Statistics 2003

Lecture Notes : Statistics

2009

CHAPTER 1:

INTRODUCTION At the end of this chapter, the students are expected to: a. Determine the nature of statistics applicable to a given situation; b. Identify population and variable in a given situation or problem; and, c. Classify the data according to variable type and appropriate level of measurement. 1.1 DEFINITION, NATURES AND IMPORTANCE OF STATISTICS The word statistics is derived from the Latin word status meaning “state”. Early uses of statistics involved compilation of data and graphs describing various aspects of the state or country. The word statistics has two basic meanings. We sometimes use this word when referring to actual numbers derived from data and the other refers to statistics as a method of analysis. DEFINITION: Statistics a collection of quantitative data, such as statistics of crimes, statistics on enrollment, statistics on unemployment, and the like. Statistics is also a science, which deals with the collection, presentation analysis, and interpretation of quantitative data. Statistics involves much more than the simple collection, tabulation and summarizing of data. Statistics is also a tool that helps us develop general and meaningful conclusions that go beyond the original data. The following are some examples of the uses of statistics:     

Surveys Consumer Preference Experiments Sampling Economics

NATURES OF STATISTICS 



DESCRIPTIVE STATISTICS It deals with the methods of organizing, summarizing and presenting a mass of data so as to yield meaningful information. INFERENTIAL STATISTICS It deals with making generalizations about a body of data where only a part of it is examined. This comprises methods concerned with the analysis of a subset of data leading to predictions or inferences about the entire set of data.

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Example: Determine whether the following statements use the area of descriptive statistics or statistical inference. 1. A bowler wants to find his bowling average for the past 12 games. 2. A manager would like to predict based on previous years’ sales, the sales performance of a company for the next five years. 3. A politician would like to estimate, based on an opinion poll, his chance for winning in the upcoming senatorial election. 4. A teacher wishes to determine the percentage of students who passed the examination. 5. A student wishes to determine the average monthly expenditures on school supplies for the past five months. 6. A basketball player wants to estimate his chance of winning the most valuable player (MVP) award based on his current season averages and the averages of his opponents. Answer: 1. 2. 3. 4. 5. 6.

___________________ ___________________ ___________________ ___________________ ___________________ ___________________

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Name:_________________________________ Course/Year/Section:_____________________

Score:___________________ Teacher:_________________

Exercise 1.1 Direction: Classify the following statements as belonging to the area of descriptive statistics or statistical inference. Write DS for descriptive statistics and write IS for inferential statistics on the space provided. _________________1. Yesterday’s records show that five (5) employees were absent due to Dengue fever. _________________2. If the present trend continues, architects will construct more contemporary homes than colonials in the next 5 years. _________________3. In a certain city, arsonists deliberately set 3% of all fires reported last year. _________________4. At least 30% of all new homes being built today are of a contemporary design. _________________5. As a result of a recent poll, most Filipinos are in favor of finding work employment abroad. _________________6. Philippines’ Gross Domestic Product (GDP) grows by 4.6% in 2002, 1.4 percentage higher than its 3.2% performance in 2001. _________________7. The average grade of 10 students in English is 89.46%. _________________8. Based from the present sales trend, it is expected that after two years, this year’s sales will be doubled. _________________9. All four provinces of ARMM are among the 10 poorest provinces in the Philippines for 2002. _________________10. During the period 1996 to 2002, unemployment rates among women were consistently higher compared to men except in 1999 and 2000.

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1.2 DEFINITION OF SOME BASIC STATISTICAL TERMS The following are terms commonly used in Statistics: 1. Population is the set of all individuals or entities under consideration or study. It may be a finite or infinite collection of objects, events, or individuals, with specified class or characteristics under consideration. Example 1. The researcher would like to determine the average age of patients infected with dengue fever for the month of June at Medical Center Imus. Population: the set of all patients with dengue fever. Example 2. The researcher would like to determine the number of BOM students at DLSU-D. Population: the set of all students in DLSU-D. 2. Variable is a characteristic of interest measurable on each and every individual in the population, denoted by any capital letter in the English alphabet. Types of Variable Qualitative Variable consists of categories or attributes, which have non-numerical characteristics. Example: classification, year level, sex and subjects enrolled Quantitative Variable consists of numbers representing counts or measurements. Variable for population 1: A = age Variable for population 2: S = sex

3. 4. 5. 6.

Classification of Quantitative Variable Discrete Quantitative Variable results from either a finite number of possible values or a countable number of possible values. Example: number of students, number of books, and number of patients Continuous Quantitative Variable results from infinitely many possible values that can be associated with points on a continuous scale in such a way that there are no gaps or interruptions. Example: height, weight, grade point average, and time Sample is part of the population or a sub-collection of elements drawn from a population. Parameter is a numerical measurement describing some characteristic of a population. Statistic is a numerical measurement describing some characteristic of a sample. Survey is often conducted to gather opinions or feedbacks about a variety of topics. Census Survey, most often simply referred to as census, is conducted by gathering information from the entire population. Sampling Survey, most often simply referred to as survey, is conducted by gathering information only from part of the population.

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Name:________________________________ Course/Year/Section:____________________

Score:___________________ Teacher:_________________

Exercise 1.2.1 Direction: Identify the population, variable of interest, and type of variable in the following: 1. The dean of COS would like to determine the average weekly allowance of BS Computer Science students. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 2. The registrar of DLSU-D would like to conduct a survey on the preferred courses of 4 th year high school students in Cavite. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 3. The dean of the CLA would like to know the number of students who are smoking. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 4. A survey by a group of students entitled “Dress Code” will be conducted to first year students to determine the fashion preferences of these students. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 5. Information will be collected to new voters for 2004 election to identify their opinion regarding politics in the Philippines. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________

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6. From all students registered this semester, the Mathematics Department would like to know how many students like mathematics. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 7. A study to be conducted by NGO would determine the Filipinos’ awareness about the war against IRAQ. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 8. A group of students taking Statistics conducted a study on the effect of boy-girl relationship to the academic performance of the students. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 9. Some parents would like to determine whether Counter Strike is good or bad to the behavior of their children. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 10. The head librarian would like to identify the book/s commonly read by DLSU-D students. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________ 11. A statistics teacher would like to determine whether the number of students in a class at the start of the semester can determine the number of failures. Population:______________________________________________________________ Variable: _______________________________________________________________ Type of variable:_________________________________________________________

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Score:___________________ Teacher:_________________

Exercise 1.2.2 Direction: Identify each statement as having discrete or continuous data. Write D for discrete and C for continuous. _____________1. Among the 1,500,000 microcomputer chips made by Motocolla, 2 are found to be defective. _____________2. Yesterday’s records show that 25 students were absent. _____________3. Radar on EDSA indicated that the driver was going 150 kph when ticketed for speeding. _____________4. The amount of time that a taxi driver spends yielding to individual pedestrians each year is 2.367 seconds. _____________5. Upon completion of a diet and exercise program, Elmer weighed 12.37 lbs. less than when he started the program. Exercise 1.2.3 Direction: Identify which of the following quantitative data would be presented by a discrete variable or a continuous variable. Write DV for discrete variable and write CV for continuous variable. _____________1. Number of students _____________2. Time (in minutes) to finish an exam _____________3. Distance (in km.) of school from place of residence _____________4. Length (in cm) of fish caught _____________5. Width of the newest brand of cellular phone _____________6. Percentage increase in enrolment this year _____________7. Number of enrollees _____________8. Monthly income of 100 randomly selected persons at KADIWA Market _____________9. Sum of points in tossing a pair of dice _____________10. Lifetime (in years) of televisions produced by ZONY

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1.3 LEVELS OF MEASUREMENT Another common way to classify data is to use four levels of measurement. The level of measurement of data determines the algebraic operations that can be performed and the statistical tools that can be applied to the data set. The following are the four levels: Level 1. Nominal is characterized by data that consist of names, labels, or categories only. The data cannot be arranged in an ordering scheme. Examples: name, religion, civil status, address, sex, degree program Level 2. Ordinal involves data that may be arranged in some order, but differences between data values either cannot be determined or are meaningless. Examples: military rank, job position, year level Level 3. Interval is like the ordinal level, with the additional property that meaningful amounts of differences between data can be determined. However, there is no inherent (natural) zero starting point. Examples: IQ Score, temperature (in ⁰C) Level 4. Ratio is the interval level modified to include the inherent zero starting point. For values at this level, differences and ratios are meaningful. Examples: height, area, width, weekly allowance

