1

INTRODUCTION All researchers used statistics to help reach their conclusions that would have been impossible to make with any degrees of scientific validity without the benefit of statistics. Researchers needed to use statistics as a tool to help them gain perspective on the particular problems of interest to them. Why learn statistics?

Statistics is an integral part of research activity Important questions and issues are addressed in research and statistics can be a valuable tool in

developing answers to these questions In conducting research, statistical analysis will prove to be a useful aid in the acquisition of knowledge Knowledge in statistics is important to help one understand and interpret the reports. A knowledge of statistical analysis helps to foster new and creative ways of thinking about problems Statistical thinking can be a useful aid in suggesting alternative answers to questions and posing new

ones Statistics helps to develop ones skills in critical thinking, with both inductive and deductive inference Science is best characterized as an interplay between theory and data and statistics serves as a bridge

between theory and data

VARIABLES Most research is concerned with variables which is a phenomenon that takes on different values of levels. In contrast, a constant does not vary within given constraints. Researchers distinguish between variables. One distinction is between an independent variable and a dependent variable. Example: Suppose a researcher is interested in the relationship between two variables: the effect of information about the gender of a job applicant on hiring decisions made by personnel managers. An experiment might be designed in which 50 personnel managers are provided with descriptions of a job applicant and asked whether they would hire that applicant. The applicant is described to all 50 managers in the same way on several pertinent dimensions. The only difference is that 25 of the managers are told that the applicant is a woman, and the other 25 managers are told that the applicant is a man. Each manager then indicates his or her hiring decision. In this experiment, the gender of the applicant is the independent variable and the hiring decision is the dependent variable. The hiring decision is termed the dependent variable because it is thought to depend on the information about the gender of the applicant. The gender of the applicant is termed the independent variable because it is assumed to influence the dependent variable and does not depend on the other variable (i.e. hiring decision). A useful tool for identifying independent and dependent variables is the phrase The effect of (independent variable) on (dependent variable). For example: in a study on the effect of psychological stress on blood pressure, the independent variable is the amount of psychological stress an individual is feeling and the dependent variable is the individual blood pressure. Similarly, if the effect of child-rearing practices on intelligence is studied, the independent variable is the type of child-rearing practice and the dependent variable is the childs intelligence. The term independent variable in general is any variable that is presumed to influence the dependent variable. The distinction between independent and dependent variables parallels cause-and-effect thinking with independent variable being the cause and the dependent variable being the effect. When reading studies or evaluating certain statistics, it is useful to make distinctions between presumed causes and the presumed effects.

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STATISTICS DEFINED

Statistical investigations and analyses of data fall into two broad categories:

THE NATURE OF DATA MEASUREMENT

A major feature of scientific research is measurement. Measurement involves translating empirical relationships between objects into numerical relationships. This frequently takes the form of assigning numbers to respondents (or objects) in such a way that the numbers have meaning and convey information about differences between respondents (or objects). The four types or levels of measurement used in sciences are (a) nominal, (b) ordinal, (c) interval and (d) ratio. However, some scientific researches, interval and ration were collapsed as one, thus, (a) nominal, (b) ordinal or ranks and (c) interval or ratio.

STATISTICS

SPECIFIC NUMBERS: numerical measurement determined by a set of

data

Twenty-three percent of people polled believed that learning statistics is difficult

METHOD OF ANALYSIS: a collection of methods for planning experiments, obtaining data, and then then organizing, summarizing, presenting, analyzing, interpreting, and

drawingconclusions based on the data

STATISTICS

(Collection, Organization, Summary,

Presentation, Analysis and

Interpretation of Data)

DESCRIPTIVE

-deals with processing data without attempting to draw any inferences/conclusions from them. It

refers to the representation of data in the form of tables, graphs and to the description of some

characteristics of the data, such as averages and deviations.

INFERENTIAL (INDUCTIVE)

-is a scientific discipline concerned with developing and using mathematical tools to make forecasts and inferences. Basic to the

development and understanding of inferential/inductive statistics are the concepts

of probability theory.

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4 LEVELS OF MEASUREMENT: NOMINAL, ORDINAL, INTERVAL AND RATIO

Nominal measurement involves using numbers merely as labels. A researcher might classify a group of people according to their religion Catholic, Protestant, Jewish and all others and use numbers 1, 2, 3, and 4 for these categories. Also gender is a nominal measurement where male might take a value of 1 and 2 for a female. In the nominal level, the numbers have no special quality about them; they are used merely as labels. In research, the basic statistics of interest for variables that involve nominal level are frequencies, proportions, and percentages. Because nominal data lack any ordering or numerical significance, they cannot be used for calculations. Numbers are sometimes assigned to the different categories (especially when data are computerized), but these numbers have no real computational significance and any average calculated with them is meaningless Ordinal measurement. A variable is said to be measured on an ordinal level when the categories can be ordered on some dimension. Suppose that a researcher is studying the effects of stress during schooling on the grades as one index of academic performance. The researcher takes several students who differ in letter grades and assigns the number 1 to a grade of A, the number 2 to a grade of B, the number 3 to the next letter grade, and the number 4 to the last and lowest letter grade. In this case, letter grade is measured on an ordinal level, which allows the students to be ordered from best to worst. Another example is when the researcher wants to know how often do teenagers aged 18 and above watch R-18 slasher/horror movies. The researcher will take respondents who differs in the extent of watching and assigns the number 4 to always, the number 3 to oftentimes, the number 2 to sometimes, the number 1 to seldom and the number 0 to never. Thus, with ordinal level, the researcher classifies individual into different categories but are ordered along a dimension of interest. Ordinal data provide information about relative comparisons, but not the magnitudes of the differences. They should not be used for calculations. Interval measurement. Interval measures have all the properties of ordinal measures but allows us to do more than order objects on a dimension. They also provide information about the magnitude of the differences between objects. For example, interval measures not only would tell us that one student is better in math than another, but also would convey a sense of how much better one student is than another. Technically speaking, interval measures have property that numerically equal distance on the scale represents equal distances on the dimension being measured. Also, in an interval level, measurements do not start from 0 starting point like the cases of number grades and calendar year. For example, a researcher might study the relationship between IQ scores and EQ intelligences. The difference between an IQ score of 50 and 100 is the same as the difference between an IQ score of 100 and 150. In both instances, the difference of 50 points corresponds to the same absolute amount of scores. Interval measures provide information about the magnitude of differences because of this useful property. However, since interval measurements have no 0 starting point,

