Quantitative Methods and Problem-Solving Process PDF

*Learning Objective*: At the end of the unit, the student should be able to: **QUANTITATIVE METHODS AND PROBLEM-SOLVING PROCESS** ---------------------------------------------------- The field of Quantitative Methods (QM) encompasses a collection of techniques and tools that utilize quantifiable d...

*Learning Objective*: At the end of the unit, the student should be able to: **QUANTITATIVE METHODS AND PROBLEM-SOLVING PROCESS** ---------------------------------------------------- The field of Quantitative Methods (QM) encompasses a collection of techniques and tools that utilize quantifiable data and mathematically oriented techniques to help decision-makers come up with a sound and proper judgment. These methodologies allow businesses to optimize results and resolve complex problems by leveraging systematic and scientific methods. It systematically utilizes past data for future uses. QM is a broad subject that includes different approaches from gathering raw data and producing valuable information for decision-making. There are three distinct classifications of QM: 1. **Mathematical QM**: includes Matrix, Determinants, Differentials, Integrals, Vectors 2. **Statistical QM:** covers Data collection, Data summarization, Sampling Analysis, Time Series, Regression 3. **Programming QM:** contains Linear Programming, Assignment Models, Decision Theory, Game Theory Although called Programming, the foundation of Programming QM methods is based on mathematical and statistical methods. However, the aid of technology/software is needed to implement the techniques. All methods under QM begin with creating models. A model is a representation of reality with only relevant details included, usually presented as an equation/mathematical form. Some types of models and their uses include: - **Descriptive Models:** Used to describe the behavior of a system based on certain information. Example: Business reporting in the form of graphs, charts and dashboards - **Explanatory Models:** Used to establish relationships between components to explain a behaviour. Example: Identifying significant barriers between technology integration and primary school teaching - **Predictive Models:** Used to identify likely outcomes based on historical data. Example: Predicting future consumer demand, given current demand - **Prescriptive Models:** Designed to find the optimal solution for a given problem. Example: Finding the best allocation of physical resources to produce multiple products Models are being utilized during the Analysis part of the Problem-Solving process, as shown in Figure 1.  **DATA** -------- Data are the raw numbers or facts we process to produce useful information. It is the foundation of all QM techniques. ### Types of Data Data may be classified as qualitative or quantitative. - **Qualitative Data:** Also known as categorical data - **Quantitative Data:** Also known as numeric data - These can be measured and not simply observed. They can be numerically represented and calculations can be performed on them. For example, data on the number of students playing different sports from your class gives an estimate of how many of the total students play which sport. This information is numerical and can be classified as quantitative. - Can be classified as **Discrete** (counted items, in integer form) or **Continuous** (measured characteristics, presented with decimal points) 2. ### Data Sources Data can also be categorized based on its source. - **Primary Data:** It is composed of new data collected for the first time for a specific purpose. This has the benefits of fitting the needs exactly, being up to date, and being reliable. - Primary data can be collected from a population (census) or a sample. - **Population** -- The group of all the people or subject that share the common characteristic of interest. Example: all Registered voters in the country - **Sample**: They are the representative or part of the population. Example: All registered voters in the city of Cabanatuan What is Sample Size? Definition - Omniconvert - **Secondary Data**: These consist of existing data collected by other organizations or for other purposes. 3. ### Levels of Measurements **Levels of measurement**, also called **scales of measurement**, tell how precisely variables are recorded. The level at which you measure a variable determines analytical methods that can be applied to the data, including descriptive or summary statistics, models, and tests that can be used to address problems. Four Levels of Measurements: - **Nominal Level**: Weakest level of measurement. Nominal data are labels or names used to identify/categorize an attribute of the variable. Can use numeric or non-numeric code as labels - Example: **Gender** -- Male or Female; **Major Island Group in the Philippines** -- 1(Luzon), 2(Visayas), 3(Mindanao) - **Ordinal Level**: Similar to Nominal Level but requires a meaningful order or rank of data. Distance between observations cannot be quantified but direction can be interpreted. - Example: Poor, Good, Very Good; 1^st^ place, 2^nd^ place, 3^rd^ place - **Interval level**: Data under this level can also be classified and ordered. It also specifies that the distances between each interval on the scale are equivalent along the scale from low to high interval, ie: differences between measures have meaning. However, there is no absolute 0 scale. - Example: Temperature in Celsius and Fahrenheit, IQ scores, pH level - **Ratio Level:** Data at this level can be categorized, ranked, evenly spaced, and has a natural zero. A natural zero implies an absence of the variable of interest. - Example: height, weight, distance, number of students ***Activity 1*** I. Identify each data presented, then, write **QL** for qualitative data and **QN** for quantitative data before the number. 