MAT-152-P2-Reviewer.pdf

MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.)...

MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) MODULE 6: A graph is a very effective visual tool as it displays data at a glance, facilitates DATA MANAGEMENT: RESPONSIBLE comparison, and can reveal trends and HANDLING OF DATA AND DATA relationships within the data such as PRESENTATIONS changes over time, frequency distribution,and correlation or relative share INTRODUCTION of a whole. Data is a set of facts that give us Graphical presentation may be in forms information about things.It doesn't show us like charts, graphs and pictures. everything about a situation—it only gives us part of the picture. Two Types of Data A. Categorical Data - means organizing Ethical awareness and responsible individuals or things into groups based on handling of data are essential to ensure that their characteristics. information is managed in a manner that Examples: gender (male or female), eye respects individuals' rights, maintains trust, color (brown, green, gray), blood type (A, B, and upholds moral principles. O), and course (Nursing, Education, Accountancy) Principles of Ethical Awareness and B. Numerical Data – is a type of data Responsible Handling of Data where exact numerical values are expected. Examples: height, weight, age, pulse rate, Keeping Information Safe number of children, typing speed Being Honest and Fair with Data Making Things Clear Classifying Numerical Data Agreeing Only When You B.1. Discrete Variables have values Understand obtained by counting. Examples: Number of children, Number of DATA PRESENTATION: male students in a class and Number of Textual, Tabular, and Graphical patients with T.B. B.2. Continuous Variables have Textual presentation combines text and values obtained by measuring. numerical facts in statistical reports and is Examples: Temperature, Distance, Area, narrative in nature. Density, Age, Height and Weight Text is any written or printed words that DEFINITIONS convey a message. It can be a single word, a sentence, a paragraph, or an entire book. DATA - Set of facts that provide a part It is a way for people to communicate ideas, picture of reality information, and stories through writing. INFORMATION - It is a data that has been Example: processed, organized, and given context, There are 42,036 barangays in the making Philippines. The largest barangay in terms of population size is Barangay 176 in Caloocan City with 247 thousand pesos. MODULE 7: Tabular presentation is often better than a INTERPRETING DIFFERENT GRAPHS textual presentation since the values are numeric and independent. INTRODUCTION A Table is a set of data arranged in rows When we want to show information in a and columns. It also consists of several picture, we must pick the right kind of boxes with information aside picture, just like choosing the right words in a story.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) Graphs are like tools for showing NOTE: The histogram represents numerical information. It is like a visual map made of data whereas the bar graph represents dots and lines. The dots represent things, categorical data. and the lines show connections between them. A pie graph or pie chart is a circular chart divided into slices to show how different There are many types of graphs parts make up a whole. Bar Graph Line Graph Histogram Pie Chart/Graph Scatter Plot A bar graph is s a type of graph that uses rectangular bars to represent data. It is great for comparing things, and help us see information easily. Scatter plots are a way to see if there's a relationship between two variables by plotting points on a graph. It uses dots to show how two different things are related. A Line graph is good for showing how things change over time, helping us spot patterns Understanding Data Ethically When analyzing graphs critically, there are several key concepts to keep in mind. Data integrity - It's important to make sure the data in the graph is accurate and and trends quickly. reliable. Transparency - It is essential for ethical interpretation. Information presented in the graph should be clear, readable, and trustworthy. Histograms are like bar graphs, but they're used to display the distribution of Graphs should clearly label axes, provide continuous data, like representing the appropriate units, and include necessary distribution of test scores in a class. context or notes. Fair representation - It means that graphs shouldn't hide anything. They should tell the whole story, not just parts of it.