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Score:___________________ Teacher:_________________

Exercise 1.3 At what level are the following variables measured? Write nominal, ordinal, interval or ratio on the space provided. ___________________1. Student number ___________________2. Weights of a sample of candies ___________________3. Zip codes ___________________4. SSS number ___________________5. Final course grades of 1.0,1.25,1.50,…. ____________________6. Instructors rated as superior, above average, average, below average, or poor ____________________7. Movies listed according to their genre such as, comedy, adventure, romance, action, suspense, or horror ___________________8. Lengths of TV commercials ( in seconds) ___________________9. The years 1896, 2000, 1776,1995 ___________________10.Attitude toward gun laws such as favorable, somewhat favorable, somewhat unfavorable.. ___________________11.Area codes ___________________12. Ideal number of children ___________________13. Family Income ___________________14. Candidate voted for in 2002 barangay elections ___________________15. Tax Identification Number ___________________16. Gender ___________________17. Average number of glasses of water consumed per day ___________________18. Blood pressure ___________________19. Height of students ___________________20. Number of clients ___________________21. Number of won cases in court ___________________22. Academic rank in High School ___________________23. Savings Account Number ___________________24. Are you a Pag-Ibig Member?(Yes/No) ___________________25. Number of books sold per day ___________________26. Weekly allowance of Engineering students ___________________27. Main source of income ___________________28. Birth order in the family ___________________29. Number of organizations involved in ___________________30. Car plate number

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CHAPTER 2:

DATA COLLECTION AND PRESENTATION Objectives: At the end of this chapter, the students are expected to: 1. Prepare a survey plan and formulate survey questions; 2. Apply the basic statistical concepts and principles in the collection of data; 3. Explore the different ways of presenting data; 4. Make observations, construct, compare, and interpret different graphs; 5. Make observations, construct, compare, and interpret statistical tables; 6. Know the different sampling methods; and 7. Determine the method of sampling that is most appropriate to use in a given population.

2.1 METHODS OF DATA COLLECTION In order to have accurate data, the researcher must know the right sources and the right way of collecting them. CHARACTERISTICS OF A GOOD QUESTION 1. A good question is unbiased. Questions must not be worded in a manner that will influence the respondent to answer in a certain way that is to favor a certain response or to be against it. An unbiased question is stated in neutral language and no element of pressure. Examples of unbiased questions: a. Do you favor the enrollment procedure employed last semester? b. Do you like classical music? Examples of biased questions: a. Do you favor the enrollment procedure employed last semester which makes long lines shorter? b. Do you listen to boring classical music? 2. A question must be clear and simply stated. A question that is simple and clear will be easier to understand and more likely to be answered truthfully. Example of a simple and clear question: a. What is your average grade last semester?

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Example of a not simple question: a. What is your academic performance last semester?

3. Questions must be precise. Questions must not be vague. The question should indicate clearly the manner how the answers must be given. Example of a precise question: a. In terms of mathematical ability, do you think male and female are equal? Example of a vague question: a. Do you think male and female are equal? 4. Good questionnaires lend themselves to easy analyses. TWO CATEGORIES OF SURVEY QUESTIONS 1. Open question- allows free response. Example: What do you think can be done to reduce crime? 2. Closed question- allows only a fixed response. Example: Which of the following approaches would be the most effective in reducing crime? Choose one. A. Get parents to discipline more. B. Correct social and economic conditions in slums. C. Improve rehabilitation efforts in jails. D. Give convicted criminals tougher sentences. E. Reform courts. TYPES OF DATA 1. Primary Data- are information collected from an original source of data, which is first-hand in nature. Examples are data collected from interviews and surveys. 2. Secondary Data- are information collected from published or unpublished sources like books, newspapers, and thesis.

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FOUR IMPORTANT POINTS TO CONSIDER WHEN COLLECTING DATA 1. If measurements of some characteristics from people (such as height) are being obtained, better results will be achieved if the researcher does the measuring instead of asking the respondent for the value. 2. The method of data collection used may expedite or delay the process. Avoid a medium that would produce low response rates. 3. Ensure that the sample size is large enough for the required purposes. 4. Ensure that the method used to collect data actually results in a sample that is representative of the population. METHODS OF DATA COLLECTION 1. Direct or Interview Method The direct or interview method of data collection use at least two persons (an interviewer and interviewee/s) exchanging information. This method will give us precise and consistent information because clarifications can be made. Also, questions not fully understood by the respondent, the interviewer could repeat the question until it suits the interviewee’s level. However, this method is time consuming, expensive and has limited field coverage. 2. Indirect or Questionnaire Method This is a method where written answers are given to prepared questions. This method requires less time and is inexpensive since the questionnaires can be mailed or hand-carrried. Also, this will give a respondent a sense of freedom in honestly answering the questions because of anonymity. 3. Registration Method This is a method enforced by certain laws. 4. Observation Method This is a method, which observes the behavior of individuals or organizations in the study. This is also used when the respondents cannot read nor write. 5. Experiment Method This method is used when the objective of the study is to determine the cause and effect of certain phenomena or event.

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Score:___________________ Teacher:_________________

Exercise 2.1 A. Answer the following questions: 1. What are the advantages and disadvantages of open questions? 2. What are the advantages and disadvantages of closed questions? 3. Choose the letter that corresponds to an unbiased option. A. Anyone is free to smoke cigarettes if he wants to. __________ Agree _________Disagree B. No sane person will burn money on cigarettes. __________ Agree _________Disagree C. Smoking may be dangerous to your health. __________ Agree _________Disagree 4. Change the following questions to make it simpler and clearer. A. What is your opinion about beauty contest being demeaning to womanhood and against the vaunted equality of the sexes? B. What is your mass measure in metric units? 5. Determine which of the following questions is best to ask about someone’s health? A. Are you a weakling? B. Are you bursting over with health? C. Can you be an advertisement for a health club? D. Are you in good health? 6. To find out how much someone likes music, which question is best to ask? A. Do you go for the present trend-all noise and sound? B. How much time do you spend listening to the radio? C. Do you prefer popular music, jazz, classics, or rock? D. Do you listen to boring classical tunes? B. From the situations given below, identify the possible errors or difficulties that may be encountered during data collection. 1. To research recognition of a certain brand of deodorant, you plan to conduct a telephone survey of 1000 consumers in the Philippines. What is wrong with using telephone directories as the population from which the sample is drawn? 2. A group of college students conduct a survey in an attempt to determine the typical annual salary of the school’s alumni. Would alumni with very low salaries be likely to respond? How would this affect the result? Identify one other factor that might affect the result.

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3. Assume that you are hired by a company to collect data on the waist sizes of college students. Why is it better to obtain direct measurements than to ask people sizes of their waists? C. Carrying out a survey 1. Get a partner/group mate. Work together to write a plan for a survey using questionnaire method. 2. Think of any topic of your interest and construct a questionnaire. 3. The plan should include:  Topic of your interest  Title of the survey/mini research  Purpose /objectives of the survey  Importance objectives of the survey  Population and Sample of the study  The questionnaire 4. Submit the plan and the questionnaire to your teacher for approval. Finalize the questionnaire. 5. Prepare the necessary survey forms and conduct the survey.

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2.2.1 SAMPLING Concept of Sampling Sampling is the process of selecting units, like people, organizations, or objects from a population of interest in order to study and fairly generalize the results back to the population from which sample was chosen. ADVANTAGES OF SAMPLING 1. Reduced Cost 2. Greater Speed 3. Greater Scope 4. Greater Accuracy 2.2.2 Probability Sampling A probability sampling method is any method of sampling that utilizes some form of random selection. In order to have a random selection method, you must set up some process or procedure that assures that the different units in your population have equal probabilities of being chosen. Humans have long practiced various forms of random selection, such as picking a name out of hat, or choosing the short straw. These days we tend to use computers as the mechanism for generating random numbers as the basis for random selection. Random selection is performed by selecting a group subjects (a sample) for the study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has a known, but possibly non-equal, chance of being included in the sample. By using a random selection, the likelihood of bias is reduced. 1. Simple Random Sampling The simplest form of random sampling is called the simple random sampling. It is the basic sampling technique where a group of subjects (a sample) is selected for a study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has an equal chance of being included in the sample. Every possible sample of a given size has the same chance of selection; i.e. each member of the population is equally likely to be chosen at any stage in the sampling process. The most common techniques for selecting simple random sample are by using strips of paper, use of printed table of random numbers, or use of random numbers generated by many computer programs or scientific calculators. 2. Stratified Random Sampling This sampling method involves dividing the population into homogeneous subgroups and then taking a simple random sample in each group. 2 Types of Stratified Random Sampling Equal Allocation (EA) - the sample sizes from the different strata are equal. That is

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Formula: Where: n = number of sample/sample size number of groups/strata Proportional Allocation (PA) – the sample sizes from the different strata are proportional to the sizes of the strata. In general, a population of size N is divided into k strata of size , and a sample of size from the first stratum is taken, a sample of size from the

stratum is taken.

