Statistics and Probability

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PREFACE Prior to the imple mentation of K-12, Statisti cs was taught in public high schools in the Philippine s typical ly in the last quarter of third year. In private schools, Statistics was taught as either an elective, or a re quired but separate subje ct outside of re gular Math classes. In college, Statistics was taught practically to everyone either as a three unit or six unit course . All colle ge students had to take at le ast three to six units of a Math course, and would typically “endure” a Statistics course to graduate . Teachers who taught these Statistics classes, whether in high school or in college, would typical ly be Math teachers, who may not necessarily have had formal traini ng in Statistics. They we re se lected out of the understanding (or misunderstandi ng) that Statistics is Math. Statistics does depend on and uses a lot of Math, but so do many discipline s, e.g. engineering, physics, accounting, che mistry, computer science. But Statistics i s not Math, not e ven a branch of Math. Hardly would one think that accounting is a branch of mathematics simply because it does a lot of calculations. An accountant would also not describe himself as a mathe matician. Math largely involves a deterministic way of thinking and the way Math is taught in schools le ads learners into a dete rministic way of examining the world around them. Statistics, on the other hand, is by and large dealing with uncertainty. Statistics uses inductive thinking (from specifics to gene ralities), while Math uses deduction (from the gene ral to the specific). “Statistics has its own tools and ways of thinking, and statisti cians are quite insistent that those of us who te ach mathematics re alize that statistics is not mathe matics, nor is it e ven a branch of mathe matics. In fact, statistics is a se parate discipline with its own unique wa ys of thinking and its own tools for approaching problems.” - J. Michael Shaughnessy, “Re se arch on Students’ Understanding of Some Big Concepts in Statistics” (2006) Statistics deals with data; i ts importance has been recognized by governme nts, by the private se ctor, and across disciplines because of the ne ed for evidence-based decision making. It has become e ven more i mportant in the past fe w years, now that more and more data is being collected, stored, analyzed and re -analyzed. From the time when humanity first walked the face of the earth until 2003, we created as much as 5 e xabytes of data ( 1 exabyte being a billion “gigabytes”). Information communications technology (ICT) tools have provided us the means to transmit and exchange data much faster, whe ther these data are in the form of sound, te xt, visual images, signals or any other form or any combination of those forms using desktops, laptops, table ts, mobile phones, and othe r gadgets with the use of the i nternet, social media (facebook, twitter). With the data deluge ari sing from using ICT tools, as of 2012, as much as 5 exabytes were being cre ated every two days (the amount of data cre ated from the beginning of history up to 2003); a year later, this same amount of data was now bei ng created every ten minutes.

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In order to make se nse of data, which is typically having variati on and uncertainty, we nee d the Science of Statistics, to enable us to summarize data for describing or explaining phe nomenon; or to make predictions (assuming trends in the data continue). Statistics is the science that studies data, and what we can do with data. Te achers of Statistics and Probability can easily spend much time on the formal methods and computations, losing sight of the real applications, and taking the excitement out of things. The eminent statistician Bradle y Efron me ntioned how diverse statistical applications are: “During the 20th Century statistical thinking and methodology has become the scie ntific framework for li terally dozens of fields including education, agriculture, economics, biology, and me dicine, and with incre asing influence re cently on the hard scie nces such as astronomy, geology, and physics. In other words, we have grown from a small obscure fie ld into a big obscure field.” In consequence, the work of a statistician has become even fashionable. Google ’s chief economist Hal Varian wrote in 2009 that “the se xy job in the next te n years will be statisticians.” He we nt on and mentioned that “The ability t o take data - to be able to understand it, to process it, to e xtract value from it, to visuali ze it, to communicate it's going to be a hugely important skill in the ne xt decades, not only at the professional le vel but e ven at the e ducational le vel for e lementary school kids, for high school kids, for colle ge kids. “ This te aching guide, pre pare d by a te am of professional statisticians and e ducators, aims to assist Se nior High School te achers of the Grade 11 se cond semester course in Statistics and Probabil ity so that the y can he lp Senior High School students discover the fun in de scribing data, and in e xploring the storie s behind the data. The K -12 curriculum provides for concepts in Statistics and Probabili ty to be taught from Grade 1 up to Grade 8, and in Grade 10, but the depth at which learners absorb the se concepts may need re inforcement. Thus, the first chapter of this guide discusse s basic tools (such as summary measures and graphs) for describing data. While Probability may have been discussed prior to Grade 11, it is also discussed in Chapte r 2, as a prelude to defining Random Variables and their Distributions. The next chapter discusses Sampling and Sampling Distributions, which bridges Descriptive Statistics and Inferential Statistics. The latter is started in Chapter 4, in Estimation, and further discussed in Chapter 5 (which deals with Tests of Hypothesis). The final chapter discusse s Regression and Corre lation. Although Statistics and Probability may be tange ntial to the primary training of many if not all Senior High School teachers of Statistics and Probability, it will be of benefit for them to se e why this course is important to te ach. After all, if the te achers themselves do not find meaning in the course, neither will the students. Work developing this set of te aching mate rials has bee n supported by the Commission on Highe r Education under a Materials Development Sub-proje ct of the K -12 Transition Project. These mate rials will also be share d with Department of Education. C h a p t e r 1 D e s c r i b i n g D a t a – L e s s o n 1