DATA

QUALITATIVE

NOMINAL ORDINAL

QUANTITATIVE

INTERVAL RATIO

4 we cannot say that a person whose IQ score of 100 is twice as intelligent, than a person whose IQ score is only 50. Likewise a temperature reading of 30oC does not mean 3 times hotter than a temperature reading of 10oC (this is also true for degrees Fahrenheit) unless the unit is degrees Kelvin. Ratio measurement. Ratio measures have all the properties of interval measures but provide even more information. Specifically, ratio measures have 0 starting point that map onto underlying dimension in such a way that ratios between the numbers represent ratios of the dimension being measured. For example, if we use inches to measure the underlying dimension of height, in the case that a child who is 50 inches tall is twice the height of a child who is 25 inches tall. Similarly, a student who got a score of 75 points in a 100-point test has thrice the score of a student who got a score of 25 points. Moreover, a runner who runs a 1-km distance in a time of 10 minutes is twice as faster than a runner who runs the same distance in a time of 20 minutes. Three Different Ways of Measuring the Heights of Four Building

We have measured height on a n interval scale. Note that on this scale, even though building B has a score of 4(that is 4 feet above the criterion) and building D has a score of 2 (2 feet above the criterion), it is not the case that building B is twice as tall as building D. We cannot make a ratio statement because all measures were taken relative to an arbitrary criterion (100 feet). Finally, we can measure each building from the ground (0 as a starting point) which is a true zero point rather than a n arbitrary criterion. Building D is 102 feet high, building B is 104 feet high, building C is 180 feet high and building A is 204 feet high (figure d). We can now state with confidence that building A is twice as tall as building D.

The figure on the left shows graphically the heights of four buildings and indicates how tall each one is. The first way of measuring the heights of these buildings is to assign the number 1 to the shortest building, the number 2 to the next shortest building, the number 3 to the nest and the number 4 to the tallest building (figure b). This assignment represents ordinal measurement. It allows us to order the buildings on the dimension of height but it does not tell us anything about the magnitude of the heights. A second method is to measure by how many feet each building exceeds the 100-feet criterion. In this case, building D is 2 feet taller than the criterion, building B is 4 feet taller than the criterion, building C is 80 feet taller than the criterion and building A is 104 feet taller than the criterion (figure c). In contrast to ordinal level, now not only can we order the buildings on a dimension of height, but also we have information about the relative magnitudes of the heights. Building B is 2 feet taller than building D, building C is 76 feet taller than building B, and so on.

5 Summary: The four levels of measurements can be thought of as a hierarchy. At the lowest level, nominal measurement allows us only to categorize phenomena into different groups. The second level, ordinal measurement, not only allows us to classify phenomena into different groups but also indicates the relative ordering of the groups on a dimension of interest. The third level, interval measurement, possesses the same properties as ordinal but, in addition, is sensitive to the magnitude of the differences in the groups on the dimension. However, ratio statements are not possible at this level since the measurement is based on some criterion which is arbitrary. The fourth and final level, ratio measurement, have all the properties of nominal, ordinal and interval measurements and also permit ratio judgments to be made (0 as a starting point).

THE MEASUREMENT HIERARCHY

The four types of measurement can be thought of a hierarchy. At the lowest level, nominal measurement allows us only to categorize or classify phenomena into different groups. The second level, ordinal measurement, not only allow us to categorize or classify phenomena into different groups but also indicates the relative ordering of the groups on a dimension of interest. Interval measurement, the third level, possesses the same properties as ordinal but in addition, is sensitive to the magnitude of the differences in the groups on the dimension. However, ratio statements are not possible at this level. It is only at the final level, ratio measurement, that such statements are possible. Ratio measures have all the properties of nominal, ordinal and interval measures and also permit ratio judgments to be made. Variables measured on the ordinal, interval, or ratio level are known as quantitative variables, whereas variables measured in nominal level are called qualitative variables.

Exercise: The following data describe the different data associated with a state senator. For each data entry, indicate the corresponding level of measurement. (1) The senators name is Carah Bao. (2) The senator is 58 years old. (3) The years in which the senator was elected to the senate are 1963, 1969, 1981, and 1994. (4) Her total taxable income last year was $78,317.19. (5) The senator sponsored a bill to protect water rights. Out of 1100 voters in her district, 400 hundred said they strongly favoured the bill, 300 said they favoured the bill, 200 said they were neutral, 150 said they did not favour the bill and 50 said they strongly did not favour the bill. (6) The senator is married now. (7) However, the senator has married three times. (8) A leading news magazine claims the senator is ranked seventh for her voting record on bills regarding public education

6 Answers:

(1) Name is nominal (2) Years of age is ratio (3) Years when the senator was elected are interval (4) Income is a ratio (5) Degree of agreement (strongly favoured, favoured, neutral, not favoured, strongly not favoured) is an

ordinal (ranks) (6) Marital status is nominal (7) Number of times the senator married constitutes counting which is ratio. (0, 1, 2, 3, . . .) (8) Rank is a nominal data

Applicants for different positions of ABC Company

1 2 3 4 5 6 7 8 9 10 Age

(years) Civil

Status Nationality Religion No. of

dependents

Degree earned

Sex Job applying

for

IQ Score Years of relative

experience (months)

24 Single Thai Christian 2 BSMath M Statistician 110 6 23 Single Thai Buddhist 0 BSMath F Statistician 128 10 28 Married Thai Buddhist 3 BSAcc M Accountant 115 16 27 Married Filipino Baptist 3 BSME M Engg Head 133 10 29 Married Filipino Catholic 4 BSME F Engg Head 110 3 28 Married American Protestant 1 BSAcc F Accountant 95 0 32 Widow American Baptist 0 BSMath F Researcher 115 12 35 Married Chinese Protestant 0 BSEE M Researcher 95 8 25 Single Chinese Catholic 0 BSCoE F Systems

Analyst 130 20

27 Single Filipino Baptist 1 BSCS M Systems Analyst

105 10

29 Single Thai Buddhist 1 BSAcc F Accountant 125 14 24 Single Chinese Catholic 0 BSME M Researcher 120 6

Answers:

1. Age Interval/Ratio

2. Civil Status Nominal

3. Nationality Nominal

4. Religion Nominal

5. Number of Dependents Interval/Ratio

6. Degree Earned Nominal

7. Sex Interval/Ratio

8. Job Applying for Nominal

9. IQ Score Interval/Ratio

10. Years of Relative Experience Interval/Ratio

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ACTIVITY No. 1 (Level of Measurements)

Identify whether the following observations are nominal, ordinal, interval/ratio. Write N for nominal, O for

ordinal, IR for interval/ratio.

_____1. Weight in pounds of new born babies

_____2. Speed of a car in miles per hour

_____3. Degree of agreement or disagreement of respondents about the appropriateness of a television

program for children below 10 years old (Strongly agree, Agree, Disagree or Strongly Disagree).