Formula: Where:

= population of each strata/stratum N = population n = sample size

Example: Given is the enrollment data of DLSU-D for first semester, SY 2009-2010. PROGRAM EA BIT 420 BCS 210 BOM 300 BSE 40 PSY 40 ENT 25 HRM 400 TOTAL(N) 1435 Select a sample of 200 students using Equal and Proportional Allocation

PA

3. Systematic Random Sampling Systematic sampling with a random start is a method of selecting a sample by taking every unit from an ordered population, the first unit being selected at random. K is called the sampling interval and the reciprocal is the sampling fraction. Formula: Where:

N = population n = sample size

4. Cluster Random Sampling This sampling method involves dividing the population into clusters, usually along geographic boundaries, then randomly taking samples of clusters, and measuring all units within sampled clusters.

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2.2.3

Non-Probability Sampling Non-probability sampling does not involve random selection of samples. It does not necessarily mean, however, that non-probability samples are not representative of the population. With nonprobability samples, the population may or may not be represented well, and it will often difficult to know how well the population has been represented. TWO TYPES OF NON-PROBABILITY SAMPLING 1. ACCIDENTAL, HAPHAZARD OR CONVENIENCE SAMPLING This is one of the most common methods of sampling, which is primarily based on the convenience of the researcher. Most common examples are interviews conducted frequently by television news programs to get quick (although non-representative) reading of public opinion. 2. PURPOSIVE SAMPLING In this type of sampling technique, samples are taken with a purpose in mind. Usually, one or more specific predefined sought. Purposive sampling can be very useful for situations where a target sample needs to be reached quickly and where sampling for proportionality is not the primary concern. With a purposive sample, it is likely to get opinions of target populations that are more readily accessible. All of the methods that follow can be considered subcategories of purposive sampling methods. Modal Instance Sampling Sampling for specific groups or types of people wherein sampling the most frequent case, or the “typical” case is sought for. This method of sampling is commonly used in informal public opinion polls. Expert Sampling Expert sampling involves the assembling of a sample of persons with known or demonstrable experience and expertise in some area. Often, expert sampling is done when it would be the best way to elicit the views of persons who have specific expertise. Quota Sampling In quota sampling, respondents are selected non-randomly according to some fixed quota. Heterogeneity Sampling Heterogeneity sampling is performed when all opinions or views about a specific topic are the primary concern and representing these views proportionately is not of major importance. Snowball Sampling In snowball sampling, the process starts by identifying someone who meets the criteria for inclusion in the study. The respondent is then asked to recommend others whom they may know who also meet the criteria.

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Score:___________________ Teacher:_________________

Exercise 2.2 Identify the type of sampling used in the following statements. 1. An engineer selects every 50th cell phone from the assembly line for careful testing and analysis. __________________________________________________________________________ 2. A reporter writes the name of each senator on a separate card, shuffles the cards, and then draws five names. __________________________________________________________________________ 3. The dean of CEAT surveys all students from each of the 12 randomly selected classes. __________________________________________________________________________ 4. A reporter obtains sample data from readers who decide to mail in a questionnaire printed in the latest issue. __________________________________________________________________________ 5. A mathematics professor selects 18 men and 18 women from each of the four classes. __________________________________________________________________________ 6. In conducting research for a psychology course, a student of DLSU-D interviews students who are leaving the JFH building. __________________________________________________________________________ 7. A reporter obtains numbered listing of the 1000 companies with the highest stock market values, uses a computer to generate 20 random numbers between 1 and 1000, and then interviews the chief executive officers of the companies corresponding to these numbers. __________________________________________________________________________ 8. A medical student at DLS-UMC interviews all diabetic patients in each of 15 randomly selected hospitals in the country. __________________________________________________________________________ 9. A researcher interviews every 45th patient in the list of in-patients. __________________________________________________________________________ 10. A student interviews school principals and classroom teachers about the implementation of the 2002 Basic Education Curriculum. __________________________________________________________________________

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2.3 Methods of Data Presentation Methods of Presenting Data: 1. Textual Method- a narrative description of the data gathered. 2. Tabular Method- a systematic arrangement of information into columns and rows. 3. Graphical Method- an illustrative description of the data. 2.3.1 The Frequency Distribution Table (FDT) An FDT is a statistical table showing the frequency or number of observations contained in each of the defined classes or categories. Parts of a Statistical Table 1. Table Heading- includes the table number and the title of the table. 2. Body- main part of the table that contains the information or figures. 3. Stubs or classes- classification or categories describing the data and usually found at the left most side of the table. 4. Caption- designations or identifications of the information contained in a column, usually found at the top most of the column. Table 1: Frequency Distribution of Staff Perception of the Leadership Behavior of the Administrator

TABLE HEADING CAPTION

STUBS/CLASSES

Perception of Leadership Behavior Strongly Favorable Favorable Slightly Favorable Slightly Unfavorable Unfavorable Strongly Unfavorable

Frequency 10 11 12 14 22 31

TOTAL

100

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TYPES OF FDT: 1. Qualitative or Categorical FDT is a frequency distribution table where the data are grouped according to some qualitative characteristics; data are grouped into non numerical categories. Example: TABLE 2: Frequency Distribution of the Gender Respondents of a Survey Gender of Respondents

Frequency

Male

38

Female

62

TOTAL

100

2. Quantitative FDT is a frequency distribution table where the data are grouped according to some numerical or quantitative characteristics. Example: WEIGHT (in kilogram)

Frequency

7-9 10-12 13-15 16-18

2 8 14 19

19-21

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TOTAL

50

STEPS IN CONSTRUCTING A FREQUENCY DISTRIBUTION TABLE 1. Determine the Range (R)

2. Determine the number of classes (K)

where N is the total number of observations in the data set. 3. Determine the class size (c) by calculating first the preliminary class size (c’).

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Preliminary class size c’: Conditions for the actual c: a. It should have the same number of decimal places as in the raw data. b. It should be odd in the last digit. 4. Enumerate the classes or categories. 5. Tally the observations. 6. Compute for values in other columns of the FDT as deemed necessary. Other columns in FDT 1. True Class Boundaries (TCB) a. Lower True Class Boundaries (LTCB): b. Upper True Class Boundaries (UTCB):

2. Class Mark (CM) – midpoint of the class interval where the observations tend to cluster about.

3. Relative Frequency (RF) – the proportion of observations falling in a class and is expressed in percentage. 4. Cumulative Frequency (CF) – accumulated frequency of the classes. a. Less than CF (CF) – total number of observations whose values are not less than the lower limit of the class. 5. Relative Cumulative Frequency (RCF) a. Less than RCF (RCF)

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Example: Construct the FDT of the given data set. Age (in years) of 40 patients confined at a certain hospital. 5

15

23

27

33

38

44

52

10

20

25

31

35

5

15

24

30

33

40

45

53

42

50

57

13

21

7

20

25

31

34

42

45

55

26

32

36

43

51 57

2.3.2 Graphical Presentation of data A graph or a chart is a device for showing numerical values or relationships in pictorial form. Advantages: 1. 2. 3. 4.

main features and implications of a body of data can be seen at once can attract attention and hold the reader’s interest simplifies concepts that would otherwise have been expressed in so many words can readily clarify data, frequently bring out hidden facts and relationships.

Qualities of a Good Graph: 1. 2. 3. 4.

It is accurate. It is clear. It is simple. It has a good appearance.