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Writers of this teaching guide re cognize that few Senior High School teachers would have formal training or applied experience with statistical concepts. Thus, the guide gives concrete suggestions on classroom activities that can illustrate the wide range of processe s behind data collection and data analysis. It would be ideal to use technology (i.e. computers) as a means to help teachers and students with computations; hence, the guide also provides suggestions in case the class may have acce ss to a computer room (particularly the use of spre adsheet applications like Microsoft Exce l). It would be unproductive for te achers and students to spend too much time working on formulas, and checking computation e rrors at the expense of gaining knowle dge and insights about the concepts behind the formulas. The guide gives a mixture of le ctures and activities, (the latter include actual collection and analysis of data). It trie s to follow suggestions of the Guidelines for Asse ssment and Instruction in Statistics Education (GAISE) Project of the American Statistical Association to go beyond lecture methods, and instead exercise conceptual learning, use active le arning strategies and focus on re al data. The guide suggests what material is optional as there is really a lot of material that could be taught, but too little time. Teachers will have to find a way of re cognizing that diverse needs of students with variable abilities and interests. This te aching guide for Statistics and Probability, to be made available both digitally and in print to se nior high school te achers, shall provide Se nior High School teachers of Statistics and Probability with much- needed support as the country’s basic education syste m transi tions into the K-12 curriculum. It is earnestly hoped that Senior High School teachers of Grade 11 Statistics and Probability can direct students into examining the context of data, ide ntifying the consequences and implications of stories behind Statistics and Probabili ty, thus becoming critical consumers of information. It is furthe r hoped that the competencies gained by students i n this course wi ll help them become more statistical literate, and more prepare d for whatever employment choices (and higher education specializations) given that employers are recognizing the importance of having thei r employee know skills on data manage ment and analysis in this very data-centric world.

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Chapter 1: Describing Data Lesson 1: Variation in Data

TIM E FRAME: 1 hour session OVERVIEW OF LESSON In this activity, students will be asked to provide some data that will be submitted for consolidation by the teacher for future lessons. Data on heights and weights, for instance, will be used for calculating Body Mass Inde x in Lesson 3. Students will also discuss the concept of statistical questions (in relation to non-statistical ones), then work in groups to discover variation in data. Students will be asked to imagine that the data they obtained in their groups would now be collected for much larger groups (the entire class, all grade 11 students in school, all grade 11 students in the district), and to discuss how data could be summarized.

LEARNING COM PETENCIES: At the end of the lesson, the learner should be able to:

distinguish statistical questions from non-statistical questions, recognize that data possess variability, identify methods for summarizing data to answer statistical questions, i.e., sort, classify, and organize data in tabular form and present this into a pictographs, bar charts, etc.

LESSON OUTLINE: 0. 1. 2. 3. 4.

Preliminaries Introduction on Statistics as the Science that Studies Data Initial Lesson: Statistical and Non Statistical Questions Main Lesson: Data and Statistics Small Data Collection Activity and Planning for Data Analysis

DEVELOPMENT OF THE LESSON (A) Preliminaries (for Future Lessons) Before the lesson and course starts, prepare a sheet of paper listing everyone’s name in class with a “Student Number” (see next page). The student number is a random number chosen in the following fashion: (a) Make a box with “tickets” listing the numbers 1 up to the number of students in class.