_____4. Length of Milkfish in a fish pond.

_____5. Eye color

_____6. Skin tone

_____7. IQ level as low, average or high

_____8. Sound intensity of the noise made by students in a cafeteria

_____9. Educational attainment

_____10. Number of children in a family

_____11. Socioeconomic status of residence in Khon Khaen City (Low, Average, High)

_____12. Population of Thailand in the year 2010

_____13. Monthly salary of employees in the College of Asian Scholars

_____14. Religious affiliation

_____15. Gender of applicants

_____16. Anxiety level whether low, moderate, high or very high

_____17. Academic performance in math (poor, fair, good, very good)

_____18. Weight in pounds of babies born in the month of December 2008

_____19. Number of coffee-break hours per day spent by executives

_____20. Length in hours of the study time spent per day by students

_____21. Military ranks

_____22. Home address of students

_____24. The year when you were born

_____25. Softdrinks preference of Thai people

_____26. Number of foreigners migrating to Thailand every year.

_____27. Length of hair of females.

_____28. The boiling point of water is 1000C.

_____29. His cellphone number is 0929-9999875.

_____30. Johns height is 168 cm.

_____31. The number of children with missing/decayed teeth in a community is 200.

_____32. The following data are the densities of sample substances taken from River Kwai (in gm/cc): 23.6,

19.8, 15.0, 7.8, 1.6 and 2.4

_____33. The average speed of motorboats crossing in a river everyday is 5 meters per second.

_____34. Anxiety level of 8 selected female students in University of Baguio

Maria Low Luisa Average Marissa Low Martha High

Lana High Maridel Low Kelly Average Sandy Low

_____35. Religion of 5 job applicants at ABC Company

Applicant A Roman Catholic Applicant D Baptist

Applicant B English Catholic Applicant E Protestant

Applicant C Seventh Day Adventist

8 _____36. Average monthly income in pesos of 5 families in Irisan, Baguio City

Family A - 23,000.00 Family D - 18,000.00

Family B - 12,000.00 Family E - 55,000.00

Family C - 14,500.00

_____37. Contents of cola softdrink in ounces (oz)

Bottle A 2.3 oz Bottle C 2.6 oz Bottle E 2.3

Bottle B 2.5 oz Bottle D 2.2 oz

_____38. The age in months of babies admitted at NDC Hospital for treatment of bronchopneumonia are as

follows:

14, 6, 29, 43, 40, 32, 60, 58

_____39. Weights in pounds of the students in Statistics

Luis 120 Lucia 200 Gerry 166

Manuel 125 Felna 145

_____40. Scores of students in Statistics Exam: 34, 56, 45, 78, 67, 98, 78, 66, 57, 75, 34, 43, 24, 77, 80

_____41. The average score of students in an English quiz is 45.8

_____42. The total area of farm lands in a certain town is 120,000 square meters.

_____43. The volume of a softdrink bottle is 1.5 liters.

_____44. The speed of a car travelling along a highway is 60 miles per hour.

_____45. The length of a snake caught in a forest is 4 meters.

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Population (N) and Sample (n) One of the goals of a statistical investigation is to explore the characteristics of a large group of items on the basis of a few. Sometimes it is physically, economically, or for some other reason almost impossible to examine each item in a group under study. In such situation the only recourse is to examine a sub-collection of items from this group. In statistics we commonly use the terms population and sample. DEFINITIONS:

Data are collections of observations (such as measurements, genders, survey responses). A population is the complete collection of all individuals (scores, people, measurements, and so on) to

be studied. The collection is complete in the sense that it includes all of the individuals to be studied.

Example: Suppose an ornithologist is interested in investigating migration patterns of birds in the Northern Hemisphere. Then all the birds in the Northern Hemisphere will represent the population of interest to him. His choice of the population restricts him, for it does not include birds that are native to Australia and do not migrate to the Northern Hemisphere. Example: Every ten years the Bureau of Census conducts a survey of the entire population of a country accounting for every person regarding sex, age, and other characteristics. In this case the entire population of the country is the population in the statistical sense. A population can be finite or infinite and is made up of study units

Example: If we are conducting a telephone interview to study all adults (our target population) in a particular city, we do not have access to those persons who do not have a telephone. Example: We may wish to study in a particular community the effect of a drug A among all men with cholesterol levels above a specified value; however short of sampling all men in the community, only those men who for some reason visit a doctors office, clinic, or hospital are available for a blood sample to be taken.

Unfortunately the target population is not always readily accessible, and we can study only that part of it that is available. There are many ways to collect information about the study population. One way is to conduct a sample. A sample is a subcollection of members selected from a population.

Population

Study Unit

Target Population

The whole group of study units which we are interested in applying our inferences or conclusions

Study Population

The group of study units to which we can legitimately apply our inferences or conclusions

10 Example: A fisheries researcher is interested in the behavior pattern of Hermit crab along the coast of the Gulf of Siam. It would be inconceivable and impossible to investigate every crab individually. The only way to make any kind of educated guess about their behavior would be by examining a small sub-collection, that is, a sample. Example: Suppose a machine has produced 10,000 electric bulbs and we are interested in getting some idea about how long the bulbs will last. It would not be practical to test all the bulbs, because the bulbs that are tested will never reach the market. So we might pick 50 of these bulbs to test. Our interest is in learning about the 10,000 bulbs and we study 50. The 10,000 bulbs constitute the population and the 50 bulbs a sample.

Relationship between population and sample

Sample data must be collected in an appropriate way, such as through a process of random selection. If sample data are not collected in an appropriate way, the data may be so completely useless that no amount of statistical torturing can salvage them. The terms population and sample are relative. A collection that constitute a population in one context may well be a sample in another context. For instance, if we wish to learn how people in Khon Khaen City feel about a certain national issue, then all the residents of Khon Khaen City would constitute the population of interest. However, assuming that Khon Khaen City represents a cross section of Thailand population, if we use the response from these residents to understand the feelings about the issue among all the Thai people, then the residents of Khon Khaen City would represent a sample.

RANDOM SAMPLING TECHNIQUES SAMPLE SIZE An important consideration in conducting research is the size of your sample. It must be large enough so that erratic behavior of very small samples will not produce misleading results. Repetition of a research or an experiment is called replication. A large sample is not necessarily a good sample. Although it is important to have a sample that is sufficiently large, it is more important to have a sample in which the elements have been chosen in an appropriate way, such as random selection. Use a sample size large enough so that we can see the true nature of any effects or phenomena, and obtain the sample using an appropriate method, such as one based on randomness.

11 RANDOMIZATION One of the worst mistakes is to collect data in a way that is inappropriate. We cannot overstress this very important point: Data carelessly collected may be so completely useless that no amount of statistical torturing can salvage them.

