Mathematics In The Modern World PDF

MATHEMATICS IN THE MODERN WORLD MODULE 2 DATA MANAGEMENT OVERVIEW This module also covers data management which highlights the applications of descriptive statistics and hypothesis testing in managing data. Data management, derived...

MATHEMATICS IN THE MODERN WORLD MODULE 2 DATA MANAGEMENT OVERVIEW This module also covers data management which highlights the applications of descriptive statistics and hypothesis testing in managing data. Data management, derived from mathematics, is used in almost all fields of human endeavor. The applications span a broad range of topics such as demographics, business, transportation, research, health, sports, education, travel, architecture, entertainment, history, psychology, environment, engineering, and criminal justice. It is designed to build on the Source: https://www.vecteezy.com/vector-art/184613-worker- concepts and skill learned in Basic Education in presenting-data-visualization-vectors processing and interpreting statistical data to describe a phenomenon and to make informed decisions. Thus, the students are expected to use a variety of statistical tools to manage data and to advocate the use of statistical data in real-life situations. Essential Questions Why is there is a need to manage data? How can data be managed to communicate information and to make decision effectively? WORK PLAN FOR MIDTERM PERIOD HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 1  MATHEMATICS IN THE MODERN WORLD SECTION 2.1 APPLICATIONS OF DESCRIPTIVE STATISTICS 2.1.1. DESCRIPTIVE STATISTICS Descriptive Statistics Inferential Statistics ▪ Concerned with ▪ Concerned with making Statistical tools derived from describing the target inferences from the mathematics are useful in processing and population sample and generalize ▪ Organize, analyze, and them to the population managing data. Selection of appropriate tools present the data in a ▪ Compare, test and efficient use of these tools can help people meaningful manner hypothesis, and predict ▪ Results are shown in future outcomes organize, analyze, and interpret data. One of the form of charts, tables, ▪ Results are shown as useful statistical tools in data management is and graphs statistics and probability ▪ Tools: measures of values descriptive statistics. Descriptive statistics central tendency, ▪ Tools: hypothesis tests, variation, and position analysis of variance deals with organization, presentation, and analysis of data that help describe, show, or Table 2.1 Descriptive Statistics vs Inferential Statistics summarize data in a meaningful way. It covers Table 1 summarizes the differences measures of central tendency, measures of between descriptive statistics and inferential variation, measures of position, and graphical statistics in terms of their nature, purpose, data presentation of data. Although these concepts presentation, and data analysis tools. appear to be the same as the statistics taught at Which of the following situations below Junior High School mathematics, the intention will require the use of descriptive statistics? of this topic is to highlight students’ skills in Situation 1: A sports analyst wants to know the selecting appropriate descriptive statistics and performance of basketball players based on the interpreting statistical results. Furthermore, tallied numbers such as points scored, blocks, this topic is focused on the applications of and rebounds during a game series. descriptive statistics in real-life situations. Situation 2: A health expert plans to investigate what public measures would prevent a disease from spreading and contribute to its eradication. Situation 3: A teacher wishes to know the achievement of students in the Mathematics in the Modern World after the midterm period. If you answered situations 1 and 3, you are correct! Source: https://www.freepik.com/free-vector/statistics-bar-doughnut-line- charts-template_3034283.htm#page=1&query=statistics&position=2 HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 2  MATHEMATICS IN THE MODERN WORLD 2.1.2. DATA PRESENTATION One of the important processes involved in data management is presentation of data. In descriptive statistics, data are summarized, organized, and communicated using a variety of visual tools such as tables, diagrams, charts, and statistical graphs. Common statistical tools include line frequency distribution, line graph, bar graph, pie graph, and scatter plots. The use of these graphs depends on the purpose of the and Source: https://www.sws.org.ph/swsmain/artcldisppage/?artcsyscode=ART- nature of the data. This topic presents ways on 20160223095529 Figure 2.1 Example of Time Series Graph how to organize and present data in charts and graphs. Bar Graph ▪ presents the data by using horizontal or vertical bars whose heights or lengths are determined by the frequencies of the data ▪ can be used to compare data for two or more groups Source: https://cyberbabble.blog/2019/04/22/the-importance-of-data- presentation/ Here are some common statistical graphs used in data presentation with illustrative examples. Time Series Graph ▪ represents data that occur over a specific period; to look for pattern or trend that occurs over the period of time Source: https://www.geekwire.com/2012/chart-pays-seattle-software- engineers/ ▪ two or more data sets can be compared on Figure 2.2 Example of Bar Graph the same time series graph HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 3  MATHEMATICS IN THE MODERN WORLD Pie Graph Frequency Distribution ▪ is a circular graph which is divided into ▪ is the organization of data in tabular form sectors or wedges according to the using classes and frequencies percentage of the frequencies in each category of the distribution of a nominal or categorical variable ▪ used in showing relationship of the parts to the whole by visually comparing the sizes of the sections Common Causes of Catastrophes in Business Source: Estrada, L. (2015). Effecting change effecting change on students’ performance and attitude toward problem solving in mathematics through schema-based instruction. Unpublished thesis. Figure 2.5 Example of Frequency Distribution Table You can make formal, creative, and presentable graphs and charts using computer Source: https://oncall258.com/business- continuity-planning-alexandria-va/ software or applications. Canva and MS Excel are Figure 2.3 Example of Pie Graph the most used tools in creating charts or graphs Scatter Plot for data presentation. You need to choose the ▪ is a graph of ordered pairs of data values most appropriate statistical graph to that is used to describe the relationship that communicate the results or information. exists between two variables Practice Makes Perfect. Identify the most appropriate statistical graph to be used in presenting the following data and state your reason. a. Common cyber crimes b. Monthly maximum temperature recorded in 2019 c. Population of ASEAN countries in 2019 d. Earth’s composition of gases e. Number of HAU Passers in Mechanical Source: https://www.mymarketresearchmethods.com/types-of-charts-choose/ Engineer Examination from 2015-2019 Figure 2.4 Example of Scatter Plot HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 4  MATHEMATICS IN THE MODERN WORLD 2.1.3. DESCRIPTIVE MEASURES Weighted Mean The previous topic shows how you can ▪ is determined when the values are not organize, present, and communicate equally represented (e.g. general weighted information using various statistical graphs. grade This topic will show you the statistical tools ▪ is obtained by multiplying each value by its that can be used to summarize quantitative corresponding weight, divided the sum of the data. The most familiar of these tools are products by the sum of the weights measures of central tendency and measures of Mode variation. These measures may be considered parameter or statistic. ▪ is the value that occurs most often in the data set PARAMETER vs STATISTIC Parameter is a Statistic is a ▪ is denoted by 𝑥̂ characteristic or characteristic or ▪ is obtained by identifying which value has the measure obtained all measure obtained from greatest frequency the values from a sample of a population. population ▪ a data set may be unimodal (has 1 mode), bimodal (has 2 modes), multimodal (has 3 or Measures of Central Tendency more modes) or no mode Measures of central tendency are also Median called measures of average. They represent the ▪ is the middle most value or midpoint of a data center of distribution or the most typical case. set These measures include mean, median, and ▪ is represented by 𝑥̃ mode. Do you still remember them? ▪ is obtained by arranging the values in Mean ascending or descending order and selecting ▪ a.k.a. arithmetic average the middle value if the number of values is ▪ is denoted by 𝑥̅ (sample mean) or 𝜇 odd or get the mean of the two middle values (population mean) if the number of values is even ▪ is obtained by getting the sum of the values In APA format, the symbols M and Mdn are used to report