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) 30 Choosing the right type of graph - Each 5 = 6 type of graph is good for showing different kinds of information. Pick the one that best Therefore, the mean score is 6. represents the data. The weighted mean, unlike a regular Respect for privacy and consent - Very mean, which treats all numbers equally, the crucial. Information about people or private weighted mean assigns different weights to things should not be disclosed. each number, making some numbers count more in the computation. Accountability and responsibility - It means always thinking about what the MEDIAN graph means. The median is the middle number when the Consider the societal impact - Reflect on dataset is arranged in order. how the information in the graph might affect different groups of people or society. Example 1: We have a set of numbers: MODULE 8: 3, 5, 7, 9, and 11 COMPUTING MEASURES OF Since we have an odd number of values, CENTRAL TENDENCY we simply get the middle number which is 7. Therefore, the median score is 7 INTRODUCTION Example 2: Measures of central tendency are We have a set of numbers: methods used to find the "middle" or 1, 3, 5, 7, 9, and 11 "center" of a dataset and this gives us a sense of where most of the data points are Since we have an even number of values, located. we get the two middle numbers and divide by 2. CENTRAL TENDENCY: 5+7 Three main measures 2 =6 Mean Therefore, the median score is 6 Median Mode MODE MEAN The mode is the number that appears most The mean represents the average value of often. a dataset. Example: It is found by adding all the numbers We have a set of numbers: together and then dividing by the number of 1, 2, 2, 3, and 4 numbers. The mode is 2 since it’s the number that Example: appears most often. We have a set of numbers: 2, 4, 6, 8, and 10 Understanding the Type of Data STEP 1: Add all the numbers Nominal - Categories with no specific 2 + 4 + 6 + 8 + 10 = 30 order (e.g., types of pets) STEP 2: Divide by the number of numbers Ordinal - Categories with a specific order (e.g., movie ratings) Since there are 5 numbers, divide it to the sum of numbers which is 30.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) Interval - Numeric scales with equal Step 6: Check Your Answer / Make intervals, no true zero (e.g., temperature in conclusion Celsius) The overall weighted average rating of the movie, considering the different numbers of Ratio - Numeric scales with equal followers for each reviewer, is approximately intervals and a true zero (e.g., height in 4.11. This calculation ensures that centimeters). reviewers with more followers have a greater influence on the average rating, Example of how to solve a measure of providing a more accurate assessment of central tendency problem: the movie's reception. A movie website wants to compute the overall average rating of a movie, MODULE 9: considering different reviewers who have rated the movie, with each reviewer having COMPUTING MEASURES different numbers of followers (weight). OF DISPERSION Reviewer A: Average rating = 4.5 (Followers = 200) INTRODUCTION Reviewer B: Average rating = 4.0 (Followers = 150) Imagine you have a bunch of numbers, and Reviewer C: Average rating = 3.5 (Followers we want to know how spread out they are. = 100) Measures of dispersion help us understand our numbers better. Answer: Step 1: Understand the Data Measures of dispersion - the statistical You need to have the average scores for tools used to quantify the spread or four subjects and the number of tests in variability of data points within a dataset. each subject. Step 2: Look at Your Data Range - the difference between the highest Each reviewer has a different number of and lowest values in a dataset. followers, so some reviews should count more in the average because they have To find the range: more followers. Step 3: Decide What You Need to Find R = HV - LV Goal: Calculate the overall average rating of where: the movie to assess its general reception. R - Range Step 4: Choose the Right Measure HV - Highest value Use the weighted mean because it accounts LV - Lowest value for the number of followers each reviewer has. Importance of Variance Step 5: Do the Math and Standard Deviation Step 5.1. Multiply values by its corresponding weight. Shows how spread out or close together Reviewer A: 4.5 × 200 = 900 Reviewer B: numbers are in a dataset 4.0 × 150 = 600 Reviewer C: 3.5 × 100 = 350 Variance - the average of the squared Step 5.2. Sum up the weighted values. differences from the mean. It shows how Total Weighted Grades = 900 + 600 + 350 = much the data points differ from the mean. 1850 Step 5.3. Sum up the weights. Sample Variance - An estimate of the Total Students = 