1. The colors of cars in a parking lot. 2. The number of students in a classroom. 3. The types of cuisines served in different restaurants. 4. The weights of fruits in a basket measured in grams. 5. The rankings of movies in a streaming platform. 6. The marital status of individuals in a survey. 7. The temperature of water in a swimming pool in °C. 8. The names of books in a library. 9. The number of steps a person takes in a day. 10. The shapes of cookies in a bakery. II. Enumerate the following. A. Distinct Classifications of Quantitative Methods 1. 2. 3. B. Four Levels of Measurements 4. 5. 6. 7. C. Sources of Data 8. 9. D. Classifications of Data 10. 11. E. QM Models 12. 13. 14. 15. III. For each statement, decide whether it refers to a population or a sample. Write **P** or **S** on the space before the number. 1. A survey is conducted on 500 students randomly selected from all the students in a university. 2. The total number of employees in a company is 1,000. You collect data from all of them. 3. A researcher measures the height of 50 out of 500 athletes in a sports tournament. 4. A teacher records the test scores of all students in a specific class. 5. A census collects data on every person living in a country. 6. A polling organization gathers opinions from 1,200 voters across a country to predict election results. 7. The average income of every household in a small village is calculated. 8. Data is collected from 100 out of 2,000 patients in a hospital to analyze treatment effectiveness. 9. A scientist studies all the birds in a sanctuary to learn about their feeding habits. 10. A market researcher surveys 300 shoppers in a mall to understand customer preferences. **UNIT 2. DATA COLLECTION** *Learning Objectives*: At the end of the unit, the student should be able to: 1. differentiate the use case of data collection methods like surveys, experiments, observation, and interviews; and 2. identify different sampling techniques. 1. **SAMPLING TECHNIQUES** ----------------------- A **Census** or **complete enumeration** involves the collection of data from every element of the population. It provides a complete information about the population. However, conducting a census becomes increasingly expensive and time-consuming as the number of elements in the population increases. For these reasons, most organizations opt to use samples instead of census. ***Advantages of Sampling*** 1. Sampling is more economical. 2. It requires less time to accomplish. 3. It allows a wider scope for the study. 4. It has fewer measurement errors and more accurate results. The **target population** is the population we want to study. The **sampled population** is the population from where the sample is selected. The **sampling frame** is a list of all members of the population. ***Types of Sampling*** 1. **Probability Sampling:** Items in the sample are chosen based on known probabilities. All members of the population have a chance of being part of the sample. 2. **Non-probability Sampling**: Items included are chosen without regard to their probability of occurrence. It includes a subjective selection of samples, thus, reliability of results is often times based on assumptions. **PROBABILITY SAMPLING METHODS** +-----------------------------------------------------------------------+ | 4. **Simple Random Sampling** | | | | **Procedure:** | | | | 1. List the elements of the population and number from 1 to ***N***. | | | | ***N*** = number of elements in the population | | | | 2. Select ***n*** numbers from 1 to ***N***, using a randomization | | technique. The sample will consist of elements corresponding to | | the selected numbers. | | | | ***n*** = number of elements in the sample | | | | **Advantage:** | | | | - Design and estimation are simple and easy to understand. | | | | **Disadvantages:** | | | | - Needs an exhaustive list of all elements in the population. | | | | - The sample size must be large. | | | | - This can lead to more resources if elements are widely spread | | geographically. | | | | **When to Use:** | | | | - Elements of population are homogenous with respect to | | characteristics under study. | | | | - Elements are not spread out geographically. | | | | **Example:** | | | | Imagine you are a quality control manager at a | | computer monitor factory, and you want to test the quality of the | | monitors. You have a large production line, and you want to select a | | sample of monitors for testing. You assign a unique serial number to | | each monitor, and then you use a random number generator to select 20 | | monitors from the entire production. This ensures that each product | | has an equal chance of being tested. | +=======================================================================+ | 1. **Systematic Sampling** | | | | **Procedure:** | | | | 1. Assign a unique number from 1 to ***N*** to each element of the | | population. | | | | 2. Determine the sampling interval ***k***. | | | | ***k = N/n*** | | | | 3. Obtain the first element in the sample using a randomization | | technique. | | | | **Advantages:** | | | | - Easy to identify the elements