the in a data set, divided by the number of mean and median of data, respectively. There is no symbol for values mode. ▪ HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 5  MATHEMATICS IN THE MODERN WORLD Measures of Variation Variance Standard Deviation The measures of variation or dispersion Population ∑(𝑥𝑖 − 𝜇)2 ∑(𝑥𝑖 − 𝜇)2 𝜎2 = 𝜎=√ tell us how the data values are dispersed or 𝑁 𝑁 scattered. They indicate the degree to which ∑(𝑥𝑖 − 𝑥̅ )2 𝑠2 = ∑(𝑥𝑖 − 𝑥̅ )2 Sample 𝑛−1 𝑠=√ quantitative data tend to spread about an 𝑛−1 average value. Three common measures of Legends: 𝜇 – population mean; N – population size; 𝑥̅ – sample mean; n – sample size; xi – individual value variation are range, variance, and standard Table 2.2 Formulas for Variance and Standard Deviation deviation. They may indicate the consistency and homogeneity of data set. Rounding Rule for the Mean, Standard Deviation, and Variance Range The mean, standard deviation, and variance ▪ is obtained by computing the difference of should be rounded to one more decimal place the highest value and the lowest value in a than that of the original data. data set, denoted by R Example Variance 1. A sample of 8 drivers shows the time (in minutes) that they spent in rush-hour traffic ▪ is obtained by getting the average of the that day. Given the data set below, compute squares of the distance each value is from the for the mode, median, mean, range, standard mean deviation, and variance. ▪ is represented by 𝜎 (population variance) 2 Driver A B C D E F G H or s2 (sample variance) Time 52 61 53 60 55 47 60 44 ▪ is the square of standard deviation Solutions: Standard Deviation a. Mode = 60 minutes ▪ is obtained by the square root of variance 60 has the highest frequency. ▪ is represented by 𝜎 (population standard b. Median = 54 minute deviation or s (sample standard deviation) Arrange the values in ascending order: 44, 47, 52, 53, 55, 60, 60, 61 The formulas for standard deviation and Since the number of values is (8) even, the variance are presented in Table 2. Take note that average of the two middle values (53 and variance is the square of standard deviation. 55) is 54. HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 6  MATHEMATICS IN THE MODERN WORLD c. Mean = 54 minutes Data Analysis in MS Excel 44 + 47 + 52 + 53 + 55 + 60 + 60 + 61 For larger population size or sample size, you 𝑥̅ = 8 can explore MS Excel in computing for the 432 mean, standard deviation, and variance. 𝑥̅ = = 54 8 Watch the video: How to Find Average and Standard Deviation in Microsoft Excel 2017 d. Range = 17 minutes https://www.youtube.com/watch?v=BGq8kuffR_Q highest value – lowest value = 61 – 44 = 17 Practice Makes Perfect. Try to answer the following problems. e. Variance = 39.4 minutes 1. Listed below are the number of calories in ∑(𝑥𝑖 −𝑥̅ )2 selected salads at several restaurants. Find Use the formula 𝑠 2 = 𝑛−1 the mean, median, and mode for the data. 𝑥𝑖 (𝑥𝑖 − 𝑥̅ )2 1165 1180 1090 900 1270 44 (44 – 54)2 = 100 47 (47 – 54)2 = 49 985 855 1120 900 1180 52 (52 – 54)2 = 4 900 1250 855 730 1595 53 (43 – 54)2 = 1 2. The grades of 10 students after completing an 55 (55 – 54)2 = 1 online course are presented below. 60 (60 – 54)2 = 36 Determine the range, variance, and standard 60 (60 – 54)2 = 36 deviation for the data. 61 (41 – 54)2 = 49 1.00 1.50 1.75 1.50 1.25 n=8 𝛴(𝑥𝑖 − 𝑥̅ )2 = 276 1.50 2.25 2.25 1.00 2.00 3. A testing laboratory wishes to test two 276 𝑠2 = ≈ 39.4 square minutes experimental brands of paint to see how long 8−1 f. Standard deviation = 6.3 minutes each will last before fading. The testing lab ∑(𝑥𝑖 −𝑥̅ )2 makes 6 trials for each brand. The results (in Use the formula 𝑠 = √ 𝑛−1 months) are shown below. Which paint brand 276 is better? Support your answer. 𝑠 = √8−1 ≈ 6.3 minutes Brand A Brand B Note that standard deviation is the square root 10 35 60 45 of variance. 