200 + 150 + 100 = 450 population variance, calculated by Step 5.4. Divide the total weighted sum averaging the squared differences from the by the total weight. sample mean. To compute for the variance:  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) Where: Step 2. Consider the Scale of Measurement The number of hours is measured on a ratio scale, as it has a true zero point (0 hours) and the differences between values are meaningful. Step 3. Assess Data Distribution We can see that the study hours vary and Note: This is important because sometimes there is a mix of lower and higher values we can't ask everyone their data, so we use without any extreme outliers. the data of a few people to estimate what all Step 4. Determine the Presence of data might look like. Outliers Standard deviation - Provides an average There are no extreme values or outliers in measure of how far the data points are from the data. the mean and is derived from the variance. Step 5. Select the Measure. Since the data has no outliers and is Sample standard deviation - Similar to relatively small, we will use standard standard deviation, but it focuses on a deviation as appropriate measures of sample. dispersion. ▪ Compute the standard deviation. To compute the standard deviation, calculate for the square root of the variance. 2 2 𝑆 = 𝑆 =𝑆 2 𝑆 - Variance 𝑆 - Standard deviation Steps in Identifying the right measure of dispersion for the data STEP 1: Understand your Data STEP 2: Consider the Scale of Measurement STEP 3: Assess Data Distribution STEP 4: Determine the Presence of Outliers STEP 5: Select the Measure Example Problem: Interpretation: The standard deviation, Imagine tracking the number of hours spent which is about 2.29 hours, tells us that most studying per week by students in a class: 5, students' study hours are close to the 7, 10, 8, 6, 9, 4, 12, 8, 6. Now, Identify and average of 7.5 hours, but they can vary by compute the most appropriate measure of around 2.41 hours. This means that some dispersion using the step- by-step process. students may study a bit more or less than others each week. Answer: Step 1. Understand Your Data MODULE 10 We have the data representing the number Tree Diagrams & Basic Probability of hours spent studying per week by Problems students in a class: 5, 7, 10, 8, 6, 9, 4, 12, 8, 6. This is a list of numerical data. Tree Diagram:: A visual representation showing all possible outcomes of events or decisions.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) Example: You might have to decide whether to go to a friend's party(Point A) or For two independent events A and B, where stay home and watch movies (Point the outcome of A does not change the B). Each of these choices leads to probability of B, the probability of A and B is more possibilities. If you choose the given by: party, maybe you'll have to decide whether to dance or just hang out. If P(A and B) = P(A) × P(B). you stay home, maybe you'll need to choose which movie to watch. Step-by-Step Process of Using Tree Diagrams STEP 1: Identify the Events. STEP 2: List the Possible Outcomes for Each Event. STEP 3: Draw the Tree Diagram, adding branches for subsequent events. Importance: STEP 4: Determine the Probabilities for Helps visualize complex probability Each Branch. scenarios. Makes calculating chances of different STEP 5: Calculate the Combined outcomes easier Probabilities for each path. Useful for breaking down multi-step problems. STEP 6: List all Possible Outcomes and their Probabilities. Basic rules of computing probability. EXAMPLE: You're ordering pizza from your favorite Probability computation: For equally likely pizzeria, and they offer three different sizes outcomes, the probability of event A is given (small, medium, large) and four different by; toppings (cheese, pepperoni, sausage, vegetarian). You want to know the probability of getting Medium with Pepperoni on top. Use the step-by-step Key rules for calculating probabilities process of creating a tree diagram and include: calculate possible probabilities. Addition Rule: Helps when events are Answer: connected, like when one thing happening affects another. It also works for events that can both happen or events that can't both happen at the same time. For two events A and B, the probability of selecting one event or another is given by: P(A or B) = P(A) + P(B) - P(A and B). Multiplication Rule: Is for when events don't affect each other, like flipping a coin and rolling a dice at the same time.