COMMON METHODS OF SAMPLING In a random sample members of the population are selected in such a way that each has an equal chance of being selected. Sampling is a process or procedure which involves taking a part of a population, making observation on this representatives and the generalizing the findings to the bigger population. (Ary, Jacob and Razavieh, 1981). Probability Sampling is a random sampling technique that each element in a population has an equal chance of being selected. Non-probability Sampling is a non-random sampling technique that each element in a population has no equal chance of being selected.

SAMPLING

TECHNIQUE

PROBABILITY SAMPLING

SIMPLE RANDOM

SAMPLING

FISH-BOWL TECHNIQUE

LOTTERY TECHNIQUE

TABLE OF RANDOM NUMBERS

SYSTEMATIC SAMPLING

STRATIFIED SAMPLING

CLUSTER SAMPLING

NON-PROBABILITY

SAMPLING

ACCIDENTAL / CONVENIENCE

SAMPLING

PURPOSIVE SAMPLING

QUOTA SAMPLING

SNOW-BALL SAMPLING

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SAMPLING STRATEGIES APPROPRIATE TO PARTICULAR FEATURES OF THE POPULATION

Personal Attributes Geographical Spread Sampling Strategies

Homogeneous Concentrated Simple Random or Systematic

Dispersed 1.) Cluster Sampling 2.) Simple Random or Systematic

Heterogeneous

Concentrated 1.) Stratified Sampling 2.) Simple Random or Systematic

Dispersed

1.) Stratified 2.) Cluster 3.) Simple Random or Systematic

Determination of sample size (n) provided that the Population size (N) is known

Slovins Formula Lynch et. al Formula

21 Ne

Nn

N = Population Size n = sample size e = margin of error (0.10, 0.05, or 0.01)

)1(

)1(22

2

ppZNd

ppNZn

Z = value of the normal variable for a reliability level Z = 1.645 (90% reliability in obtaining the sample size)) Z = 1.96 (95% reliability in obtaining the sample size) Z = 2.575 (99% reliability in obtaining the sample size) p = 0.50 (proportion of getting a good sample) (1 p) = 0.50 (proportion of getting a poor sample) d = 0.01, 0.025, 0.05, or 0.10 (choice of sampling error) N = population size n = sample size

Example:

Find a minimum sample n if a population size N is 5000 with a margin of error due to sampling of 5%.

Given : N = 5000 e = 5% = 0.05 Slovins Formula: Find a minimum sample n if a population size N is 5000 with a margin of error due to sampling of 5% and a 95% reliability in obtaining the sample size. Given: N = 5000 d = 5% = 0.05

z = 1.96 (95% reliability) Modified Lynch et. Al Formula:

35775.3564604.13

4802

9604.05.12

4802

)96.1)(25.0()05.0)(5000(

)96.1)(5000)(25.0(

)25.0(

)25.0(22

2

22

2

zNd

Nzn

37037.3705.13

5000

5.121

5000

)05.0)(5000(1

5000

1 22

Ne

Nn

13 Stratified Sampling: The following are the population from 5 different communities. Use Modified Lynch et al. to find the sample size for each community with a margin of error due to sampling of 5% and a 99% reliability in obtaining the sample size.

Community Population Size (N)

A 800 B 400 C 500 D 600 E 700

Total N = 3000

Community Population Size (N)

Ratio i = nN 5433000 = 0.181

Sample size per community

A 800 0.181 800 144.8 = 145 B 400 0.181 400 72.4 = 72 C 500 0.181 500 90.5 = 91 D 600 0.181 600 108.6 = 109 E 700 0.181 700 126.7 = 127

Total N = 3000 n = 544 Note: The minimum sample size n was 543, however in the computation the value of n is 544 which is accepted as long as it is not less than 543.

Community Population Size (N)

A 145 B 72 C 91 D 109 E 127

Total n = 544 A researcher wants to know the study habits of the students in a particular school. Determine the size of the sample units from each level using 2% margin of error with 95% reliability in obtaining the sample size.

Gender Year Level Total Freshman Sophomore Junior Senior

Male 750 600 550 500 2400 Female 580 650 450 670 2350 Total 1330 1250 1000 1170 N = 4750

Ratio i = n N = 1595 4750 = 0.3358 (up to 4 decimal places for accuracy)

54304.54315765625.9

96875.4972

65765625.15.7

96875.4972

)575.2)(25.0()05.0)(3000(

)575.2)(3000)(25.0(

)25.0(

)25.0(22

2

22

2

zNd

Nzn

159585.15948604.2

9.4561

9604.09.1

9.4561

)96.1)(25.0()02.0)(4750(

)96.1)(4750)(25.0(

)25.0(

)25.0(22

2

22

2

zNd

Nzn

14 Male Freshman 0.3358 750 = 251.85 = 252 Female Freshman 0.3358 580 = 194.76 = 195 Male Sophomore 0.3358 600 = 201.48 = 201 Female Sophomore 0.3358 650 = 218.27 = 218 Male Junior 0.3358 550 = 184.69 = 185 Female Junior 0.3358 450 = 151. 11 = 151 Male Senior 0.3358 500 = 167. 9 = 168 Female Senior 0.3358 670 = 224.99 = 225 -------- -------- 806 789 Sample size n = 806 + 789 = 1595

Gender Year Level Total Freshman Sophomore Junior Senior

Male 252 201 185 168 806 Female 195 218 151 225 789 Total 447 419 336 393 1595

NON-PROBABILITY SAMPLING

1.) Accidental/Convenience Sampling Simply use results that are readily available or accessible. Usually the first person who comes along who typifies a unit of analysis serves as the respondent of the study.

2.) Purposive Sampling Implemented with the researcher defining a criterion or set of criteria for

determining the respondents of the study. It is the researchers judgment that becomes the basis for selecting an element or group that will serve as the unit of analysis. It is useful in qualitative or exploratory studies. The objective is not to have many respondents but to make sure that the person who would be interviewed will provide a wealth of information. The aim is not to quantify but to characterize an event being studied.

3.) Quota Sampling - Similar to stratified sampling except that the selection of the elements per stratum is

done through the application of random sampling strategy. Quota sampling entails grouping elements according to certain characteristics and ensuring that each group is represented. Quota sampling is helpful if the sampling frame is not available per group or stratum. It refines the application of convenience sampling since there is conscious intent on the part of the researcher to view the probable differences of every stratum or group with regard to the critical variables of the study.

4.) Snowball or Referral Sampling Involves having a respondent refers other people who are in a position

to answer some of the questions of the researcher. This is a particularly helpful in the study of highly sensitive topics where the identity of respondents is difficult to divulge or may even be unknown to many. In other words, if the sampling frame cannot be provided and the topic has security implications, a researcher could obtain referrals from the first respondent to the other respondents who may be willing to talk.

A sampling error is the difference between a sample result and the true population result; such an error results from chance sample fluctuations. A nonsampling error occurs when the sample data are incorrectly collected, recorded, or analyzed (such as by selecting a biased sample, using a defective measurement instrument, or copying the data incorrectly).