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(b) Shake the box, get a ticket, and give the number in the ticket to the first person in the list. (c) Shake the box again, get another ticket, and give the number of this ticket to the ne xt person in the list. (d) Do (c) until you run out of tickets in the box. Once the list (see next page) is finished, make sure to inform students confidentially of their student numbers. Perhaps, when the attendance is called, each student can be provided a separate piece of paper that lists her/his name and student number. Tell students to remember their student number, and to always use this throughout the class whenever data are requested of them. Explain to students that in data collection specific identities are not required, especially because people have a right to confiden tiality, but there should be a way to develop and maintain a database to check quality of data provided, and verify from respondent in data collection activity the data provided. Explanatory Note: These preliminary steps for generating a student number and informing students confidentially of their student number are e ssential for the “data collection” activities to be taken in this lesson and other lessons so that students can be uniquely identified, without having to obtain their names. In statistical activities, facts are collected from respondents for purposes of getting aggregate information, but confidentiality should be protected. This way, re spondents can be truthful in giving information, and the researcher can give a commitment to respondents that the data they provide will never be released to anyone in a form that will identify them without their authorization.

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Student Name

Student Number

1. 2, 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32, C h a p t e r 1 D e s c r i b i n g D a t a – L e s s o n 1

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(B) Introduction Provide students their “student numbers” and ask them to fill out Activity Sheet 1-01a. After 3-5 minutes, tell students to submit the Sheet to you so you can put all records on the Class Recording Sheet. Explain to students that compiling all the se records from everyone in the class is an example of a census since data has been gathered from every student in class. Mention that the government, through the Philippine Statistics Authority (PSA), conducts censuses to obtain information about socio-economic conditions in the country. This helps government makes plans, such as how many schools and hospitals to build. Censuses of population and housing are conducted every 10 years on years ending in zero (e.g., 1990, 2000, 2010) to obtain population counts, and demographic information about all Filipinos. Mid decade population censuses have also be en conducted since 1995. Censuses of agriculture, and of Philippine business and industry, are also conducted by the PSA to obtain information on production and other relevant economic information. Inform students that the student numbers they were given are meant to identify them without having to know their specific identities in the class recording sheet (which will contain the consolidated records everyone provided). This helps protect confidentiality of information. Mention that the PSA is bound by law to protect th e confidentiality of information provided by responden ts. Even market research organizations in the private sector and individual researchers also guard confidentiality as they merely want to obtain aggregate data. Ask students what comes to their minds when they hear the term “ data” (which may be viewed as a collection of facts from experiments, observations, sample surveys and censuses, and administrative reporting systems). Give them a follow up question about whether data, such as the facts they gave in activity sheet 1-01a, would be the same or varying from person to person. Even if the data varies, some numbers may show up more than once in the entire data set. The frequency of a particular data value is the number of times the data value occurs.

(C) Initial Lesson: Statistical Questions and Non-Statistical Questions Tell students that data are collected to answer statistical questions, the answers of which can change depending on who it is asked to, and when it is asked (Non-statistical questions are questions that anticipate a single answer.) Give a 5 minute exercise to the students on distinguishing statistical questions from non-statistical questions. C h a p t e r 1 D e s c r i b i n g D a t a – L e s s o n 1

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Ask students which of the following are statistical questions and why:

How old is student number 3 (in the class list)? (Not a statistical question since only a particular fact, the age of student number 3, is of interest) How old are the people w ho watch the most recent episode of the television show “Maalaala Mo Kaya ?” (Statistical question since this will require getting data on ages of all viewers of the tv show) Do dogs eat more than cats? (Statistical question since this will require getting data on the amount of food eaten by dogs and cats, maybe a sample of them, for a reference period, say past week, or past month) Is the vehicle of the Mayor of our city/town/municipality bigger than the vehicle used by the President of the Philippines? (Not a statistical question since only a particular fact, size of mayor’s car in relation to size of vehicle of president, is of interest) How many days are there in December? (Not a statistical question since only a particular fact, number of days in December, is of interest) Does it rain more in Cebu than in Davao? (Statistical question since this will require getting rainfall data on the two cities in a reference period, say past month, past year) Do I have a college degree? (Not a statistical question since only a particular fact, whether or not I have a college degree, is of interest) How much was the Supreme Court Chief Justice’s last paycheck for? Not a statistical question since only a particular fact, the amount of income received by the SC CJ in the last paycheck, is of interest) Do math teachers earn more than science teachers? (Statistical question since this will require getting data on income/wages of either all math and science teachers, or even a sample of these teachers) How many searches on Google do residents in M akati City (or some other city near the school) conduct each day? (Statistical question since this will require getting data on the frequency of Google searches of residents of M akati or whatever city of interest) What is the weight of Student A, say Ana (or whatever random name you can get from the list of student names)? (Not a statistical question since only a particular fact, weight of Student A is being asked) W hat is the proportion of students in class w ho are underweight or overweight for their age? (Statistical question since this will require getting data on the weights of all students in class, comparing these w eights to a reference weight for student ages, and determining the percentage of students that are underweight or overweight for their age)