Common Types of Graph: 1. Scatter Graph – a graph used to present measurements or values that are thought to be related. 2. Line Chart – graphical presentation of data especially useful for showing trends over a period of time. 3. Pie Chart- a circular graph that is useful in showing how a total quantity is distributed among a group of categories. The “pieces of pie” represent the proportions of the total that fall into each category. 4. Column and Bar Graph- like pie charts, column charts and bar charts are applicable only to grouped data. They should be used for discrete, grouped data of ordinal or nominal scale.

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Graphical Presentation of the Frequency Distribution Table 1. Frequency Histogram – a bar graph that displays the classes on the horizontal axis and the frequencies of the classes on the vertical axis. 2. Frequency Polygon- a line chart that is constructed by plotting the frequencies at the class marks and connecting the plotted points by means of straight lines. 3. Ogives- graphs of the cumulative frequency distribution a. Ogive- the >CF is plotted against the LTCB

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Name:_________________________________

Score:___________________

Course/Year/Section:_____________________

Teacher:_________________

Exercise 2.3 1. A survey taken at a hotel in Malate indicated that 40 guests preferred the following means of transportation: car car bus plane train bus bus plane car plane plane bus plane car car train train car car car car plane plane car bus car bus car plane car plane plane car car car bus train car bus car Construct a categorical distribution showing the frequencies corresponding to the different means of transportation. Interpret the results. 2. The following are the body weights (in grams) of 50 rats used in a study of vitamin deficiencies: 136 92 115 118 121 137 132 120 104 129 125 119 115 101 129 87 108 110 133 124 135 126 127 103 110 126 118 82 104 113 137 120 95 146 126 119 119 105 132 95 126 118 100 113 106 125 117 102 146 148 Construct the FDT of the given data set and write a brief report about it. 3. The following are the number of customers a restaurant served for lunch on 60 weekdays: 50 64 55 51 60 41 71 53 63 64 49 59 66 45 61 57 65 62 58 65 55 61 60 55 53 57 58 66 53 56 64 46 59 49 64 60 58 64 42 47 59 62 56 63 61 68 57 51 61 51 60 59 67 52 52 58 64 43 60 62 Construct the FDT of the given data set and write a brief report about it. 4. Construct a graph for the given FDT and write a brief interpretation. The 2002 Purchases by A Car Rental Agency CAR MAKER NUMBER OF PURCHASES 1. Chevrolet Cavalier 45 2. Ford Mustang 30 3. Ford Taurus 60 4. Pontiac Grand Am 15 5. Toyota Camry 30

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CHAPTER 3:

MEASURES OF CENTRAL TENDENCY Objectives: At the end of this chapter, the students are expected to: a. Employ summation and apply operations involving the summation; b. Compute and interpret the different measures of central tendency and location; c. Compare the different measures of central tendency and location; d. Recognize the advantage and disadvantages of each measure of average; e. Make generalizations from a given set of data; f. Use measures of average in making predictions and decisions; and, g. Apply their knowledge to real life situations; 3.1 Summation Notation Suppose that a variable X is a variable of interest, and that n measurements are taken. The notation will be used to represent the n observations. Let the Greek letter

indicate the “summation of”, thus we can write the sum of the

observations as:

The numbers 1 and n are called the lower and upper limits of summation, respectively. Example: Write out the following in full, that is, without summation signs:

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Rules on Summation Rule # 1: The summation notation is distributive over addition.

Rule # 2: If c is a constant, then

Rule #3: If c is a constant, then

Examples: A. Use the rules on summation to write out the expansion of the given expression:

B. Write each of the following expressions in summation with appropriate limits.

C. Given: Find the value of the following expressions:

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Exercise 3.1 Evaluate the following: 1. Given:

2. Given:

3. Given:

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Measures of Central Tendency In statistics, an average is a measure of central tendency. It is a single number that represents a set of data.

Definition: A measure of central tendency is any single value that is used to identify the “center” of the data or the typical value. It is often referred to as the average. 3.2.1 The Arithmetic Mean  The most common average and sometimes simply referred to as the mean  The sum of all the values of the observations divided by the number of observations  Denoted by a Greek letter μ (mu) for population  the sample mean, used to estimate the population mean μ, is computed as:

Examples: 1. The numbers of employees at 5 different stores are 4, 8, 10, 12, and 6. Find the mean number of employees for the 5 stores. Solution:

2.

Scores in Algebra for the first long quiz for a sample of 10 students are as follows: 84, 75, 90, 98, 88, 79, 95, 86, 93, and 89. Solution:

3.2.2 The Median  The positional middle of an array  In an array, one-half of the values precede the median and one-half follow it

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If N is odd:

If N is even:

Example: Find the median of the given data set: 75, 75, 67, 71, 72 Solution:

3.2.3 The Mode  It is the observed value that occurs most frequently.  It locates the point where the observation values occur with the greatest density.  It does not always exist, and if it does, it may not be unique. A data set is said to be unimodal if there is only one mode, bimodal if there are two modes, trimodal if there are three modes, and so on.  It is not affected by extreme values.  It can be used for qualitative as well as quantitative data. Examples: Identify the mode(s) of the following data sets. Data Set 1. 2

5

2

3

5

2

1

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Data Set 2. 2

5

5

2

2

5

1

3

5

4

2

5

5

3

3

2

1

2

3

1

4

4

5

5

Blue

White Yellow

Red

Green

Blue

Orange

Data Set 3. 1

2

Data Set 4. Red

Blue

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 3.2 1. The grades of a student on seven examinations were 85, 96, 72, 89, 95, 82 and 85. Find the student’s average grade. 2. The salaries of 4 employees were P12,000, P10,000, P15,000 and P18,000. What is the average salary? 3. Out of 100 numbers, 20 were 5’s, 40 were 4’s, 35 were 6’s and 5 were 2’s. Find the mean. 4. Find the median of the set of numbers: 5 3 7 3 8 2 1 5. Find the median of the set of numbers: 11

25

18

79

12

13

6. A student received grades of 89, 72, 92 and 35. What is the mode of the given grades? 7. The reaction times of an individual to certain stimuli were measured by a psychologist to be 0.23, 0.52, 0.25, 0.52, 0.26, 0.25, 0.39, and 0.22 seconds. Determine the modes of the given reaction times. 8. The numbers of incorrect answers on a true-false test for 15 students were recorded as follows: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, 2. Find the mean, median and mode. 9. The following are the response times in seconds of a smoke alarm after the release of smoke from a fixed source: 12, 9, 11, 7, 9, 14, 6,10. Find the mode. 10. A bridge is designed to carry a maximum load of 150,000 pounds. Is the bridge overloaded if it is carrying 18 vehicles having a mean weight of 4,630 pounds? 11. For three rounds of golf Peter scores 88, 79, and 82. What fourth-round score would he need to reduce his mean score to 81 for all rounds? 12. The average IQ of 10 students in Math is 114. If 9 of the students have IQ scores of 101, 118, 128, 106, 115, 99, 118, 109 and 125. What must be the other IQ?

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3.3 Measures of Location or Fractiles Measures of location or fractiles are values below which a specified fraction or percentage of the observations in a given set must fall. 3.3.1 Percentiles Percentiles are values that divide a set of observations in an array into 100 equal parts. Thus, , read as first percentile, is the value below which 1% of the values fall, , read as second percentile, is the value below which 2% of the values fall,…..,

, read as

ninety-ninth percentile, is the value below which 99% of the values fall. Formula:

Example: The following were the scores of 10 students in a short quiz: 2

8

6

9

7

5

8

10

10

1

Find the 64th percentile. Solution: First, arrange the data from lowest to highest. 1

2

5

6

7

8

8

9

10

10

observation = 7.04 or the 8th observation

Other forms of fractiles: 3.3.2 Deciles Deciles are values that divide the array into 10 equal parts. Thus, decile, is the value below which 10% of the values fall, value which 20% of the values fall,…,

, read as first

, read as second decile, is the

, read as ninth decile, is the value below which 90%

of the values fall.

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Formula:

Example: From the given set of scores in a quiz, find the 4th decile or 3

8

9

11

12

18

19

Solution: Since the data is already arranged from lowest to highest then we may proceed in finding the 4th decile. 3

8

9

11

12

18

19

3.3.3 Quartiles Quartiles are values that divide the array into 4 equal parts. Thus, quartile, is the value below which 25% of the values fall,..,

, read as first

, read as third quartile, is the

value which 75% of the values fall. Example: From the given set of scores in a quiz, find the 3rd quartile or 3

8

9

11

12

18

.