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) D. The standard normal distribution is a continuous probability distribution where the area under the curve represents probability, and the total area always equals 1. E. A normal distribution can be standardized, allowing us to compare values from different distributions on the MODULE 11 same scale. This is done by transforming it into a standard normal distribution using the Probabilities of Normal Distribution z-score formula: 1. Normal Distribution - A symmetric probability distribution where most values cluster around a central mean, forming a bell-shaped curve. 2. Standard Normal Distribution - A special type of normal distribution with: - Mean = 0 - Standard deviation = 1 -Used for standardizing values from 4. Standard Score (z-score) different distributions. These z-scores help us quickly see who performed below, at, or above the mean and Characteristics of a Standard Normal by how much in terms of standard Distribution deviations A. All normal distributions are symmetric and have bell-shaped density curves with a single peak. The ends of the curve approach the x-axis but never touch it; these are called asymptotes. Probability Cases: B. The standard normal distribution is normal distribution with a mean of 0 and 1. Case 1: P(z < a) standard deviation of 1 - The probability that the z-score is less than a given value (a). C. The standard normal distribution has: - Example: What percentage of students mean = median = mode scored below 100 on an IQ test? symmetry about the center 50% of values less than the mean 2. Case 2: P(z > a) and 50% greater than the mean - The probability that the z-score is greater than a given value (a). -Example: What percentage of students scored above 120 on an IQ test?  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) 3. Case 3: P(a < z < b) MODULE 12 - The probability that the z-score falls between two specific values (a) and (b). Pearson-r Correlation Coefficient, Linear - Example: What percentage of students Relationship and Regression Line scored between 95 and 125 on an IQ test? Linear relationship - means that as one thing changes, another thing changes in a way that forms a straight line on a graph. Linear correlation is the term we use to describe how closely thisstraight-line relationship is followed. The Pearson correlation coefficient, denoted as r, measures the strength of the relationship between two variables. The closer r is to 1 or -1, the stronger the linear relationship. A graph with one variable on the x-axis and the other on the y-axis shows how well they align to form a straight line. Pearson Product Moment Coefficient of Correlation, r Formula Where: r = Pearson Product Moment Coefficient of Correlation x = the observed data for the independent variable y = the observed data for the dependent variable ∑xy = the summation of the product of x and y ∑x ∑y = the product of the summation of x and the summation of y ∑x 2 = summation of the squares of x ∑y 2 = summation of the squares of y n = number of paired observations The value of r is interpreted using a correlation scale. An r of +1 indicates a perfect positive relationship, while -1 shows a perfect negative relationship. An r of 0 means no linear relationship.  MAT 152 P2 Exam Reviewer 1st Sem, A.Y. 2024-2025 (*This material shall only be used as a reviewer for the exam and not as a tool for cheating, note that the contents of this reviewer may or may not cover the entire contents of the exam.) NOTE: GO BACK TO RECORDED Value of r Descriptive Equivalent LECTURES OF MODULES 11 AND 12 ∓1.00 Perfect positive(negative) FOR EXAMPLES, AND PRACTICE correlation SOLVING DIFFERENT PROBLEMS USING YOUR SCIENTIFIC CALCULATOR. ∓0.91 to 0.99 Very high positive(negative) ONCE AGAIN, THIS REVIEWER MAY OR correlation MAY NOT COVER THE ENTIRE SCOPE ∓0.71 to 0.90 High positive(negative) OF THE EXAM SO IT’S BETTER THAT correlation YOU SHOULD STILL BROWSE OTHER MATERIALS FOR YOUR REVIEW. ∓0.51 to 0.70 Moderately positive(negative) GOODLUCK AND GODBLESS!!! correlation ∓0.31 to 0.50 Low positive(negative) correlation From your MAT 152 Teacher, ∓0.01 to 0.30 Very low positive(negative) correlation -Sir Nico Gamboa 0.00 No correlation When two variables are correlated, we can use one to make predictions about the other. This ability to forecast is aided by regression analysis, a tool for prediction, estimation, and forecasting. If two variables, x (independent) and y (dependent), are correlated, we can predict y using the regression line that defines their relationship. The equation for this line is given by the regression formula: where: ŷ = the predicted value x = the independent variable a and b = are found using the formulas below The formulas for finding a and b are the following and,