GED 102 Mathematics in the Modern World PDF

GEd 102 MATHEMATICS IN THE MODERN WORLD Ms. Christine V. Aranda, LPT ©2024 Batangas State University 1  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distr...

GEd 102 MATHEMATICS IN THE MODERN WORLD Ms. Christine V. Aranda, LPT ©2024 Batangas State University 1  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 2 ©2024 Batangas State University  General Fields of Statistics: Descriptive Statistics and Inferential Statistics Descriptive Statistics. Descriptive statistics focus on summarizing and describing the basic features of a dataset, providing an overview of the data's central tendency, variability, and distribution. Inferential Statistics. Inferential statistics involve using sample data to make conclusions or predictions about a larger population. It helps answer questions about the population based on a representative sample.  General Fields of Statistics: Descriptive Statistics and Inferential Statistics Descriptive Statistics. Measures of central tendency (mean, median, mode) Measures of variability (range, variance, standard deviation) Data visualization (histograms, box plots, scatter plots) Inferential Statistics. Hypothesis testing (t-tests, ANOVA, regression) Confidence intervals (estimating population parameters) Regression analysis (modeling relationships between variables)  General Fields of Statistics: Descriptive Statistics and Inferential Statistics Descriptive Statistics. The average height of the 100 students in our sample is 165 cm. Inferential Statistics. Based on our sample, we estimate that the average height of all Filipino students is between 163 cm and 167 cm with 95% confidence.  MEASUREMENT It essentially means quantifying an observation according to a certain rule. For instance, the presence of fever can be quantified by using a thermometer. Body weight can be determined by using a weighing scale. Or the mental ability can be quantified by using written examination that can generate scores. The quantification sometimes can be done is simply counting. In quantifying an observation, there are two types of quantitative informations: variable and constant. A variable is something that can be measured and observed to vary. While a constant is something that does not vary, and it only maintains a single value.  SCALES OF MEASUREMENT Nominal Scale : Categorical Data Ordinal Scale : Ranked Data Interval Scale : Measurement Data Ratio Scale : Measurement Data  SCALES OF MEASUREMENT Nominal Scale. It concerns with categorical data.  SCALES OF MEASUREMENT Ordinal Scale. It concerns with ranked data.  SCALES OF MEASUREMENT Interval Scale. It deals with measurement data.  SCALES OF MEASUREMENT Ratio Scale.  Key Concepts in Statistics Population. A population can be defined as an entire group people, things, or events having at least one trait in common (Sprinthall, 1994). A common trait is the binding factor in order to group a cluster and call it a population. Merely having a clustering of people, things or events cannot be considered as a population. At least one common trait must be established to make a population. But, on the other hand, adding too many common traits can also limit the size of the population.  Key Concepts in Statistics A group of students (this is a population, since the common trait is “students”) A group of male students. A group of male students attending the Statistics class A group of male students attending the Statistics class with iPhone A group of male students attending the Statistics class with iPhone and Earphone  Key Concepts in Statistics Parameter. In gauging the entire population, any measure obtained is called a parameter. Sample. The small number of observation taken from the total number making up a population is called a sample.  Key Concepts in Statistics Graphical Representation Distribution of Scores 120 Graphs. It is another way to visually show the behavior 110 of data. To create a graph, distribution of scores must be 105 105 organized. 