19

Solution: Since the data is already arranged in ascending order, then we may proceed in finding the 3rd quartile. 3

8

9

11

12

18

19

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 3.3 1. A psychologist obtained the IQ scores of 10 students. The IQ scores are as follows: 110 95 85 140 132 100 95 70 85 100 Find Interpret the values. 2. The number of absences in a semester of 25 randomly selected students were obtained by a teacher: The number of absences were as follows: 3 5 2 6 3 8 2 3 6 2 1 3 8 5 2 1 3 1 0 2 0 1 2 0 0 Find Interpret the results.

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CHAPTER 4

MEASURES OF DISPERSION Objectives: At the end of this chapter, the students are expected to: a. b. c. d. e. f. g. h.

Compute and interpret the different measures of dispersion; Compare the different measures of dispersion; Relate the measure of central tendency to the measure of dispersion; Discuss the characteristics of the samples based on their measures of dispersion Recognize the advantages and disadvantages of each measure of dispersion Make generalizations from a given set of data; Use measures of average and spread in making predictions and decisions; Apply their knowledge to real life situations.

Measures of Dispersion indicate the extent to which individual items in a series are scattered about an average. It is used to determine the extent of the scatter so that steps may be taken to control the existing variation. It is also used as measure of reliability of the average value. General Classifications of Measures of Dispersion 1. Measures of Absolute Dispersion 2. Measures of Relative Dispersion 4.1 Measures of Absolute Dispersion The measures of absolute dispersion are expressed in the units of the original observations. They cannot be used to compare variations of two data sets when the averages of these data sets differ a lot in value or when the observations differ in units of measurement. 4.1.1 The Range The range of a set of measurement s is the difference between the largest and smallest values.

Example: The IQ scores of 5 members of the Morales’ family are 108, 112, 127, 116, and 113. Find the range. 4.1.2

The Standard Deviation and Variance

For a finite population of size N, the population variance is

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And the population standard deviation is

The population variance

, can be estimated by the sample variance

, where

And the population standard σ deviation can be estimated by the sample standard deviation s, where

Example: A sample of 5 households showed the following number of household members: 3, 8, 5, 4, and 4. Find the standard deviation. 4.2 Measures of Relative Dispersion Measures of Relative Dispersion are unitless and are used when one wishes to compare the scatter of one distribution with another distribution. 4.2.1 Coefficient of Variation The coefficient of variation, CV, is the ratio of the standard deviation to the mean and is usually expressed in percentage. It is computed as

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 4.1 1. Find the range of the following data sets: Data set a: 15, 16, 7, 5, 14, 10 Data set b: 110, 125, 98, 85, 62, 100 2. Find the standard deviation of each set in #1. 3. On a final examination in Statistics, the average grade of 100 students of DS School was 80 and the standard deviation was 8. In another school, DF School, the average grade of 120 students was 75. And the standard deviation was 7.9. In which school was there a greater a. Absolute deviation b. Relative deviation 4. Find the coefficient of variation of each set in #3. 5. On 16 days, a restaurant had the following numbers of orders for chicken and steak: Chicken: 46 55 43 48 54 65 36 40 51 53 64 32 41 46 53 47 Steak: 39 41 25 30 46 36 37 23 30 33 50 44 41 28 35 37 Calculate the mean, median, mode, standard deviation, variance and CV and determine which item the number of order is relatively more variable.

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CHAPTER 5:

PROBABILITY Objectives: At the end of this chapter, the students are expected to: a. b. c. d.

List down the elements of sample spaces and events; Perform operations on events; Demonstrate greater ability in making lists in a systematic organized way; Demonstrate ability in using certain formulas and procedures to determine “how many”; e. Demonstrate ability in recognizing situations in real life wherein counting techniques and formulas are applicable; f. Demonstrate knowledge and understanding of the probability of events and outcomes; and, g. Use probability to analyze and understand real world situations and problems. 5.1 Random Experiments, Sample Spaces, Events Random Experiment is a process or procedure, repeatable under basically the same condition, leading to well-defined outcomes. It is random because we can never tell in advance what the realization is going to be even if we can specify the possible outcomes. Example: 1. Tossing an ordinary coin once 2. Recording the number of cars pulling up at a service station for gasoline per day 3. Rolling a die Sample Space is the set of all possible outcomes of a random experiment. It is denoted by Greek letter omega (Ω) or S. It is also known as universal set. Types of Sample Spaces A finite sample space is a sample space with finite number of possible outcomes. An infinite sample space is a sample space with infinite number of possible outcomes. Natures of Sample Spaces Discrete Sample Space is a sample space with countable (finite or infinite) number of possible outcomes.

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Continuous Sample Space is a sample space with a continuum number of possible outcomes. An event is a subset of the sample space denoted by any letter in the English alphabet. An event is an outcome of a random experiment. Types of Events: 1. 2. 3. 4.

Elementary Event – an event consisting of one possible outcome. Impossible Event - an event consisting of no outcome and is denoted by { } or ∅. Sure Event – an event consisting of all the possible outcomes. Complement of an event – is the set of all elements of the sample space which are not in the event A. Denoted by A’. Example: 1. Set up the sample space for the single toss of a pair of fair dice. 2. From your sample space in number 1, noting the dots on each die, list the elements of the following events: a. A = event of obtaining a sum of 7 or 11. b. B = event of obtaining a sum of at least 10. c. C = event of obtaining a sum of at most 6. d. D = event of obtaining a product of 24. e. E = event of obtaining a 3 in exactly one of the dice. f. F = event of obtaining a 3 on either die. g. G = event of obtaining a sum of 7 and a product of 12. 3. If two fair coins are tossed, are the outcomes in Coin 1 and Coin 2, independent or mutually exclusive events? Why? 4. An experiment consists of asking 3 women at random if they wash their dishes with brand X dishwashing liquid. a. List the elements of the sample space using the letters Y for “yes” and N for “no”. b. List the elements of S corresponding to the event A that at least two of the women use brand X. 5. An experiment involves tossing a pair of dice, one black and one red. a. List the elements of the sample space b. List the elements of the sample space that corresponds to: A = event that the sum is less than 5 B = event that 6 occurs on either die C = event that a number greater than 4 appears on the black die D = event that the sum is greater than 8 and a number greater than 4 comes up on the black die

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5.2 Counting Techniques 5.2.1 The fundamental Principle of Counting If an operation can be performed in ways, and if for each of these a second operation can be performed in

ways, then the two operations can be performed in

•

ways.

Example: 1. How many sample points are in the sample space when a pair of dice is tossed once? 2. Miguel is going to watch a basketball game at the Araneta Coliseum. The Coliseum has four gates where he can enter. If he cannot pass through the same gate twice, in how ways can he enter and leave the Coliseum? 3. How many different ways are there to arrange the letters in the word LOGIC? 5.2.2 Permutation A permutation is an arrangement of all part of a set of objects. Linear Permutation The number of permutations of n distinct objects is n! (read as “n factorial”). Example: Consider the letters a, b, c. List down the possible permutations. Permutation of n elements taken r at a time Theorem: The number of permutation of n distinct objects taken r at a time is

Example: Two raffle tickets are drawn from 20 for the first and second prizes. Find the number of sample points in the sample space S. Circular Permutation The number of permutations of n distinct objects arranged in a circle is (n-1)!. Example: In how many ways can 5 different plants be planted in a circle? Permutation of Things Not All Different The number of distinct permutations of n objects of which second kind,

, of a

kind is

40

are one of a kind,

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2009

Example: How many distinct permutations can be made from the letters of the word “Mississippi”? 5.2.3 Combination A combination is a selection of r objects from n without regard to order. Theorem: The number of combinations of n distinct objects taken r at a time is

Example: An experimenter must select 3 animals from 10 available animals to be used as a control group. In how many ways can the control group be selected?