to be included in the sample. | | | | - Sample is distributed evenly over the entire population. | | | | - Can be done even without an available list of all elements in the | | sample. (example: choosing households in a certain community) | | | | **Disadvantages:** | | | | - Requires information on the arrangement of the elements in the | | sampling frame. | | | | - Periodic irregularities in the list will affect the reliability | | of the results. | | | | **When to Use:** | | | | - The arrangement of elements in the sampling frame is according to | | the magnitude. | | | | - There is no available list of all elements but arrangement is | | known. | | | | **Example:** | +-----------------------------------------------------------------------+ | 2. **Stratified Random Sampling** | | | | **Procedure:** | | | | 1. Divide the population into non-overlapping | | strata (i.e.: every element will only belong to one and only one | | stratum) according to a common characteristic (stratification | | variable). | | | | 2. Obtain a simple random sample for each stratum, with sample sizes | | proportional to strata sizes. (i.e.: if Strata 1 has 40% of the | | population, then 40% of the sample should also be selected from | | Strata 1) | | | | **Advantages:** | | | | - More reliable results if strata are heterogenous with each other. | | | | - Assured representation of items across the entire population. | | | | - Can facilitate administration of data collection. | | | | **Disadvantage:** | | | | - Need information on the stratification variable to identify the | | stratum of each element. | | | | **When to Use:** | | | | - Population is heterogeneous with respect to the characteristics | | under study. | | | | - The analysis is for certain subpopulations. | | | | **Example:** | | | | We want to observe consumer behavior in one of the municipalities in | | Region 3. We check the proportion of females, the proportion of young | | / old, and the proportions according to average income in the sample. | | Then, we divide the population in sub-groups according to gender, age | | and income. After that, we apply SRS or systematic sampling method to | | select a certain number of people from each subgroup we have | | created.  | | | | The aim is to ensure the same sub-group proportions in the sample. If | | there are 10% of young females with high income in the population, | | then we want 10% of our sample to be young females with high income. | +-----------------------------------------------------------------------+ | 3. **Cluster Sampling** | | | | **Procedure:** | | | | 1. Divide the population into nonoverlapping clusters, each a | | representative of the population. | | | | 2. Select a sample of clusters using simple random sampling. | | | | 3. The sample consists of all the elements in the selected cluster. | | | | **Advantages:** | | | | - Only needs a list of clusters, not a list of elements. | | | | - More cost-effective in terms of transportation and listing, | | especially if the population is geographically widespread. | | | | **Disadvantage:** | | | | - Often requires a larger sample size than other probability | | sampling techniques for the same level of precision. | | | | **When to Use:** | | | | - No available list of elements. | | | | **Example:** | | | | When studying a network's performance, you can randomly select | | specific departments or servers as clusters to analyze the | | performance metrics. | | | |  | +-----------------------------------------------------------------------+ **In Excel, we can use the functions:** **= RAND()** - Yields a value from 0 to 1 **= INT(RAND()\*N) + 1** - Generates a random whole number from 1 to N. **= RANDBETWEEN (\[min\], \[max\])** - Returns a random integer between the specified numbers. **NON-PROBABILITY SAMPLING METHODS** - **Convenience Sampling**: The sample consists of selected elements that are the most accessible or easiest contact. Subjects who are available and willing to participate at the time of study. Mostly used in research in biology and social sciences as participation is limited for these areas of study. **Example**: A researcher wants to study the eating habits of college students. Instead of randomly selecting participants from the entire student population, the researcher surveys students in their own class because they are easily accessible. This method is quick and inexpensive but may introduce bias since the sample may not represent a broader population. - **Purposive Sampling:** The selection of sampled elements is based on the researcher's judgment of who qualified as a representative sample. **Example:** A researcher is studying the impact of leadership styles on employee productivity in tech startups. To gather data, the researcher specifically selects CEOs of tech startups with fewer than 50 employees and at least 5 years of operation. These criteria ensure the participants are relevant to the study\'s objectives. This method ensures the sample is tailored to the research goals but can introduce bias since it relies on the researcher\'s judgment. - **Quota Sampling**: Similar to Stratified sampling but the selection of sample within the stratum does not use probability sampling method. There is just a set quota or number of sampling units included for each group but uses convenient or purposive sampling to select units within each group. **EXAMPLE**: A marketing researcher wants to study the purchasing habits of a city's residents and ensures the sample represents