15

ACTIVITY No. 2 Determining Sample Size and Stratified Sampling

Use Slovins and Lynch et al formulas in determining the sample size of the following problems and use stratified sampling if necessary.

1. A researcher uses a 5% margin of error in computing for his sample size. If the population size is 15,000 what is the sample size with 95% reliability?

a. Slovin Formula b. Lynch et. al Formula

2. The following is a table about a population in a certain community:

Gender Age in Years Row Total

11 20 21 30 31 40 41 50 Male 240 400 350 260 1250

Female 250 300 400 250 1200 Column Total 490 700 750 510 N = 2450

a. What would be the required sample size with 95% reliability at 5% margin of error? (Use Lynch et.

Al formula) b. Use stratified sampling to find the minimum sample size in each stratum.

Gender Age in Years Row Total

11 20 21 30 31 40 41 50 Male

Female

Column Total

n =

16

METHODS OF PRESENTATION OF DATA Statistical data collected should be arranged in such a manner that will allow a reader to distinguish their essential features. Depending on a type of information and the objectives of the person presenting the information, data may be presented using one or a combination of three forms: TEXTUAL, TABULAR, and GRAPHICAL. TEXTUAL FORM The textual or paragraph form is utilized when the data to be presented are purely qualitative or when very few numbers are involved. This method is, generally, not desirable when too many figures are involved as the reader may fail to grasp the significance of certain quantitative relationships, but it becomes an effective device when the objective is to call the readers attention to some data that require special emphasis.

Example: From a newspaper report, it was gathered that China has a population of 707 million, India has 505 million, US has 207 million, USSR (before the break-up) has 245 million, and Indonesia has 125 million. That more than half of the worlds people, about 2.1 billion live in Asia, 456 million in Europe, 354 million in North America, 195 million in South America, and 20 million in Oceana. Shanghai has 10,820,000; Tokyo has 8,841,000; New York has 7,895,000; and Moscow has 7,050,000. TABULAR FORM A more effective device of presenting data because the data are presented in more concise and systematic manner. People who want to make some comparisons and draw relationships usually find tabular arrangement more convenient and understandable than the textual presentation. The data are presented through tables consisting of vertical columns and horizontal rows with headings describing these rows and columns. Example:

Continent/Region Population Country Population Cities Population Asia 2,100,000,000 China

India Indonesia

707,000,000 505,000,000 125,000,000

Shanghai Tokyo

10,820,000

8,841,000 North America 354,000,000 USA 207,000,000 New York 7,895,000 Europe 465,000,000 USSR 245,000,000 Moscow 7,050,000 South America 195,000,000 Oceana 20,000,000

GRAPHICAL OR PICTORIAL FORM Among the different methods of presenting data, the graph or chart is perhaps the most effective device for attracting peoples attention. Readers who look for comparisons and trends may skip statistical tables but may pause to examine graphs. Graph has a great advantage over tables because graph conveys quantitative values and compares more readily than tables.

17

MEASURES OF CENTER (CENTRAL TENDENCY)

Definitions:

A measure of central tendency is a single value that is used to identify the center of the data A representative or average value that indicates where the middle of the data set is located.

o It is thought of as a typical value of the distribution. o Precise yet simple o Most representative value of the data

There are several different ways to determine the center, so we have different definitions of measures of center, including the mean, median, and mode.

Mean The arithmetic mean of a set of values is the number obtained by adding the values and dividing the total by the number of values. The (arithmetic) mean is generally the most important of all numerical descriptive measurements, and it is what most people call an average.

Median The median of a data set is the middle value when the original data values are

arranged in order of increasing (or decreasing) magnitude.

Mode The mode of a data set is the value that occurs most frequently. When two values occur with the same greatest frequency, each one is a mode and the data set is bimodal. When more than two values occur with the same greatest frequency, each is a mode and the data set is said to be multimodal. When no value is repeated, we say that there is no mode and the data set is said to be nonmodal.

Procedures for Finding

Measures of Center USES OF MEAN, MEDIAN AND MODE 1. When a quantitative data is measured on a level that at least approximates interval characteristics and the distribution of observations is not too skewed, all three measures of center are meaningful. 2. When a distribution is skewed, both the mean and the median should be reported. 3. When a quantitative (or some qualitative) data is measured on an ordinal level that departs markedly from interval characteristics, the mean is not an appropriate index of center but the mode or median must be used instead. Other uses of the Mean, Median and Mode 4. When a qualitative data is measured (that is, nominal measures), the mean or median are meaningless because these concepts require ordering objects along a dimension. In this case, the mode (that is, the most frequency occurring category) is the only applicable descriptor of center. 5. When a quantitative data that contain some outliers (extreme values that fall outside the overall pattern), trimmed mean will be used. Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. The trimmed mean is more resistant.

18 RESISTANT MEASURE: A resistant measure is one that is not influenced by extremely high or low data values (outliers). A measure of center that is more resistant than the mean but still sensitive to specific data values is the trimmed mean. To compute the 5% trimmed mean, order the data from the smallest to largest, delete the bottom 5% of the data, and then delete the top 5% of the data. Finally compute the mean of the remaining 90% of the data. THE MODE: The mode )(x is the most frequent, most typical, or most common value in a distribution.

For example, there are more Catholics in the Philippines than people of any other Christina religion; and so we refer to this religion as the mode. Similarly, if at a given university, Nursing is the most popular course, this too would represent the mode. The mode is the only measure of center available for nominal-level variables and it can be used to describe the most common score in any distribution regardless of the level of measurements. To find the mode, find the score or category that occurs most often in a distribution. It can be easily found by inspection, rather than by computation. Example: Scores: 1, 2, 3, 1, 1, 6, 5, 4, 1, 4, 4, 3 The mode is 1 because it is the number that occurs more than any other scores in the set (it occurs four times). Note: The mode is not the frequency of the most frequent score (f = 4), but the value of the most frequent score ( x = 1) Example: Scores: 6, 6, 7, 2, 6, 1, 2, 3, 2, 4 Some frequency distributions contain two or more modes. In the following set of data above, the scores 2 and 6 both occur most often. Graphically, such distributions have two points of maximum frequency. These distributions are referred to as being bimodal in contrast to the more common unimodal variety, which has only a single point of maximum frequency. THE MEDIAN )~(x : When ordinal or interval data are arranged in order of size, it becomes possible to locate

the median, the middlemost point in a distribution. Thus, the median is a measure of center that cuts the distribution into two equal parts. If the number of cases in a distribution is odd, the median falls exactly in the middle of the distribution but if the number of cases in a distribution is even, the median is always that point above which 50% of the cases fall and below which 50% of the cases fall. It means that we add the two middlemost values and divided by 2. The data should be in order from low to high (or high to low) in order to locate the median. Example: Scores: 1, 2, 3, 1, 1, 6, 5, 4, 1, 4, 4

Array: 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 6 Number of cases: n = 11 (odd) Position of median = (n + 1)/2 (for odd) = (11 + 1)/2 = 6th position either from left or right (top or bottom) The median x~ = 3 is the 6th score in the distribution counting from either end.

