ➔ The word statistics originated from existential) - A statement that the word "status," meaning "state." logically follows from the major ➔ It is the science that deals with the and minor premises. collection, classification, analysis, ➔ It is also denoted by All p are q, r is and interpretation of numerical p, Therefore, r is q facts or data, in such a way that ➔ Example: All fish are sea creatures valid conclusions and meaningful Every shark is a fish Therefore, every predictions can be drawn from shark is a sea creature them. GENERAL PURPOSES OF STATISTICS EXAMPLES OF DEDUCTIVE REASONING ➔ Statistics are used to organize and ➔ All College of Arts and Sciences summarize the information so that faculty members have a Master’s the researcher can see what degree. Mr. Lozada is a College of happened in the research study Arts and Sciences professor. and can communicate the results Therefore, he was a Master’s degree to others. ➔ All even numbers are divisible by 2, ➔ Statistics help the researcher to The number 506 is even. Therefore, answer the questions that initiated it is divisible by 2 the research by determining ➔ Reflex angles are angles that are exactly what general conclusions more than 180 degrees. The angle is are justified based on the obtained. 235 degrees. Therefore, it is a reflex angle METHODS OF DATA GATHERING LESSON 3 DATA MANAGEMENT 1. Direct or Interview DATA MANAGEMENT ➔ It is a person-to-person encounter ➔ It is development, execution, and between the source of information, supervision of plans, policies, the interviewee, and the one who programs, and practices that gathers information, the control, protect, deliver, and interviewer. enhance the value of. It is an administrative process by which the required data is acquired, validated, stored, protected, and Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD 2. Indirect or Questionnaire systematically manipulated by the investigator. ➔ It is the technique in which a ★ Dependent variable (DV): The questionnaire is used to elicit the dependent variable in an information or data needed. experiment is the variable that the investigator measures to 3. Registration determine the effect of the independent variable. ➔ It obtains data from the records of government agencies authorized SCIENTIFIC METHOD by law to keep such data or information and made these ➔ The data from the experiment available to researchers. force a conclusion consonant with ➔ Example: Registration of birth, reality. Thus, scientific Registration of marriage, and methodology has a built-in Registration of death safeguard for ensuring that truth assertions of any sort about reality 4. Observation must conform to what is demonstrated to be objectively ➔ It is the technique in which data, true about the phenomena before particularly those pertaining to the the assertions are given the status behaviors of individuals or groups of scientific truth. of individuals during the given situation. Descriptive Statistics ➔ To notice using a full range of appropriate senses. To see, hear, ➔ Statistics involves the collection feel, taste, and smell. and classification of data. ➔ This is also used when the ➔ Abowler may want to find his respondents cannot read nor bowling average for the past 10 write. games. ➔ A teacher may wish to determine 5. Experimental the percentage of students who passed an examination. ➔ It is a system used to gather data from the results of performed series Inferential Statistics of experiments on some controlled and experimental variables. This is ➔ It involves the analysis and commonly used in scientific interpretation of that data. inquiries. ➔ A manager might predict, based on ★ Independent variable (IV): The previous years' sales, the sales independent variable in an performance of a company for the experiment is the variable that is next five years. Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ➔ A politician could estimate, based n= 𝑁 1 + 𝑁𝑒² on an opinion poll, his chances of winning in an upcoming senatorial ➔ Where: n = sample size election. ➔ N = population size ➔ e = desired margin of error (usually POPULATION 0.05 or 5%) ➔ Example : Compute the sufficient ➔ A population is the set of sample size of a target population measurements corresponding to consisting of 1, 524 sixth- graders in the entire collection of units for a given school district using sample which information is sought. It determination formula. represents the group of objects or ➔ Given: N = 1524 e = 0.05 subjects about which conclusions are to be drawn. n= 𝑁 1 + 𝑁𝑒² ➔ Example: ★ The scores of the entire students of Given: N = 1524 e = 0.05 Senior High School in EAC-Cavite represent the population for that Solution: specific group of students. 𝑁 ★ All children of any age who have n= 1 + 𝑁𝑒² older or younger siblings in 1524 Barangay Lucsuhin form the n= 1 + [1524 (0.05)² population for that group of 1524 children. n= 1 + [1524 (0.0025) SAMPLE n = 316.84 ≈ 317 ➔ A sample is a set of individuals Determine the sample size for each grade selected from a population, usually level given in the table below intended to represent the population in a research study. Grade Level Population Size per Grade Level Example: Grade 7 800 ★ The scores of 50 students of Senior High School in EAC-Cavite. Grade 8 650 ★ The 40 children who actually Grade 9 725 participated in one specific study about siblings in Barangay Grade 10 300 Lucsuhin. Total 2,475 SAMPLE SIZE Given: N = 2,475 e=0.05 ➔ Sample determination Formula Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD 𝑁 n= 1 + 𝑁𝑒² TYPES OF DATA n= 2,475 Qualitative Data 1 + 2,475 (0.05)² n= 2,475 ➔ Data that deal with categories or 1 + 2,475 (0.0025) attributes n= 2,475 ➔ Color of skin 1 + 6.1875 ➔ Courses in Computer Engineering n = 344.35 ≈ 344 ➔ It is considered as Categorical ★ Grade 7 = 800 x 344 = 111.19 ≈ 111 Quantitative Data 2,475 650 ★ Grade 8 = 2,475 x 344 = 90.34 ≈ 90 725 ➔ Data that deal with numerical ★ Grade 9 = 2,475 x 344 = 100.77 ≈ 101 values 300 ★ Grade 10 = 2,475 x 344 = 41.70 ≈ 42 ➔ Number of units in one semester ★ Total Sample size = 111 + 90 + 101 + ➔ Grade point average 42 = 344 ➔ It is considered as Numerical DATA TYPES OF QUANTITATIVE DATA ➔ Data are measurements or observations. A data set is a Discrete Data collection of measurements or observations. ➔ Discrete Data refers to data that ➔ A datum is a single measurement are obtained by counting. or observation and is commonly ➔ It results from either a finite called a score or raw score. number of possible values or a ➔ The measurements that are made countable number of possible on the subjects of an experiment values. are also called data. ➔ Examples ➔ Usually, data consist of the ★ Number of students in the measurements of the dependent classroom variable or of other subject ★ Number of cars in the parking lot characteristics, such as age, ★ Number of students, number of gender, number of subjects, and so books, number of patients on. The data as originally measured are often referred to as raw or Continuous Data original scores. ➔ Continuous Data refers to data that are obtained by measuring. ➔ It can assume any of an infinite number of values and can be associated with points on a continuous line interval. Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ➔ Examples VARIABLE ★ Area of a mango farm in Pampanga ➔ It is any property or characteristic ★ Volume of water in a pool in Pansol, of some event, object, or person Laguna that may have different values at ★ Height, weight, volume different times depending on the conditions PARAMETER Qualitative Variable ➔ A parameter is a value, usually a numerical value, that describes a ➔ A qualitative variable describes a population. characteristic and is often referred ➔ It is typically derived from to as a categorical variable measurements of the individuals because it can be separated into within that population. distinct categories. ➔ For instance, in the National ➔ Qualitative variables are typically Achievement Test of High School in descriptive but can sometimes be SY 2011-2012, the mean percentage assigned numeric values. score in Mathematics is 46.37. Quantitative Variable STATISTIC ➔ A quantitative variable has a ➔ A statistic is a value, usually a value or numerical measurement numerical value, that describes a to which mathematical operations sample. can be applied. ➔ It is typically derived from ➔ For example, variables such as measurements of the individuals age, height, and weight are within that sample. considered quantitative. ➔ For example, the average number of points earned by students in a LEVEL OF MEASUREMENTS particular math class at the end of the term is a statistic. ➔ Nominal, ordinal, interval, and ratio (Interval and ratio are SAMPLING ERROR sometimes called continuous or scale) ➔ Sampling error is naturally occurring discrepancy, error, that THE HIERARCHY OF LEVELS OF exists between a sample statistics MEASUREMENT and the corresponding population parameter. Nominal ➔ Attributes are only named, weakest Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ➔ Label qualitative data into mutually ★ Example 2: Find the mean of the exclusive categories following set of scores; 2.5, 3.4, 4.2, ➔ For example: What is your civil 5.1, 4.8 status? Where do you live? ★ x̄ = 2.5+ 3.4 + 4.2 + 5.1 + 4.8 5 x̄ = 4 ★ Example 3: The following frequency Ordinal distribution lists the results of a 10-point quiz given by a professor in ➔ Attributes can be ordered Statistics class with 42 students. ➔ Ranks qualitative data according Find the mean (Given: n=42, ∑ f x = to its degree 294) ➔ For example: How satisfied are you ★ Solution: with our food service? What is your 294 ★ x̄ = x̄ = 7.00 level of anxiety? 42 ★ Example 4: Determine the student’s Interval GWA given the following grades last summer ➔ Distance is meaningful ★ Given: ➔ Numerical data that has order and its differences can be determined Course Units Grade do not have a “true” zero ➔ For example: Temperature Math 3 3.00 Ratio English 3 2.00 P.E 2 1.25 ➔ Absolute zero ➔ A numerical data that has order, n=8 differences can be determined and has a “true” zero Solution: ➔ For example: Speed, Height, and Weight x̄ = 3(3.00)+3(2.00)+2(1.25) x̄ = 2.1875 8 LESSON 4 MEASURES OF CENTRAL TENDENCY MEDIAN MEAN ➔ The middlemost score ➔ Also known as the “average” or ➔ Middle value of a distribution “arithmetic mean” ➔ The sum of all values in dataset Examples: divided by the total number of observations ★ Example 1: 106, 118, 125, 119, 142, 106, ➔ Examples: 118, 119, 125, 145 Median = 119 ★ Example 1: Consider the following: ★ Example 2: 200, 551, 448, 315, 218, 16, 20, 2, 35, 18, 22, 17 367, 200, 218, 315, 367, 448, 551 315 + 367 ★ Mean = 16 + 20 + 2 + 35 + 18 +22+17 = 18.57 ★ Median = 2 = 341 7 Villanueva, Juliana G. ┃DMED1 LESSON 1-5┃SCHOOL OF DENTAL MEDICINE - MATHEMATICS IN THE MODERN WORLD ★ Example 3: 1,2,3,4,5,6 Median = 4 ★ Example 4: 2,3,4,5,6,7 4+5 ★ Median = 2 = 4.5 MODE ➔ Most frequent value in dataset/score Examples: ★ Example 1: 45, 22, 63, 78, 99, 22 ★ Mode = 22 ★ Example 2: 67, 89, 34, 90, 23, 53, 22, 34, 53 ★ Mode = 34, 53 ★ Example 3: 68, 97, 61, 85, 38, 29, 11 ★ Mode = none APPLICATIONS (OTHER EXAMPLES) ➔ Data scores of 14 students in Statistics and Probability Midterm exam, Find the mean, median, and mode 72 83 84 83 72 80 79 80 76 80 85 79 90 91 ➔ The average score of 14 students who took the Statistics and Probability Midterm exam. The mean is 80.93, Mean ➔ Seven out of fourteen or 50% of students has a score above 80 or seven out of fourteen or 50% of students has a score below 80 so the median is 80 ➔ The most frequent score is 80 so the mode is 80 Villanueva, Juliana G. ┃DMED1