100 X – Raw Score F – Frequency of Occurrence 120 1 100 110 1 95 120, 65, 110, 105 2 90 100 2 90 75, 105, 80, 95 1 90 105, 85, 100, 90 3 85 85, 100, 90, 85 2 85 80 1 95, 90, 90 75 1 80 65 1 75 65  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 2 ©2024 Batangas State University  MEASURES OF CENTRAL TENDENCY Mean.  MEASURES OF CENTRAL TENDENCY Median. It is the middle point or midpoint of any distribution.  MEASURES OF CENTRAL TENDENCY Mode. It is the most frequently occurring score in a distribution.  Appropriate Use of the Mean, Median and Mode Distribution of monthly income per household in a certain municipality.  A class of 13 students takes a 20-item quiz on Science 101. Their scores were as follows: 11, 11, 13, 14, 15, 18, 19, 9, 6, 4, 1, 2, 2. a. Find the mean. b. Find the median c. Find the mode. A day after, the of 13 students mentioned in problem 1 takes the same test a second time. This time their scores were: 10, 10, 10, 10, 11, 13, 19, 9, 9, 8, 1, 7, 8. a. Find the mean. b. Find the median c. Find the mode. d. Was there a difference in their performance when taking the test a second time? For the set of scores: 1000, 50, 120, 170, 120, 90, 30, 120. a. Find the mean. b. Find the median c. Find the mode. d. Which measure of central tendency is the most appropriate, and why?  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 2 ©2024 Batangas State University  Measures of Dispersion: Measures of Variability The Range of a set of data values is the difference between the greatest data value and the least data value. Machine 1 Machine 2 9.52 8.01 6.41 7.99 10.07 7.95 5.85 8.03 8.15 8.02 Mean = 8.0 Mean = 8.0  Measures of Dispersion: Measures of Variability The Standard Deviation. The life-blood of the variability concept. It provides measurement about σ(𝑥 − 𝜇)2 𝜎= how much all of the scores in the distribution 𝑛 normally differ from the mean of the distribution. σ(𝑥 − 𝑥)ҧ 2 𝑠= 𝑛−1  Measures of Dispersion: Measures of Variability The Standard Deviation. The life-blood of the variability concept. It provides measurement about how much all of the scores in the distribution normally differ from the mean of the distribution. Machine 2 X - Mean 8.01 8.01 – 8 7.99 7.99 – 8 7.95 7.95 – 8 8.03 8.03 – 8 8.02 8.02 – 8 Mean = 8.0 0  Measures of Dispersion: Measures of Variability The following numbers were obtained by sampling a population. σ(𝑥 − 𝑥)ҧ 2 𝑠= 2, 4, 7, 12, 15 𝑛−1 x ഥ 𝒙−𝒙 ഥ)𝟐 (𝒙 − 𝒙 2 4 7 12 15  Measures of Dispersion: Measures of Variability The Variance. Variance is another technique for assessing disparity in a distribution. In the simplest sense, variance is the square of the standard deviation.  At ABC University, a group of students was selected and asked how much of their weekly allowance they spent in buying mobile phone load. The following is the list of amounts spent: Php 120, 110, 100, 200, 10, 90, 100, 100. Calculate the mean, the range, and the standard deviation. At XYZ University, another group of students was selected and asked how much of their weekly allowance they spent in buying mobile phone load. The following is the list of amounts spent: Php 200, 180, 30, 20, 10, 160, 150, 80. Calculate the mean, the range, and the standard deviation. Consider the data in problems 1 and 2, in what way do the two distribution differ? Which group is more homogeneous?  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 2 ©2024 Batangas State University  Measures of Relative Position The z- Score Case A  Measures of Relative Position The z- Score Case B  Measures of Relative Position The z- Score Case C  Measures of Relative Position The z- Score Case D  Measures of Relative Position z-Scores The z-Scores for a given data value 𝑥 is the number of standard deviations that 𝑥 is above the mean of the data. 