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(D)M ain Lesson: Data and Statistics Suggest to students that data may be viewed as the facts (counts, measureme nts, or opinions) obtained to answer a statistical que stion. Define Statistics is a science that studies data , and what we can do with data. Suggest that this involves processes from collecting, processing (including performing quality checks), analyzing, interpreting and communicating data. Trivia: The word “statistics” actually comes from the word “state”— because governments have been involved in the statistical activities, especially the conduct of censuses either for military or taxation purposes. The ne ed for and conduct of censuses are recorded in the pages of holy texts. In the Christian Bible, particularly the Book of Numbers, God is reported to have instructed Moses to carry out a census. Another census mentioned in the Bible is the census ordered by Caesar Augustus throughout the entire Roman Empire before the birth of Christ. Inform students that uncovering patterns in data involves not just science but also art, and this is why some people may think “Stat is ee eks!” and may view any statistical procedures and results with much skepticism. (See Figure 1-1.)

Figure 1-1. Cartoon: " ...recommended by 4 out of 5 quacks!" (Source: http://www.cartoonstock.com)

Make known to studen ts that statistical methods enable us to C h a p t e r 1 D e s c r i b i n g D a t a – L e s s o n 1

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characterize persons, objects, situations, and phenomena; explain relationships among variables ; formulate objective assessments and comparisons; and, make evidence-based decisions and predictions.

and that the main tasks of a statistician include:

Designing the collection of data to answer statistical questions in a way that maximizes information content and minimize s bias; Verifying the quality of the data after it is collected Examining data so that insight and meaningful information can be produced to support decision making.

(E) Small Data Collection Activity and Planning for Data Analysis Divide students into groups of five. Tell students to share with each other some opinions and facts to answer about three questions (that will not yield a “sensitive” response). They may use questions for Activity Sh eet 1-01a:

What is their height (in cm?) and weight (in kg)? mother? What is the ir favorite color? How do they feel today?

the age of their

Or other questions (that will not yield a sensitive response):

Are the five students satisfied with the way the mayor does his/her job? (yes, no, unsure, no opinion) How many hours did the students watch television during the past seven days? How many hours did they go on facebook yesterday?

Explanatory Note about Data Collection Activity: The questions listed above are indicative to help students learn that data has variation. It is crucial to ask questions of interest that are not culturally sen sitive. After 5 minutes of sharing answers with each other, ask students whether the answers shared with each other were the same, or whether they varied. Tell students to imagine that these same questions would now be asked of all grade 11 students in the entire school, or in the entire district. Ask them how they would summarize the information collected. Possible Answers:

Histogram for heights and weights of students Histogram for ages of their fathers and for their mothers

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Bar chart or Pie Chart for (Distribution of) favorite color Pictogram/Pie chart/Bar chart on satisfaction with the mayor Bar chart/Pie chart for hours spent watching television in the past 7 days Bar chart for hours spent on facebook yesterday

KEY POINTS

Difference between a statistical question and a non -statistical question The bedrock of statistics (the science that studies data) is data, which is characterized by variation. We can summarizing data collected to answer a statistical question by way of o Graphs (pictographs, bar graphs) o Summary numbers (median, mode)

REFERENCES Albert, J. R. G. (2008).Basic Statistics for the Tertiary Level (ed. Robe rto Padua, Welfredo Patungan, Nelia Marquez), published by Rex Bookstore. Workbooks in Statistics 1: 11 th Edition, Institute of Statistics, UP Los Banos, College Laguna 4031 https://www.khanacademy.org/math/probability/statisticalstudies/statistical-questions/v/statistical-questions https://www.illustrativemathematics.org/content-standards/tasks/703