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 5.2 1. How many numbers consisting of 5 digits can be made from 1, 2, 3, 4, 5, 6, 7, 8, 9? 2. In how many ways can 5 people be seated on a sofa, if there are only three seats available? 3. In how many ways can 6 children be seated in a round table? 4. An organization sponsored a raffle where they were to sell all 500 tickets sequentially numbered from 001 to 500. How many of the tickets a. Has the same three digits b. Ends in 3 c. Contain the digits 1, 2, 3 (not necessarily in that order) 5. The geographical distribution of the hometown of 80 students of CvSU-CBE is given as: 50 from Luzon, 10 from Visayas, and 20 from Mindanao. How many ways can three students be selected at random such that a. All of them come from any three places b. No student comes from Luzon and Mindanao 6. In how many ways can 10 individuals be selected from 25 individuals? 7. If a student can answer any 6 questions from an exam with 10 questions, how many ways can he answer the exam? 8. If there are 9 horses in a race, in how many ways can they finish first, second, and third? 9. How many ways can 10 students line up in a food counter? 10. How many choices do we have if we are going to bet in the Lotto 6/42 draw? 11. In how many ways can 6 boys and 7 girls be seated in a row of 13 chairs? What if the boys and the girls must alternate? 12. If in an examination consisting of 24 questions a student may omit 6, in how many ways can he select the problems he will answer? 13. How many three-digit numbers greater than 300 can be formed from 0, 1, 2, 3, 4, 5 and 6, if each digit can be used only once? 14. How many sample points or elements are there when simultaneously a coin is tossed once, two dice are thrown and a card is selected at random from an ordinary deck of 52 cards? 15. In how many ways can a window dresser display four shirts in a circular arrangement? 16. How many distinct permutations can be made from the word “COMMUNICATION”? 17. How many ways can a manager select the top three employees based on performance from 8 of his employees?

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18. In how many ways can 10 students be split into 2 groups containing 4 and 6 students, respectively? 19. In how many ways can the 5 starting positions in a basketball team be filled with 8 men who can play any position? 20. How many distinct permutations can be made from the letters of the word “REACT”? 21. In how many ways can a jack, a queen and a king be chosen from a deck of 52 cards? 22. How many ways are there to select 3 candidates from 8 equally qualified recent graduates for openings in Intel, Philippines? 23. In how many ways can the letters in the word REPUBLIC be arranged if 2 letters are used at a time?

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5.3 Basic Concept and Properties of Probability The probability of an event A, denoted by P(A), is the chance that the event will occur. It is the sum of the probabilities of mutually exclusive outcomes that constitute the event. It must satisfy the following properties:  for any event A   Example: A coin is tossed twice. What is the probability that at least one head occurs? Theorem: If an experiment can result in any one of N different equally likely outcomes and if exactly n of these outcomes corresponds to event A, then the probability of event A is:

Example: If a card is drawn from an ordinary deck, find the probability that it is a heart. Solution: Let H be the event of obtaining a heart N= 52 cards n= number of hearts Therefore, the probability that the card is heart, is . Theorems on Probabilities of Events Theorem: The Additive Rule If A and B are two events, then

.

Example: 1. The probability that a student passes History is , and the probability that he passes English is . If the probability of passing at least one course is , what is the probability that he will pass both courses? Solution:

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2. What is the probability of drawing a red or a face card if a card is selected from an ordinary deck of 52 cards? Solution:

Therefore, the probability of obtaining a red or face card is

.

Corollary 1: If A and B are mutually exclusive events, then Corollary 2: If

are mutually exclusive, then

Theorem: If A and are complementary events, then Exercises: 1. What is the probability of getting a total of 7 or 11 when a pair of dice is tossed? 2. What is the probability of drawing an Ace or a King, if a card is randomly chosen from a deck of 52 cards? 3. The probability of passing History is , what is the probability of failing the subject?

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Conditional Probability Definition: The probability of an event B occurring when it is known that some event A has occurred is called a conditional probability. It is defined by:

Example: A card is drawn from a standard deck. Suppose we are told that the card picked is spade. What is the probability that the card drawn is the ace of spades? Solution:

Thus,

Probability of Independent Events Definition: Two events are said to be independent if any one of the following conditions are satisfied:  P(A   Otherwise, the events are said to be dependent. Example: The probability that Jack will correctly answer the toughest question in an exam is . The probability that Rose will correctly answer the same question is . Find the probability that both will answer the question correctly, assuming that they do not copy from each other. Solution:

*events are said to be independent since they do not copy from each other.

Therefore, the probability that Jack and Rose will both correctly answer the toughest question is .

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 5.3 1. If two dice are thrown and the sum of dots are noted, what is the probability that the sum is: a. 5 b. 11 or 8 c. At least 10 d. 5 and 6 e. Even? 2. If a card is drawn at random from a deck of 52 cards, what is the probability that the card is : a. Red b. Even c. Face card d. Red or face e. Red and face? 3. An organization sponsored a raffle where they were to sell all 500 tickets sequentially numbered from 001 to 500. Find the probability that the winning number a. Has the same three digits b. Ends in 3 c. Contain the digits 1,2,3 (not necessarily in that order) 4. The geographical distribution of hometown of 80 students of CvSU-CBE is given as: 50 from Luzon, 10 from Visayas, and 20 from Mindanao. Suppose three students are selected. Find the probability that a. They all come from any of the three places b. No student comes from Luzon and Mindanao 5. A poker hand consists of 52 cards, the order is conventionally disregarded from a wellshuffled deck of 52 cards. What is the probability of having a. 3 Kings and 2 Queens b. 4 red cards and 1 black card c. 4 aces and a Jack 6. The probability that a student will pass Statistics is 0.50 and the probability that he will pass English is 0.80. The probability that he will pass both is 0.60. What is the probability that the student will pass at least one of the two subjects? 7. What is the probability that an even number or a 3 will appear when a fair die is tossed?

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8. The probability that it will rain tomorrow is 0.73. What is the probability that it will be sunny tomorrow? 9. A box contains 6 red and 4 black balls. Two balls from the box are drawn one at a time without replacement. What is the probability that the second ball is red if it is known that the first is red? 10. The balls used in selecting the numbers for bingo carry the numbers 1,2,3,…75. If one of the balls is selected at random, what is the probability that it is a number less than or equal to 15? 11. If three of twenty tires are defective and four of them are randomly chosen for inspection, what is the probability that one of the defective tires will be included? 12. If an airline’s records show that 468 of 600 of its jets from San Francisco to Phoenix arrived on time, what is the probability that any one of the airline’s jets from San Francisco to Phoenix will not arrive on time? 13. Harry Potter is faced with the problem of opening a safe with 10 buttons numbered from 0 to 9 without using his magic. The safe can be opened by pressing three buttons, not necessarily distinct, in correct order. a. What is the probability that Harry will hit the right combination? b. Realizing that the probability of getting the right combination is too small Harry thought of a way of swinging the odds in his favor. He pulverized the lead from a pencil and blew the powder onto the buttons, revealing three buttons that have been pressed many times before. What is the probability that Harry will be able to open the safe in a single trial? 14. A bowl contains 15 red beads, 30 white beads, 20 blue beads, and 7 black beads. If one of the beads is drawn at random, what are the probabilities that it will be a. Red beads b. White or blue beads c. Black beads d. Neither white nor black 15. If the probabilities are respectively, 0.92, 0.33 and 0.29 that a person vacationing in Boracay, will visit Palawan, or both, what is the probability that a person vacationing there will visit at least one of the two areas? 16. The probabilities that a student will get an A or a B or a C in a Math course is 0.09, 0.15, and 0.53. What is the probability that a student will get a grade lower than C? 17. If the probability of passing Statistics is 0.65, what is the probability of failing the subject? 18. If a coin is tossed twice, what is the probability of getting a head on the second toss if the outcome on the first toss is also a head?

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19. A jeepney driver buys spare part X and Y for his vehicle at a bargain shop. The probability that X is defective is 0.10 and the probability that Y is defective is 0.05. Determine the following probabilities: a. Both parts are defective b. Both parts are good c. X is good, Y is defective 20. A box contains 12 black balls and 8 white balls. Two balls are drawn in succession. a. Find the following probabilities if the balls are drawn with replacement: a.1. A black ball is drawn then another black ball is drawn. a.2 A black, then a white ball is drawn. a.3 Two white balls are drawn. a.4 Both balls are of the same color. b. Answer the questions in (a) if the balls are drawn without replacement.