19 Example: Scores: 6, 6, 7, 2, 6, 1, 2, 3, 2, 4

Array: 1, 2, 2, 2, 3, 4, 6, 6, 6, 7 Number of cases: n = 10 (even) Position of median = n/2 (for even)

= 10/2 = 5th position (from left and right) or (top and bottom) ) 3 = 5th position from left 4 = 5th position from right Median x~ = (3 + 4)/2 = 7/2 = 3.5 THE MEAN )(x : By far the most commonly used measure of center, the arithmetic mean, is obtained by adding

up a set of scores and dividing by the number of scores. Thus, mean is defined formally as the sum of a set of scores divided by the total number of scores in the set.

By formula:

Population Mean Sample Mean

Example:

Respondent X (IQ) Computation

1. Albert 125 2. Beth 92

3. Connie 72

4. Drake 126

5. Elmer 120

6. Fritz 99

7. Gertz 130

8. Henry 100

X = 864

Unlike the mode, the mean is not always the score that occurs most often. Unlike the median, the mean is not necessarily the middlemost point in a distribution. The mean is the point in a distribution around which the scores above it balance with those scores below it. Thus, the mean is a balance point that the sum of the deviations that fall above the mean is equal in absolute value to the sum of the deviations that fall below the mean. The Weighted Mean Researchers sometimes find it useful to obtain a mean of means that is, to calculate a total mean for a number of different groups. Suppose the students in three different sections of Sociology class receive the following mean scores on their final examinations for the course: Section 1 Section 2 Section 3 Mean: 85 72 79 Number of cases: (n) 28 28 28

N

X

n

XX

1088

864

n

XX

20 Because exactly the same numbers of students were enrolled in each section of the course, it is quite simple to calculate a total mean score: When groups differ in size, you must weight each group mean it its size (n). The weighted mean may be calculated by first multiplying each group mean by its respective number of cases (n) before summing the products, and then dividing by the total number in all groups: Where: = mean of a particular group

= number of cases in a particular group

= number in all cases combined (n1 + n2 + n3 + + nk)

= weighted mean

Section 1 Section 2 Section 3 Mean: 85 72 79 Number of cases: (n) 28 40 32 Thus, the mean final grade for all sections combined was 77.88 Weighted mean can also apply in relation to Likert Scale (1 = Strongly Disagree, 2 = Disagree, 3 = Agree, 4 = Strongly Agree) Example: Suppose a survey was conducted regarding their extent of watching horror films, the following data were gathered: Question: Always

(5) Oftentimes

(4) Sometimes

(3) Seldom

(2) Never

(1) To what extent do you watch horror film movies? n = 28 n = 39 n = 15 n = 26 n = 12 Range Verbal interpretation Thus, the weighted mean of 3.375 suggests that on average the 4.21 5.00 Always respondents sometimes watch horror films. 3.41 4.20 Oftentimes 2.61 3.40 Sometimes 1.81 2.60 Seldom 1.00 1.80 Never

67.783

797285

3

321

XXX

total

groupgroup

wN

XnX

groupX

groupn

totalN

wX

88.77100

7788

100

252828802380

324028

)79(32)72(40)85(28

wX

375.3120

405

1226153928

)1)(12()2)(26()3)(15()4)(39()5)(28(

total

groupgroup

wN

XnX

21

Comparing the Mode, Median and Mean The time comes when a researcher chooses a measure of center for a particular research situation. Will he/she employ the mode, the median, or the mean? The decision involves several factors, including the following: 1. Level of measurement 2. Shape or form of the distribution of data 3. Research objective

OBTAINING THE MODE, MEDIAN AND MEAN FROM A SIMPLE FRREQUENCY DISTRIBUTION Example: Suppose a researcher conducted personal interviews with 20 lower-income respondents in order to determine their ideal conceptions of family size. Each respondent was asked: Suppose you could decide exactly how large your family should be. Including all children and adults, how many people would you like to see in your family? Raw Data: 2, 3, 3, 2, 2, 1, 4, 4, 6, 5, 7, 8, 9, 3, 7, 3, 7, 6, 8, 7 These data can be rearranged as a simple frequency distribution as follows X f 1 1 2 3 3 4 Mode 4 2 5 1 6 2 7 4 Mode 8 2 9 1 ----------- n = 20 The median is the middlemost score in the ordered list of scores. If there is an odd number of cases, the median is the score in the exact middle of the list; if there is an even number of cases, the median is halfway between the two middlemost scores.

n = 20 n/2 = 20/2 = 10th score

4 = 10th score from top 5 = 10th score from bottom

x~ = (4+ 5)/2 = 4.5

22 Determine the sum of the scores = 97 X f fX Calculate the mean 1 1 (1)(1) = 1 2 3 (3)(2) = 6 3 4 (4)(3) = 12 4 2 (2)(4) = 8 5 1 (1)(5) = 5 6 2 (2)(6) = 12 7 4 (4)(7) = 28 8 2 (2)(8) = 16 9 1 (1)(9) = 9 ------------------------------------ n = 20 fX = 97 Summary: Modes )(x = 3 and 7

Median )~(x = 4.5

Mean )(x = 4.85

There is a wide range of family size preferences, from living alone (1) to having a big family (9). Using either the mean = 4.85 or the median = 4.5, we might conclude that the average respondents ideal family contained between four and five members. Knowing that the distribution is bimodal, however, we see that there were actually two ideal preferences for family size in this group of respondents one for a small family (Mode = 3) and the other for a large family (Mode = 7). Example: Given the impact of television on childrens attitudes and behaviour, an important concern of behavioural scientists is the amount of time children of various ages spend watching television. The following data are the weekly viewing times (in hours) of 12-year-olds. Describe or interpret the data set using the measures of center.

18 17 22 20 25 20 16 19 18 22 26 23 23 23 24 24 22 21 19 20 20

Solution: Arrange the data either in ascending or descending order

X f fX 16 1 (1)(16) = 16 17 1 (1)(17) = 17 18 2 (2)(18) = 36 19 2 (2)(19) = 38 20 4 (4)(20) = 80 21 1 (1)(21) = 21 22 3 (3)(22) = 66 23 3 (3)(23) = 69 24 2 (2)(24) = 48 25 1 (1)(25) = 25 26 1 (1)(26) = 26 ______________________________ n = 21 fX = 442

85.420

97

n

X

n

fXX

23 Mode: The highest frequency (f) is 4 which corresponds to 20. Thus the modal time is 20 hours Median: Since n = 21 which is odd Position of median = (n + 1)/2 (for odd) = (21 + 1)/2 = 22/2 = 11th position either from left or right (top or bottom) Median time = 21 hours Mean:

05.2121

442

n

X

n

fXx

Interpretation: The weekly viewing times (in hours) of 12-year-olds ranges from 16 to 26 hours. Most of the 12 year-old children spent watching television for 20 hours in a week (mode). Half of the children spent watching television in a week for 21 or more hours (median). On average, a 12-year old child spent about 21 hours watching television in a week.