gender distribution. Based on population data, they decide on a quota of 60% females and 40% males. The researcher surveys the first 600 females and 400 males they encounter in shopping malls until the quota is met. This method ensures representation of specific subgroups but can introduce bias since participants are not randomly selected. Do Activity 2.1 **DATA COLLECTION METHODS** --------------------------- The process of gathering and analyzing accurate data from various sources to find answers to research problems, trends and probabilities, etc., to evaluate possible outcomes is known as **data collection**. Knowledge is power, information is knowledge, and data is information in digitized form, at least as defined in IT. Hence, data is power. But before you can leverage that data into a successful strategy for your organization or business, you need to gather it. **Observation** **Observation** was done when the population consists of machines, animals, files, documents, or any other inanimate objects. This was used mainly for reaction and behavior-related studies. Observers can be human or automatic recorders. The observer watches some activities and records what happens by: - Counting the occurrence of events - Taking some measurements - Seeing how something works **Surveys** **Surveys** are done by asking people questions or administering questionnaires which usually contains close-ended questions. They can be in the form of interviews, postal surveys, or longitudinal surveys. **Interviews** **Interviews** can be more personal and involve face-to-face discussions about a topic between the researcher and the participants. The researchers might share the questions with the participants before the interview sessions to allow them to decide if they feel comfortable in taking part in the interview. This method may include gathering consent forms for video or audio recordings.  Interview involves asking questions verbally. It is one of the most reliable ways of getting data but costly since you have to travel to the location of the participants to conduct the interview. Almost anyone can come up with a list of questions, but the key to efficient interviews is knowing what to ask and adding follow-up questions, making them more customized. Interviews can be done in person, via calls or a web chat interface. **Usability Testing** Usability testing is a user-centered design technique where you evaluate a product/app/website by testing it on a group of people with no prior exposure to it. The goal is to measure the intuitiveness of your design, user flows, and content with users who closely represent your target market. In plain English, usability testing is about finding a group of real people, similar to potential users of your app or website, and then having them test it out in various ways. Usability testing helps you eliminate assumptions and get real data on the user experience of your app, website, or product. Example: **Use of Secondary Sources** Secondary data (also known as second-party data) refers to any dataset collected by any person other than the one using it. It usually comes from previous studies of individuals, private organizations, and government agencies. ***Activity 2.1*** ***Instruction: Read the descriptions below and decide what type of sample selection has taken place.*** 1. School children, some with foster parents and some with natural parents are identified from school records.  Method: children are selected randomly within each of the two groups and the number of children in each group is representative of the total child population for this age group. 2. A survey of the attitudes of mothers with children under one year.  Method: interviewers stop likely-looking women pushing baby strollers in the street. The age bands and social classes of the respondents are almost the same.  3. A survey of attitudes of drug users to rehabilitation services.  Method: drug users are recruited by advertising in the local newspaper for potential respondents.  4. A postal survey of the attitudes of males to the use of male contraceptives.  Method: all male adults whose Patient ID ends in \'5\' are selected for a survey.  5. A study of the length of stay of patients at Anytown General Hospital.  Method: all patients admitted to wards 3, 775, and 10 in a hospital are selected for a study. ***Activity 2.2*** **Instruction**: Read the scenario below. Then, answer the questions presented. The university Academic Council wants to understand the perception of students about online, hybrid, and face-to-face classes as this can help them decide what modes of classes they should offer.   A list of the 3,000 current students who were able to experience all three modes of classes is already available, arranged alphabetically, and with information about the degree program. This will serve as the Sampling Frame. The Office of Student Affairs intends to take a probability sample of 750 students and project the results from the sample to the entire population of voters. **Questions:** 1. Can Simple Random Sampling (SRS) be used to get the 750 samples needed? Why or why not? *ANSWER:* 2. Can Systematic Sampling be used to get the 750 samples needed? Why or why not? *ANSWER:* 3. If the sampling frame available can be divided into separate alphabetical lists based on degree program (for example: a list for BSIT students, a list for BS Architecture students), what type of sampling technique can be used? Discuss your choice. *ANSWER:*