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CHAPTER 6:

PROBABILITY DISTRIBUTION Objectives: At the end of this chapter, the students are expected to: a. List down the possible values of random variables; b. Find the probability distribution of a random variable; c. Demonstrate knowledge and understanding of the probability of events and its distribution; and, d. Use probability to analyze and understand real world situations and problems. 6.1 Concept of a Random Variable A function whose value is a real number determined by each element in the sample space is called a random variable. It is a quantity resulting from an experiment that, by chance, can assume different values. The term random variable is used to describe the value that corresponds to the outcome from a given experiment. A capital letter is used to denote a random variable. A random variable may be a continuous random variable or a discrete random variable. Example: 1. A coin is tossed three times. List down the elements of the sample space. List down the possible values of a random variable Y, the number of heads that fall. Solution: S=

2. Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. List down the elements of the sample space and possible values of a random variable X, the number of red balls. Solution:

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3. Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of a sample space for the three tosses of the coin and to each sample point assign a value for the random variable W. Solution: S=

If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a discrete sample space. A random variable defined over a discrete sample space is called discrete random variable.

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called continuous sample space. A random variable defined over a continuous sample space is called continuous random variable.

Example: Determine the following whether discrete or continuous random variable. 1. A pair of dice is rolled and X is a random variable that represents the sum of the spots on the two dice. 2. An economist is interested in the random variable C, the number of persons filing for employment at the Labor Office. 3. The proprietor of a hamburger franchise is interested in the volume of Coke sold per day at her franchise. 4. A businessman is interested in the random variable Q, the total overtime accumulated by his employees each week. 5. An experiment consists of observing the random variable L, where L is the length of time that it takes for IT student to finish a major examination.

Discrete and Continuous Probability Distribution A probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. It has the following characteristics: 1. The probability of a particular outcome is between 0 and 1, inclusive. 2. The sum of the probabilities of all mutually exclusive outcomes is 1. Definition: The table or formula listing of all the possible values that a random variable can take on, along with the associated probabilities, is called discrete probability distribution. Note: The probabilities associated with all possible values of a discrete random variable must sum to 1. Example: 1. A coin is tossed three times. List down the elements of the sample space. List down the possible values of a random variable Y, the number of heads that fall.

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Solution: S=

The discrete probability distribution of the random variable Y is:

Note: 2. Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of a sample space for the three tosses of the coin and to each sample point assign a value for the random variable W. Solution: S= The discrete probability distribution of the random variable W is:

Note:

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2009

Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 6.1 1.

Construct the probability distribution of the sum of the numbers when a pair of dice is tossed.

DISCRETE PROBABILITY DISTRIBUTION BINOMIAL DISTRIBUTION An experiment often consists of repeated trials, each with two possible outcomes, which may be labeled success or failure. This is true in flipping of a coin 5 times, where each trial may result in a head or tail. We may choose to define either outcome as a success. It is also true if 5 cards are drawn in succession from an ordinary deck and each trial is labeled “success” or “failure”, depending on whether the card is red or black. Experiments of this type are known as binomial experiments. A binomial experiment is one that possesses the following properties: 1. The experiment consists of n repeated trials. 2. Each trial results in an outcome that may be classified as a success or a failure. 3. The probability of a success, denoted by p, remains constant from trial to trial. 4. The repeated trials are independent. DEFINITION: BINOMIAL DISTRIBUTION If a binomial trial can result in a success with probability p and a failure with probability q= 1-p, then the probability distribution of the binomial random variable X, the number of successes in n independent trials is

n b(x; n, p)    p x qn  x , x

for x=0,1,2,…….,n.

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SAMPLE PROBLEM: 1. Find the probability of obtaining exactly three 2’s if an ordinary die is tossed 5 times. Solution: From the given problem, n = 5, x = 3, b(3;5; ) = 5C3

0.032

Therefore, the probability of obtaining exactly three 2’s when a die is tossed 5 times is 0.032. 2. The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that (a) at least 10 survive ;(b) from 3-8 survive; and (c) exactly 5 survive? Given: Solution: a.

= 0.034 Therefore, the probability that at least 10 people will survive is 0.034. b.

Thus, the probability that from 3 to 8 patients will survive is 0.88. c. Thus, the probability that exactly 5 patients will survive is 0.19.

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POISSON DISTRIBUTION Experiments yielding numerical values of a random variable X, the number of outcomes occurring during a given time interval or in a specified region, are often called Poisson experiments. The given time interval may be of any length, such as a minute, a day, a week, a month, or even a year. Hence a Poisson experiment might generate observations for the random variable X representing the number of telephone calls per hour received by an office, or the number of days school is closed due to snow during winter. The specified region could be a line segment, an area, a volume, or perhaps a piece of material. In this case X might represent the number of typing errors per page, or the number of bacteria in a given culture. A Poisson experiment is one that possesses the following properties: 1. The number of outcomes occurring in one time interval or specified region is independent of the number that occur in any other disjoint time interval or region of space. 2. The probability that a single outcome will occur during a very short time interval or in a small region is proportional to the length of the time interval or the size of the region and does not depend on the number of outcomes occurring outside this time interval or region. 3. The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible. The number X of outcomes occurring in a Poisson experiment is called a Poisson random variable and its probability distribution is called the Poisson distribution. Since its probabilities depend only on μ , the average number of outcomes occurring in the given time interval or specified region, we shall denote them by a symbol p(x; μ) .

DEFINITION: POISSON DISTRIBUTION The probability distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region, is px; μ 

e μμx x!

,

for x = 0,1,2,…..,

Where μ is the average number of outcomes occurring in the given time interval or specified region and e=2.71828….

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EXAMPLE: 1. The average number of days school is closed due to snow during the winter in a certain city in the eastern part of United States is 4. What is the probability that the schools in this city will close for 6 days during a winter? Solution: Given:

Thus, the probability that the schools in the United States will close in 6 days due to winter, is 0.10. 2. The average number of field mice per acre in a 5-acre wheat field is estimated to be 10. Find the probability that a given acre contains 12 mice. Solution: Given:

Therefore, the probability that a 5-acre wheat field contains 12 mice is 0.095.

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 6.1.1 1. A baseball player’s batting average is 0.250. What is the probability that he gets exactly 1 hit in his next 5 times at bat? 2. A multiple-choice quiz has 15 questions, each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guesswork yields from 5 -10 correct answers? 3. The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive? 4. A study conducted at George Washington University and the National Institute of Health examined national attitudes about tranquilizers. The study revealed that approximately 70% believe that “tranquilizers don’t really cure anything, they, they just cover up the real trouble.” According to this study, what is the probability that at least 3 of the next 5 people selected at random will believe that tranquilizers actually do cure the problem rather than just cover it up? 5. On the average a certain intersection results in 3 traffic accidents per month. What is the probability that in any given month at this intersection (a) exactly 5 accidents will occur? (b) less than 3 accidents will occur? 6. A secretary makes 2 errors per page on the average. What is the probability that on the next page she makes (a) 4 or more errors? (b) no error? 7. A certain area of the eastern United States is, on the average, hit by 6 hurricanes a year. Find the probability that in a given year this area will be hit by (a) fewer than 4 hurricanes; (b) anywhere from 6 to 8 hurricanes.

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b.

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THE NORMAL DISTRIBUTION AND ITS APPLICATION

-3

-2

-1

0

1

2

3

DEFINITION: A continuous random variable X is said to be normally distributed if its density function is given by:

for

and

for

constants

µ

and

σ,

where

Notation: If X follows the above distribution, we write The graph of the normal distribution is called normal curve. Properties of the normal curve: 1. The curve is bell-shaped and symmetric about a vertical axis through the mean µ. 2. The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean. 3. The total area under the curve and above the horizontal axis is equal to 1. DEFINITION: The distribution of a normal random variable with mean zero and standard deviation equal to 1is called a standard normal distribution.

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If

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, then X can be transformed into a standard normal random variable

through the following transformation:

If X is between the values

, the random variable Z will fall between the

corresponding values:

Therefore, Examples: 1. Let Z be a standard normal random variable. That is,

. Find the following

probabilities: (see the z-table for the probabilities) a. b.

c. d.

2. Let Z be a standard normal random variable. That is a.

b.

59

. Find the value of a.


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c.

3. Let X be a normal random variable with

Solution: Given: a.

X is a normal random variable

Therefore, the b.

60

. Find the following probabilities:


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Therefore, the c.

Therefore, the 4. Given a test with a mean of 84 and a standard deviation of 12. a. What is the probability of an individual obtaining a score of 100 or above in this test? b. What score includes 50% of all the individuals who took the test? c. If 654 students took the examination, then how many students got a score below 60? Solution: Given: µ=84, σ=12 a.