=====================================================================================

Activity No. 3

Measures of Center

Problem: Tuitions at private colleges and universities vary quite a bit. Below are lists of tuitions per unit of

basic subjects at accredited colleges and universities in Thailand. A sample of 30 colleges and universities

showed annual tuition per unit (Baht) as follows:

270 290 345 295 300 245 240 325 300 295 310 265 275 285 330

295 270 285 270 265 275 320 310 335 345 335 265 280 245 260

Describe or interpret the data set using measures of center.

24

MEASURES OF DISPERSION/VARIABILITY/VARIATION In summarizing a given set of data, sometimes, the measures of center (central tendency) alone are not sufficient to give useful information. They have to be supplemented by other measures of description, and such description is the MEASURES OF VARIABILITY. A measure of variability indicates the extent to which values in a distribution are spread around the central tendency. A measure of variation is a single value that is used to describe the spread of the distribution. A measure of central tendency alone does not uniquely describe a distribution

INTERPRETING AND UNDERSTANDING STANDARD DEVIATION We understand that the standard deviation measures the variation of values about the mean. Values close together will yield a small standard deviation, whereas values spread farther apart will yield a larger standard deviation. Because variation is such an important concept and because the standard deviation is such an important tool in measuring variation, there are ways of developing a sense for values of standard deviations. CONCEPTS: Variation refers to the amount that values vary among themselves Values that are relatively close together have lower measures of variation, and values that are spread farther apart have measures of variation that are larger

(1) Range (R)

Difference between the highest and the lowest observed values in a distribution. A very rough measure of spread Provides useful but limited information since it depends only on the extreme values

(2) Sample Variance (s2)

Important measure of variation Shows variation about the mean

Measures of Variability or Dispersion

Measures of Absolute Variation

Range Variance Standard Deviation

Measures of Relative Dispersion

Quantiles

Median Quartiles Deciles Percentiles

Coefficient of Variation

25

RAW DATA (UNGOUPED DATA)

Population variance Sample variance Formula 1:

Formula 2: ( )

( )

(3) Sample Standard Deviation (SD) Most important measure of variation Square root of Variance Has the same units as the original data

RAW DATA (UNGOUPED DATA)

Population Standard Deviation Sample Standard Deviation

s= X2-(X)2

n(n-1)

Remarks: If there is a large amount of variation, then on average, the data values will be far from the

mean. Hence, the SD will be large. If there is only a small amount of variation, then on average, the data values will be close to the

mean. Hence, the SD will be small.

Comparing Standard Deviations Example: Team A - Heights of five marathon players in inches

Mean = 65 S = 0

65 65 65 65 65

N

X

2

2

1

2

2

n

XXS

N

X

2

1

2

n

XXS

26

( )

( )

Height (X) ( ) 65 (65 65)2 = 0 65 (65 65)2 = 0 65 (65 65)2 = 0 65 (65 65)2 = 0 65 (65 65)2 = 0

Height (X) X2 65 652 = 4225 65 652 = 4225 65 652 = 4225 65 652 = 4225 65 652 = 4225

X = 325 ( )

X = 325 X2 = 21125

( )

( ) ( ) ( )

( )

Example: Team B - Heights of five marathon players in inches

Mean = 65 S = 4.0

62 67 66 70 60

( )

( )

Height (X) 2XX 62 (62 65)2 = 9 67 (67 65)2 = 4 66 (66 65)2 = 1 70 (70 65)2 = 25 60 (60 65)2 = 25

Height (X) X2 62 622 = 3844 67 672 = 4489 66 662 = 4356 70 702 = 4900 60 602 = 3600

X = 325 ( )

X = 325 X2 = 21189

( )

( ) ( ) ( )

( )

1

2

2

n

XXS

1

2

2

n

XXS

1

2

2

n

XXS

04

0

15

02

S

1

2

2

n

XXS

164

64

15

642

S

27

OBTAINING THE SAMPLE VARIANCE AND STANDARD DEVIATION FROM A SIMPLE FREQUENCY DISTRIBUTION

Example: Suppose a researcher conducted personal interviews with 20 lower-income respondents in order to determine their ideal conceptions of family size. Each respondent was asked: Suppose you could decide exactly how large your family should be. Including all children and adults, how many people would you like to see in your family? Raw Data: 2, 3, 3, 2, 2, 1, 4, 4, 6, 5, 7, 8, 9, 3, 7, 3, 7, 6, 8, 7 These data can be rearranged as a simple frequency distribution as follows: X f fX fX2= (fX)(X)

1 1 (1)(1) = 1 (1)(1) = 1 2 3 (3)(2) = 6 (6)(2) = 12 3 4 (4)(3) = 12 (12)(3) = 36 4 2 (2)(4) = 8 (8)(4) = 32 5 1 (1)(5) = 5 (5)(5) = 25 6 2 (2)(6) = 12 (12)(6) = 72 7 4 (4)(7) = 28 (28)(7) = 196 8 2 (2)(8) = 16 (16)(8) = 128 9 1 (1)(9) = 9 (9)(9) = 81 n = 20 fX = 97 fX2 = 583 Solving for the sample variance: Solving for the sample standard deviation

43.292.52 ss

WHEN THE VARIOUS MEASURES OF VARIABILITY ARE USED

1. Range. This is the least reliable of the measures and is used only when one is in a hurry to get a measure of variability. It may be used with ordinal, interval, or ratio data.

2. Standard Deviation and the Variance. The standard deviation is used whenever a distribution approximates a normal distribution. It is the basis for much of the statistics. As the most reliable measure of variability it is used with interval and ratio data. Use standard deviation or variance when two means are equal.

)1(

22

2

nn

fxfxnS

)1(

22

nn

fxfxnS

92.5

380

2251

380

940911660

)19(20

97)583(20

)1(

222

2

nn

fxfxnS

28

THE SAMPLE STANDARD DEVIATION (s) and SAMPLE VARIANCE (s2)

RELATIONSHIP BETWEEN THE STANDARD DEVIATION AND VARIANCE

Variance = (Standard deviation)2 s2 = (s)2

Standard deviation =

STANDARD DEVIATION: A MEASURE OF DISTANCE

Theres an important difference between the standard deviation and its co-measure, the mean. The mean is a measure of position but the standard deviation is a measure of distance (on either side of the mean of the distribution)

(1) Majority within one standard deviation for most frequency distribution, a majority (as often as 68%) of all observations are within one standard deviation on either side of the mean.