50 30 30 35 s = √39.4 ≈ 6.3 40 40 20 25 HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 7  MATHEMATICS IN THE MODERN WORLD Supplemental Readings Descriptive vs. Inferential Statistics: What’s the Difference? by Statology https://www.statology.org/descriptive-inferential-statistics/ Descriptive statistics: Use these tools to analyze data vital to practice-improvement projects by Corner, B. & Johnson, E. Source: https://www.myamericannurse.com/wp-content/uploads/2017/11/ant11-Research-101- 1017a-1.pdf Presenting Numerical Data by the Student Learning Development, University of Leicester Source: https://www2.le.ac.uk/offices/ld/all-resources/numerical-data/numerical-data Difference between descriptive and inferential statistics https://www.differencebetween.com/difference-between-descriptive-and-vs-inferential-statistics/ https://statisticsbyjim.com/basics/descriptive-inferential-statistics/ How to describe charts, graphs, and diagrams in the presentation https://preply.com/en/blog/2018/08/17/charts-graphs-and-diagrams-in-the-presentation/#scroll- to-heading-5 References Singh, S. (2018). Descriptive and inferential Singh. https://towardsdatascience.com/statistics- descriptive-and-inferential-63661eb13bb5 Blumann, A. (2016). Elementary statistics: A step by step approach, A brief Version (7 th ed.). McGraw- Hill Education. Rosenthal, G., & Rosenthal, J. (2012). Statistics and Data Presentation for Social Work (Par I Introduction and Descriptive Statistics). http://web.a.ebscohost.com/ehost/ebookviewer/ebook/ZTAwMHh3d19fNDMyMjc1X19BTg 2?sid=679681db-61d8-4f59-8127-f3ec8727e9c5@sdc-v- sessmgr01&vid=5&format=EB&rid=1 HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 8  MATHEMATICS IN THE MODERN WORLD SECTION 2.2 APPLICATIONS OF HYPOTHESIS TESTING 2.2.1. HYPOTHESIS TESTING Statistics, as a mathematical tool, is designed to help the researcher or statistician in different fields to make decisions about the outcomes of potentially important body of knowledge for planning, development, and evaluation. Inferential statistics is most appropriate in situations that call for testing, comparing, and predicting outcomes. One crucial process in conducting inferential Source: statistics is hypothesis testing. In this topic, you will explore the applications of hypothesis testing. Statistics in Real World When you were in Senior High School, Today many higher education institutions offer online courses because of the impact of the you already had an introductory course on Coronavirus disease 2019 (COVID-19) pandemic. With parametric statistics through the mathematics the increased use of this learning platform, some subject “Statistics and Probability.” In this topic questions about the effectiveness of online learning have been raised. For instance, your knowledge and skills about statistical ▪ Do students perform better in online learning hypothesis testing will be further enhanced. setting than in the face-to-face classroom Hypothesis testing can be likened to a court setting? trial. We assume that the defendant is either ▪ Do gender, types of interaction, and technology innocent or guilty, and the defendant will be facility influence student satisfaction with online learning? convicted or acquitted. ▪ Is there is a significant relationship between self- Your MMW teacher assumes that you are efficacy and online learning environments? equipped with basic knowledge about statistical To answer these questions, we need to work with hypothesis testing. Let us begin with defining data set and perform data analysis using various statistical tests through hypothesis testing some terms related to hypothesis testing. HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 9  MATHEMATICS IN THE MODERN WORLD Definition of Terms Type II Error Hypothesis ▪ occurs when a false null hypothesis is ▪ is an assumption, a claim, or a proposed accepted or not rejected, a.k.a. Beta error explanation for an observation or One-Tailed Test phenomenon ▪ is a test that indicates that the null Null Hypothesis hypothesis should be rejected when the ▪ is a claim that indicates the absence of test statistic value is in the critical relationship or difference between two region on one side of the mean, a.k.a. values, denoted by symbol Ho directional test Alternative Hypothesis Two-Tailed Test ▪ is a claim that indicates the presence of ▪ is a test that indicates the null hypothesis relationship or difference between values, should be rejected when the test statistic denoted by symbol Ha or H1 value is in either of the two critical regions, a.k.a. non-directional test Hypothesis Testing Confidence Level ▪ is a decision-making process for evaluating or testing claims about a ▪ is the probability that a parameter lies population based on information obtained within the specified interval estimate of from samples the parameter Level of Significance Critical Value ▪ is the maximum probability of committing ▪ is a value that separates a critical region type I error, denoted by the Greek (rejection region) from acceptance region alphabet alpha (𝛼) in a hypothesis test, usually presented in tables Type I Error Statistic ▪ occurs when a true null hypothesis is