𝑥−𝜇 𝑥−𝑥ҧ Population: 𝑧𝑥 = Sample: 𝑧𝑥 = 𝜎 𝑠 In which exam did you do better? Grade 𝝁 𝝈 (population mean) (population SD) Physics 95 85 10 Biology 85 75 5 In which exam did you do better? Grade 𝝁 𝝈 (population mean) (population SD) Physics 95 85 10 Biology 85 75 5 -4 -3 -2 -1 0 +1 +2 +3 +4 Physics 45 55 65 75 85 95 105 115 125 Biology 55 60 70 80 75 80 85 90 95 Here is how to interpret z-scores: A z-score of less than 0 represents an element less than the mean. A z-score greater than 0 represents an element greater than the mean. A z-score equal to 0 represents an element equal to the mean. A z-score equal to 1 represents an element, which is 1 standard deviation greater than the mean; a z-score equal to 2 signifies 2 standard deviations greater than the mean; etc. A z-score equal to -1 represents an element, which is 1 standard deviation less than the mean; a z-score equal to -2 signifies 2 standard deviations less than the mean; etc. If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2 and about 99% have a z-score between -3 and 3. Example 1: Raul has taken two tests in his chemistry class. He scored 72 on the first test, for which the mean of all scores was 65 and the standard deviation was 8. He received a 60 on a second test, for which the mean of all scores was 45 and the standard deviation was 12. In comparison to the other students, did Raul do better on the first test or the second test? Example 2: A consumer group tested a sample of 100 light bulbs. If found that the mean life expectancy of the bulb was 842 h, with a standard deviation of 90. One particular bulb from the DuraBright Company has a z-score of 1.2. What was the life span of this light bulb? A percentile refers to a point in the distribution below which a given percentage of scores fall. Percentile for a Given Data Value Given a set of data and a data value 𝑥. number of data values less than 𝑥 Percentile of score 𝑥 = ∙ 100 total number of data values Example 3: On a reading examination given to 900 students. Elaine’s score of 602 was higher than the scores of 576 of the students who took the examination. What is the percentile for Elaine’s score? number of data values less than 602 Percentile = ∙ 100 total number of data values 576 = ∙ 100 900 = 64 𝐸𝑙𝑎𝑖𝑛𝑒 ′ 𝑠 𝑠𝑐𝑜𝑟𝑒 𝑜𝑓 602 𝑝𝑙𝑎𝑐𝑒𝑠 ℎ𝑒𝑟 𝑎𝑡 𝑡ℎ𝑒 64𝑡ℎ 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒. Example 4: On an examination given to 8600 students, Hal’s score of 405 was higher than the scores of 3952 of the students who took the examination. What is the percentile for Hal’s score? Quartiles. Quartiles divide the distribution into quarters. Example 5: The following table lists the calories per 100 milliliters of 25 popular sodas. Find the quartiles for the data. Calories, per 100 milliliters, of Selected Sodas 43, 37, 42, 40, 53, 62, 36, 32, 50, 49, 26, 53, 73, 48, 45, 39, 45, 48, 40, 56, 41, 36, 58, 42, 39 26, 32, 36, 36, 37, 39, 39, 40, 40, 41, 42, 42, 43, 45, 45, 48, 48, 49, 50, 53, 53, 56, 58, 62, 73  𝑄1 𝑄2 26, 32, 36, 36, 37, 39, 39, 40, 40, 41, 42, 42, 43, 45, 45, 48, 48, 49, 50, 53, 53, 56, 58, 62, 73 𝑸𝟐 = 𝟒𝟑 𝑄3 𝟑𝟗 + 𝟑𝟗 𝑸𝟏 = = 𝟑𝟗 𝟐 𝟓𝟎 + 𝟓𝟑 𝑸𝟑 = = 𝟓𝟏. 𝟓 𝟐 Example 6: The following table lists the weights, in ounces, of 15 avocados in a random sample. Find the quartiles for the data. 12.4, 10.8, 14.2, 10.2, 11.4, 12.6, 12.8, 13.1, 15.6, 9.8, 11.4, 12.2, 16.4, 14.5 9.8, 10.2, 10.8, 11.4, 11.4, 12.2, 12.4, 12.6, 12.8, 13.1, 14.2, 14.5, 15.6, 16.4 Box-and-Whisker Plots A box-and-whisker plot sometimes called a box plot is often used to provide a visual summary of a set of data. A box-and-whisker plot shows the median, the first and third quartiles, and the minimum and maximum values of data set. Box-and-Whisker Plots A box-and-whisker plot sometimes called a box plot is often used to provide a visual summary of a set of data. A box-and-whisker plot shows the median, the first and third quartiles, and the minimum and maximum values of data set. Construction of a Box-and-Whisker Plot 1. Draw a horizontal scale that extends from minimum data value to the maximum data value. 2. Above the scale, draw a rectangle (box) with its left side at Q1 and its right side at Q3. 3. Draw a vertical line segment across the rectangle at the median Q2. 