Therefore, the probability of an individual obtaining a score of 100 or above on this test is 0.0918 or 9.18%. b. In notation form, the statement is equivalent to:

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Finding the corresponding z-score of the probability 0.50, z = 0.00 From the transformation formula,

Therefore, the score that includes 50% of those who took the exam is 84. c. Given: µ=84, σ=12, N= 654

The number of students who got a score lower than 60 is equal to the product of the probability and the total number of students.

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 6.2 1. Let Z be a standard normal variable. Find the following probabilities: a. b. c. d. 2. Given a normal distribution with µ= 82 and

find the probability that X assumes

a value a. Less than 78 b. More than 90 c. Between 75 and 80 3. The mean weight of 500 male students at a certain college is 151 pounds. And the standard deviation is 15 pounds. Assume that the weights are normally distributed. a. How many students weigh between 120 and 155 pounds? b. What is the probability that a randomly selected male student weighs less than 128 pounds?

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CHAPTER 7:

ESTIMATION Objectives: At the end of this chapter, the students are expected to: 1. Demonstrate knowledge and understanding of the basic concepts of estimation; 2. Compute and interpret the estimates of the mean, variance and standard deviation; and 3. Relate knowledge of estimation to real life situations. Basic Concepts of Estimation Definition of terms: Estimator- any statistic whose value is used to estimate an unknown parameter. Estimate- a realized value of an estimator. Point Estimate- a single value used to represent the parameter of interest. Interval Estimator- a rule that tells us how to calculate two numbers based on a sample data, forming an interval within which the parameter is expected to lie. The pair of numbers (a,b) is called interval estimate or confidence interval. Level of Confidence or confidence coefficient- the degree of certainty to an interval estimate for the unknown parameter Point Estimation of the mean and the Standard Deviation A statistic is used to estimate parameters. The following are used to estimate the parameters given below:

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Parameter

Statistic

Population mean (µ) Population Standard Deviation (σ) Interval Estimation of the Mean for a Single Population Confidence Interval for µ ,σ is known If

is the mean of a random sample of size n from a population with known variance confidence interval for µ is given by µ

Note: For small samples selected from nonnormal populations, we cannot expect our degree of confidence to be accurate. However, for small samples of size , regardless of the shape of most population, sampling theory guarantees good results. To compute a

confidence interval for µ, it was assumed that

this is generally not the case,

shall be estimated by s, provided

is known. Since

Example: A survey of the delivery time of 100 orders worth P20,000 from WILLIAM’S PIZZA yielded a mean of 55 minutes with a standard deviation of 12 minutes. Assuming that the delivery time follow a normal distribution, construct a 95% confidence interval for the true mean. Solution: Given:

minutes,

12 minutes, n = 100 orders, µ

Substituting the values in the formula: µ

65

= 5%


2009

we obtained:

Conclusion: The WILLIAM’S PIZZA is 95% confident that the true mean delivery time is between 52.648 minutes and 57.352 minutes. Error in Estimating the Population Mean If

is used as an estimate of µ, we can be

confident that the error will

not exceed Example: The heights of a random sample of 50 college students showed a mean of 174.5 cm and a standard deviation of 6.9 cm. What can we assert with 98% confidence about the possible size of our error if we estimate the mean height of all college students to be 174.5? Solution: Given: = 174.5 cm, = 6.9 cm, n= 50 students,

= 2%

The possible size of the error can be obtained by using

Substituting the values in the formula:

Conclusion: We can therefore conclude that we are 98% confident that the sample mean differs from the true mean height by 2.27 cm. Sample Size for Estimating the Population Mean If

is used as an estimate of µ, we can be

not exceed a specified amount e when the sample size is Example:

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confident that the error will .


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The monthly wage of new employees at a certain broadcasting company is said to follow a normal distribution with a standard deviation of P1,000. How large sample would be needed to be 99% confident that the sample mean will be within P300 of the true mean. Solution: Given:

,

,

= 1%

by substitution:

Conclusion: Therefore we can conclude that the sample size should be 74 employees to be 99% confident that the sample mean will be within P300 of the true mean wage. Small-Sample Confidence Interval for µ, If

is unknown

and s are the mean and standard deviation respectively, of a random sample of size

from an approximate normal population with unknown variance

,

confidence interval for µ is given by

where

is the t value with

degrees of freedom.

Note: Values for t are found in the Table of T-values Example: A random sample of 8 cigarettes of a certain brand has average nicotine content of 3.6 milligrams and a standard deviation of 0.9 milligrams. Construct a 99% confidence interval for the true average nicotine content of this particular brand of cigarettes, assuming an approximate normal distribution.

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Solution: Given:

,

0.9 milligrams, n = 8 cigarettes,

= 1%

with by substitution:

we obtained:

Conclusion: Therefore we can conclude that we are 99% confident that the true average nicotine content of a certain brand of cigarette is within 3.2818 milligrams and 3.9182 milligrams.

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Name:_________________________________

Score:___________________


Teacher:_________________

Exercise 7. 1. An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed, with a standard deviation of 40 hours. If a random sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm. How large a sample is needed if we wish to be 96% confident that our sample mean will be within 10 hours of the true mean? 2. The contents of 7 similar containers of sulfuric acid are 9.8, 10.2, 10.4, 9.8, 10.0, 10.2 and 9.6 liters. Find a 95% confidence interval for the mean content of all such containers, assuming an approximate normal distribution for container contents. 3. A random sample of 100 PUJ (Public utility jeep) shows that a jeepney is driven on the average 24,500 km per year, with a standard deviation of 3,900 km. a. Construct a 99% confidence interval for the average number of kilometer a jeepney is driven annually. b. What can we assert with 99% confidence about the possible size of our error if we estimate the average number of km driven by jeepney drivers to be 23,500 km per year? 4. Suppose that the time allotted for commercials on a primetime TV program is known to have a normal distribution with a standard deviation of 1.5 minutes. A study of 35 showings gave an average commercial time of 10 minutes. Compute for the maximum error. Construct a 95% confidence interval for the true mean. 5. A random sample of 12 female students in a certain dorm showed an average weekly expenditure of P750 for snack foods, with a standard deviation of P175. Construct a 90% confidence interval for the average amount spent each week on snack foods by female students living in this dormitory, assuming the expenditures to be approximately normally distributed. 6. The mean and standard deviation for the quality grade point averages of a random sample of 28 college seniors are calculated to be 2.6 and 0.3 respectively. Find the 95% confidence interval for the mean of the entire senior class. How large a sample is required if we want to be 95% confident that our estimate of µ is not off by more than 0.05? 7. To estimate the average serving time at a fast food restaurant, a consultant noted the time taken by 40 counter servers to complete a standard order (consisting of 2 burgers,

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2 large fries and 2 drinks). The servers averaged 78.4 seconds with a standard deviation of 13.2 seconds to complete the orders. What can the consultant assert with 95% confidence about the maximum error if he uses seconds as an estimate of the true average time required to complete this standard order? 8. A company surveyed 4400 college graduates about the lengths of time required to earn their bachelor’s degrees. The mean is 5.15 years, and the standard deviation is 1.68 years. Based on these sample data, construct the 99% confidence interval for the mean time required by all college graduates. 9. In a time-use study, 20 randomly selected managers were found to spend an average of 2.4 hours each day on paperwork. The standard deviation of the 20 observations is 1.30 hours. Construct a 95% confidence interval for the mean time spent on paperwork by managers. 10. In a study of physical attractiveness and mental disorders 231 subjects were rated for attractiveness, and the resulting sample mean and standard deviation are 3.94 and 0.75, respectively. Determine the sample size necessary to estimate the sample mean, assuming you want a 95% confidence and a margin of error of 0.05. 11. The number of incorrect answers on a true-false test for a sample of 15 students was recorded as follows: 2, 1, 3, 0, 1, 3, 6, 0, 3, 3, 5, 2, 1, 4, 2. Estimate the variance. 12. In a study of the use of hypnosis to relieve pain, sensory ratings were measured for 16 subjects, with the results given below. Use these sample data to estimate the mean. 8.8 6.2 7.7 7.4 6.4 6.1 6.8 9.8 8.3 11.9 8.5 5.2 6.1 11.3 6.0 10.6

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Statistics 2003

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