rejected, a.k.a. Alpha error ▪ is a value obtained by using all data value of aI sample HOLY ANGEL UNIVERSITY SCHOOLfrom OF aARTS population AND SCIENCES 10  MATHEMATICS IN THE MODERN WORLD Steps in Hypothesis Testing For the sample size, according the Central Table 2.3 summarizes the steps to be Limit Theorem, at least 30 samples is undertaken when conducting statistical recommended. The test statistics to be employed hypothesis testing. Every hypothesis-testing depends on the nature of data and the purpose of context starts with declaration of hypothesis. test. What are the scales of measurement of the There are two types of statistical hypotheses data? Are data continuous? Is it a test of for each context: the null hypothesis and the association, comparison, or difference? In the alternative hypothesis. next subtopic, common statistical tests will be presented. The numerical value obtained from a Steps in Statistical Hypothesis Testing statistical test (called the computed value or test 1 State the null hypothesis and alternative statistic) is compared with critical value to hypothesis 2 Set the level of significance and determine the decide on the rejection or acceptance of null direction of the tests hypothesis. There are four possible outcomes as 3 Collect data 4 Calculate a test statistic and determine the shown in Table 2.4 in hypothesis-testing critical value situation. 5 Decide on the rejection or acceptance of null hypothesis Finally, based on the decision, we have to 6 Draw a conclusion draw a conclusion. When drawing a conclusion Table 2.3 Formulas for Variance and Standard Deviation or summary of results of a statistical study, it is Moreover, in hypothesis testing, the important to state whether there is enough researcher decides what level of significance evidence to reject the null hypothesis or support (𝛼) to use. In research 𝛼 is usually set at 0.01 or the alternative hypothesis. 0.05. After a significance level is chose, the type For you, which hypothesis-testing step is of test is determined: one-tailed test or two- difficult to accomplish? Why? tailed test. The type of test depends on the statement of alternative hypothesis. If the True Ho False Ho alternative hypothesis involves the symbol > or Type I Correct Reject Ho Error Decision 2.026), we reject the null hypothesis. 6 Conclusion: There is a significant difference in the https://cyfar.org/types-statistical-tests mathematics achievement of the two groups of students. Table 2.7 Example of Hypothesis Testing using T-Test HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 15  MATHEMATICS IN THE MODERN WORLD Activity Supplemental Readings Normal Distribution Hypothesis testing is a formal https://www.dummies.com/education/math/statistics procedure for investigating our ideas about /understanding-the-statistical-properties-of-the- normal-distribution/ the world using statistics. Statistical tests are Hypothesis Testing Examples used in hypothesis testing. Choosing the right https://online.stat.psu.edu/statprogram/reviews/stati statistical tests will make our hypothesis stical-concepts/hypothesis-testing/examples testing valid. In this activity, you must know Videos the uses of various statistical tests for How to Perform Pearson Correlation Test in Excel comparison, relationship, and regression and https://www.youtube.com/watch?v=Ev86DMtLXOk provide illustrative examples for each Chi Squared Test statistical test. Complete the table below by https://www.youtube.com/watch?v=qYOMO83Z1WU& t=2s identifying the conditions required and uses of each statistical test and give a situation where Choosing which Statistical Test to Use – Statistics Help https://www.youtube.com/watch?v=rulIUAN0U3w it can be applied. Statistical Conditions References Examples Tests and Uses z-test Bluman, A. (2016). Elementary statistics: A step by step approach, A brief Version (7th ed.). Paired t-test McGraw-Hill Education. Gonzales, J. (2015). Essential Statistics. MaxCor Independent t- Publishing. tests Chapter 8: Introduction to Hypothesis Testing. ANOVA https://www.sagepub.com/sites/default/files /upm-binaries/40007_Chapter8.pdf Pearson r Rosenthal, G., & Rosenthal, J. (2012). Statistics and Spearman rho Data Presentation for Social Work (Par II Inferential Statistics and Data Presentation). Chi-square http://web.a.ebscohost.com/ehost/ebookvie wer/ebook/ZTAwMHh3d19fNDMyMjc1X19B Tg2?sid=679681db-61d8-4f59-8127- f3ec8727e9c5@sdc-v- sessmgr01&vid=5&format=EB&rid=1 HOLY ANGEL UNIVERSITY I SCHOOL OF ARTS AND SCIENCES 16