4. Draw a horizontal line segment, called a whisker, that extends from Q1 to the minimum and another whisker that extends from Q3 to the maximum. Example 7: Construct a box-and-whisker plot for the following data. Number of Rooms Occupied in a Resort during 18-Day Period 86, 77, 58, 45, 94, 96, 83, 76, 75, 65, 68, 72, 78, 85, 87, 92, 55, 61 Stem-and-Leaf Diagrams Construction of Stem-and Leaf Diagram 1. Demonstrate the stems and list them in a column from smallest to largest or largest to smallest. 2. List remaining digit of each stem as a leaf to the right of the stem. 3. Include a legend that explains the meaning of the stems and the leaves. Include a title for the diagram. Example 8: History test scores: 65, 72, 96, 86, 43, 61, 75, 86, 49, 68, 98, 74, 84, 78, 85, 75, 86, 73 STEMS LEAVES 4 3, 9 6 1, 5, 8 7 2, 3, 4, 5, 5, 8 8 4, 5, 6, 6, 6 9 6, 8  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 5 ©2024 Batangas State University  Normal Distributions Frequency Distributions and Histograms Download time Number of (in seconds) subscribers 0-5 6 5-10 17 10-15 43 15-20 92 20-25 151 25-30 192 30-35 190 35-40 149 40-45 90 45-50 45 50-55 15 A Histogram for the frequency distribution 55-60 10 A Grouped Frequency Distribution with 12 Classes  Normal Distributions Frequency Distributions and Histograms Download time Number of (in seconds) subscribers 0-5.06 5-10 1.7 10-15 4.3 15-20 9.2 20-25 15.1 25-30 19.2 30-35 19.0 35-40 14.9 40-45 9.0 45-50 4.5 50-55 1.5 A Histogram for the frequency distribution 55-60 1.0 A Relative Frequency Distribution  Normal Distributions A normal distribution forms a bell-shaped curve that is symmetric about a vertical line through the mean of the data. Properties of a Normal Distribution The distribution curve is bell-shaped. The curve is symmetrical about its center. The mean, median and mode coincide at the center. The tails of the curve flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve asymptotic to the base line. The total area under the normal curve is equal to 1 or 100%. Two Factors that the graph of the normal distribution may depend on: 1. Mean 2. Standard Deviation Large values of SD makes the normal curve short and wide. Small value of SD yields a skinnier and taller normal curve. The Standard Normal Curve It is a normal probability distribution that has a mean 𝜇 = 0 and a standard deviation 𝜎 = 1 unit. Example 1: Find the area of the standard normal distribution between 𝑧 = −1.44 and 𝑧 = 0. Example 2: Find the area of the standard normal distribution between 𝑧 = −0.67 and 𝑧 = 0. Example 3: Find the area of the standard normal distribution to the right of 𝑧 = 0.82. Example 4: Find the area of the standard normal distribution to the left of 𝑧 = − 1.47. Example 5: A soda machine dispenses soda into 12-ounce cup. Test show that the actual amount of soda dispensed is normally distributed, with a mean of 11.5 oz and a standard deviation of 0.2 oz. a. What percent of cups will receive less than 11.25 oz of soda? b. What percent of cups will receive between 11.2 oz and 11.55 oz of soda? c. If a cup is filled at random, what is the probability that the machine will overflow the cup? Example 6: A study shows that the lengths of the careers of professional football players are nearly normally distributed, with a mean of 6.1 years and a standard deviation of 1.8 years. a. What percent of professional football players have a career of more than 9 years? b. If a professional football players is chosen at random, what is the probability that the player will have a career between 3 and 4 years? Translating the raw score into the z-score. Case A. When the percentage of cases is between the raw score and the mean. The normal distribution of physics scores has mean of 85 and a standard deviation of 10. What percentage of scores will fall between the physics score of 95 and the mean? Translating the raw score into the z-score. Case B. When the percentage of cases fall below a raw score. Using the same example, on a normal distribution of scores in physics class, with a mean of 85 and a standard deviation of 10, what percentage