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ACTIVITY SHEET NUMBER 1-01a Students should completely fill out the following: 1. Student Number : _________________________ 2. Sex (put a check or cross): ________ Male

________ Female

3. Number of siblings : _________________________ 4. Weight (in kilograms) :

_________________________

5. Height (in meters) : _________________________ 6. Age of mother (as of her last birthday in years) : (if mother deceased, provide age if she were alive) 7. Daily allowance in school (in pesos) :

_________________________

_________________________

8. Daily food expenditure in school (in pesos) : _________________________ 9. Usual number of text messages sent in a day : _________________________ 10. Favorite color (put a check or cross; choose only one): ____White ____Red ____ Pink ____ Orange ____Yellow ____Green ____Blue ____Purple ____Brown ____Gray ____Black

11. Usual Sleeping Time (on weekdays): _________________________ 12. On a scale from 1 (very unhappy) to 10 (happiest), how do you feel today? : ____________

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ACTIVITY SHEE T 1-01b. Groups should fill out responses to some questions : Student ID Question 1.

2.

3.

Are the responses varying or the same?

How can we summarize the data collected for each of the questions above (especially if we consolidate the data from the entire class) ?

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CLASS RECORDING SHEET 1-01a (for the Teacher) (PRINT M ORE THAN 1 copy, if necessary) Sex ( 1= male; 2 femal e)

Stude nt Numb er

Numb er of siblin gs

Report ed Weigh t (in kg)

Report ed Height (in m)

Age of moth er (in years )

. Daily allowa nce in school

Daily food expendit ure in school

Usual numbe r of text messa ges sent in a day

Usual Rating Sleepin on g Time Feeling (on Today weekda (1 very ys) unhap py, 10 happie st)

Actu al Weig ht (in kg)*

Actu al Heig ht (in m)*

* to be obtained in next lesson

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ASSESSMENT 1. Name at least one difference between a statistical question and a non statistical question. ANSWER: Statistical questions are answered by collecting data with variation (and consequently summaries for the data will be require d), while non -statistical questions are questions where answer require specific facts (and not data with variation), so summary statistics and graphs will not be needed for nonstatistical que stions. 2. Ten persons were randomly selected and asked how many le tters were in their middle names, and we received the following data: 6, 6, 7, 12, 15, 7, 8, 7, 6, 7

Ask studen ts to make a bar graph with this data

ASSESSMENT 1. Name at least one difference between a statistical question and a non statistical question. ANSWER: Statistical questions are answered by collecting data with variation (and consequently summaries for the data will be require d), while non -statistical questions are questions where answer require specific facts (and not data with variation), so summary statistics and graphs will not be needed for nonstatistical que stions. 2. Ten persons were randomly selected and asked how many le tters were in their middle names, and we received the following data: 6, 6, 7, 12, 15, 7, 8, 7, 6, 7 Ask studen ts to make a bar graph with this data

Answer: 4

3

2

1

0

5

10 letters

15

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3. Martin collected data over the last 10 days on the amount of hours of sleep he had every night, and made a line plot of these data. Hours of Sleep per Nig ht:

x x

6

x

x

x

x

x

x

x

7

8

9

10 11

x 12

x – one night

What was the most sleep he got in one night? __________________ Answer: 12 hrs What was the least amount of slee p he got in one night? ________________ Answer: 7 hrs What is the most common amount of sleep I get? _________________ Answer:8 hrs How many nights did he sleep less than 9 hours? _________________ Answer: 6 nights How many nights did he sleep more than 9 hours? _____________ Answer: 2 nights

4. Ronald collected information about favorite sports among his friends Frequency

6

Favorite Sport

5 4 3 2 1 0

Chess

Basketball

Football

Volleyball Sport

What is the most popular type of dog from the data? Answer: Basketball How many of Ronald’s friends that we re questioned do not consider

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Basketball or Football as the ir favorite sport? Answer: 5 How many friends did Ronald question? Answer : 12

Explanatory Note: Teachers have the option to just ask this assessment orally to the entire class, or to group students and ask them to identify answers, or to give this as home work, or to use some questions/items here for a chapter examination.

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