of physics scores fall below a score of 95? Translating the raw score into the z-score. Case C. When the percentage of cases is above a raw score. On a normal distribution of scores in physics class, with a mean of 85 and a standard deviation of 10, what percentage of physics scores above a score of 95? Translating the raw score into the z-score. Case D. When the percentage of cases is between raw scores. On a normal distribution of physics scores, the mean is 85 and the standard deviation is 10. Your physics score is 95 and your friends score is 80. You wanted to determine how many students got a score between your friend’s score of 80 and your score of 95.  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 5 ©2024 Batangas State University The Product-Moment Correlation Coefficient or Pearson r is an statistical tool that can determine the linear association between two distributions or groups. This tool can only establish the strength of association or correlation but it can never justify any causal relation that may appear or seemed obvious. The Formula for Pearson-r Linear Correlation σ 𝑿𝒀 − (ഥ 𝒙)(ഥ 𝒚) 𝒓= 𝑵 𝑺𝑫𝒙 𝑺𝑫𝒚 𝒙 refers to one variable and 𝑦 refers to another variable ഥ and 𝒚 𝒙 ഥ refers to the mean of 𝒙 and the mean of 𝒚 𝑺𝑫𝒙 and 𝑺𝑫𝒚 refers to the standard deviation of 𝒙 and 𝒚 respectively 𝑵 refers to the numbers of variable ∑ it is the symbol for summation Determine if there is a correlation between hours of study and grades of students last semester Student Hours of Study (x) Grade (y) A 15 2.75 B 35 1.25 C 05 3.00 D 20 2.50 E 30 1.50 F 40 1.00 G 20 2.25 H 25 1.75 I 25 2.00 J 08 3.00 Student Hours of Study (x) Grade (y) A 15 2.75 B 35 1.25 C 05 3.00 D 20 2.50 E 30 1.50 F 40 1.00 G 20 2.25 H 25 1.75 I 25 2.00 Guilford’s Interpretation for the values of r r value Interpretation Less than.20 Almost negligible relationship.20 -.40 Definite but small relationship.40 -.70 Substantial relationship.70 -.90 Marked relationship.90 – 1.00 Very dependable relationship Seven randomly selected participants were given both math and music tests. Their scores are as follows: Math Music 16 14 6 7 17 15 11 14 12 12 4 6 13 11 Is there math ability related to their music ability?  DATA MANAGEMENT 5.1 The Data 5.2 Measures of Central Tendency 5.3 Measures of Dispersion 5.4 Measures of Relative Position 5.5 The Normal Distributions 5.6 The Linear Correlation: Pearson r 5.7 The Least-Squares Regression Line 5 ©2024 Batangas State University Independent and Dependent Variable Number of Hours of Students Temperature Study Number of Electricity Bill Grade Teachers  Hours of Study Student Grade (y) (x) A 15 2.75 B 35 1.25 C 05 3.00 D 20 2.50 E 30 1.50 F 40 1.00 G 20 2.25 H 25 1.75 I 25 2.00 J 08 3.00 Regression Analysis is a reliable statistical method of estimating how a response variable depends on one or more predictors. Regression line is used to model the relationship of the variables.  3.5 3 Student Hours of Study (x) Grade (y) A 15 2.75 2.5 B 35 1.25 C 05 3.00 2 D 20 2.50 Grade Grade E 30 1.50 1.5 Linear (Grade) F 40 1.00 G 20 2.25 1 y = -0.0632x + 3.5096 H 25 1.75 R² = 0.959 I 25 2.00 0.5 J 08 3.00 0 0 10 20 30 40 50 Hours of Study The Formula for the Least-Squares Line The equation of the least-squares line for the 𝑛 ordered pairs 𝑥1 , 𝑦1 , 𝑥2 , 𝑦2 , 𝑥3 , 𝑦3 , … , 𝑥𝑛 , 𝑦𝑛 is 𝑦 = 𝑚𝑥 + 𝑏, where 𝑛 σ 𝑥𝑦−(σ 𝑥)(σ 𝑦) σ 𝑦 −𝑚(σ 𝑥) 𝑚= and 𝑏 = 𝑛 σ 𝑥 2 −(σ 𝑥)2 𝑛  Student Hours of Study (x) Grade (y) x2 xy A 15 2.75 B 35 1.25 C 05 3.00 D 20 2.50 E 30 1.50 F 40 1.00 G 20 2.25 H 25 1.75 I 25 2.00 J 08 3.00 𝒚 = 𝒎𝒙 + 𝒃, Slope (m) = 𝒏 σ 𝒙𝒚−(σ 𝒙)(σ 𝒚) σ 𝒚 −𝒎(σ 𝒙) 𝒎= and 𝒃 = y intercept (b) = 𝒏 σ 𝟐 σ 𝒙 −( 𝒙) 𝟐 𝒏  𝒚𝒑𝒓𝒆𝒅 = 𝒎𝒙 + 𝒃 x mx +b =y 37 22 8  GROUPINGS The Least-Squares Regression Line a. Adult men Stride length (m) 2.5 3.0 3.3 3.5 3.8 4.0 4.2 4.5 Speed (m/s) 3.4 4.9 5.5 6.6 7.0 7.7 8.3 8.7 b. Dogs Stride length (m) 1.5 1.7 2.0 2.4 2.7 3.0 3.2 3.5 Speed (m/s) 3.7 4.4 4.8 7.1 7.7 9.1 8.8 9.9 c. Camels Stride length (m) 2.5 3.0 3.2 3.4 3.5 3.8 4.0 4.2 Speed (m/s) 2.3 3.9 4.4 5.0 5.5 6.2 7.1 7.6