(2) Minority deviate outside two standard deviation for most frequency distribution, a small minority (often as small as 5%) of all distributions deviate more than two standard deviations on either side of the mean.

(3) Usual or normal within two standard deviation for most frequency distribution the usual or normal values (as often as 95%) of all observations are within two standard deviations on either side of the mean.

MAJORITY OF THE DISTRIBUTION In a normal distribution, majority of the scores/values lie within one standard deviation from the left and right of the mean. This is based on the principle that majority (64.26%) of sample values lie with 1 standard deviation of the mean. Majority of Scores/Values = (mean) 1(standard deviation) = Lower Range: Upper Range:

29 USUAL OR NORMAL VALUES RANGE RULE OF THUMB (Rough estimates of the minimum and maximum usual sample values) The Range Rule of Thumb is based on the principle that for many data sets, the vast majority (95.44%) of sample values lie within 2 standard deviations of the mean. For interpretation: If the standard deviation s is known/given, use it to find rough estimates of the minimum and maximum usual sample values as follows: Lower Range: Minimum usual value = (mean) 2(standard deviation) Upper Range: Maximum usual value = (mean) + 2(standard deviation)

SKEWNESS Definition: A distribution of data is skewed (asymmetric) if it is not symmetric and if it extends more to one side than the other. (A distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half)

Definition: Skewness is a degree of asymmetry (or departure from symmetry) of a distribution.

Lopsided to the right = Skewed to the left = Negatively Skewed Lopsided to the left = Skewed to the right = Positively Skewed Data not lopsided = Symmetric = Zero Skewness

For skewed distributions, the mean tends to lie on the same side of the mode as the longer tail. Thus a measure of the asymmetry is supplied by the difference: mean minus mode. This can be made dimensionless if we divide it by a measure of dispersion, such as standard deviation. To avoid using the mode, we can employ the empirical formula (mean mode) = 3(mean median). Thus the coefficient of skewness (I) is given by the formula:

( )

Where : I = index of skewness The equation above is called Pearsons second coefficient of skewness. Intervals I 1.00 data can be considered to be significantly skewed to the right I 1.00 data can be considered to be significantly skewed to the left Example: Find the Pearsons second coefficient of skewness for the Ages of Oscar-winning Best Actors and Actresses (Mathematics Teacher magazine)

30 Actors: 32 37 36 32 51 53 33 61 35 45 55 69 76 37 42 40 32 60 38 56 48 48 40 43 62 43 42 44 41 56 39 46 31 47 45 60 Actresses: 50 44 35 80 26 28 41 21 61 38 49 33 74 30 33 41 31 35 41 42 37 26 34 34 35 61 60 34 24 30 37 31 27 39 34 Summary: Actor Actress Mean 45.97 38.94

Median 43.5 35 Mode 32 34 Standard Deviation 11.08 13.55

Actor Pearsons second coefficient of skewness: I = 3(45.97 43.5) 11.08 = 0.6687725663 0.67 Interpretation: approximates a normal distribution Actress Pearsons second coefficient of skewness: I = 3(38.94 35) 13.55 = 0.872324723 0.87 Interpretation: approximates a normal distribution

31 Level of acceptability of a four-year Fish Technology course along the area of Marketability as perceived by the Community, Local Government and the Academe To compute for the standard deviation based on the number of items (Community)

Majority Range: Usual or Normal Range: Lower Range: 4.12 0.31 = 3.81 Lower Range: 4.12 2(0.31) = 3.50 Upper Range: 4.12 + 0.31 = 4.43 Upper Range: 4.12 + 2(0.31) = 4.74 Interpretation: The 100 community respondents who perceived the level of acceptability of a four-year Fish Technology course along the area of Marketability, has an overall mean rating of 4.12 with a standard deviation of 0.31. Based on these two results, it implies that majority of these community respondents who perceived the level of acceptability, their mean response ranges from 3.81 (acceptable) to 4.43 (very acceptable). Likewise, it is expected that it is usual or normal for these community respondents that their perceived mean ratings ranges from 3.50 (acceptable) to 4.74 (very acceptable).

Indicators Mean Response Community (N = 100)

LGU (N = 50)

Academe (N = 100)

1 4.00 4.24 4.04 2 4.33 4.38 4.36 3 4.65 4.06 4.60 4 3.74 3.60 4.18 5 4.18 4.06 4.16 6 3.81 4.28 3.93 7 4.11 3.82 4.01 Overall Mean

4.12

4.06

4.18 Standard Deviation

Indicators

( )( )

Using Variance Formula:

( )

( ) ( ) ( )

( )( )

Standard Deviation:

Community (N = 100)

X X2 1 4.00 (4.00)2 = 16.0000 2 4.33 (4.33)2 = 18.7489 3 4.65 (4.65)2 = 21.6225 4 3.74 (3.74)2 = 13.9876 5 4.18 (4.18)2 = 17.4724 6 3.81 (3.81)2 = 14.5161 7 4.11 (4.11)2 = 16.8921

n = 7 (number of items)

X = 28.82

X2 = 119.2396

32

Activity No. 4 Exploratory Data Analysis

Male

X X2 3

10 5 4 2 6 7 8 4 3 5 4

12 4 8 5 5 9 7 5

10 6

10 3

X = X2 =

The following are the number of cigarettes smoke on an average day according to gender on Status of Cigarette Smoking and Drinking Liquor among ESL Teachers in Baguio City Korean Schools. Answer the following:

(1) What is the mean and median number of cigarettes for male group?

(2) What is the standard deviation for the data set? (3) What is the coefficient of skewness for the data set? (4) The majority of the male group smoke cigarettes

between what two values? (5) The male group usually smokes cigarettes between

what two values?

33

HYPOTHESIS TESTING

One of the principal objectives of research is comparison: How does one group differ from another? This typical question can be handled by the primary tools of classical statistical inference estimation and hypothesis testing. The unknown characteristic, or parameter, of a population is usually estimated from a statistic computed from sample data. Ordinarily, a researcher is interested in estimating the mean and the standard deviation of some characteristic of the population. The purpose of statistical inference is to reach conclusions from sample data and to support the conclusions with probability statements. With such information, a researcher will be able to decide whether an observed effect is real or is due to chance. Testing the significance of the difference between two means, two standard deviations, two

proportions/percentages is an important area of inferential statistics. Comparison of two or more variables

often arises in research or experiments and to be able to make valid conclusions, one has to apply an

appropriate test statistic.

Fundamentals of Hypothesis Testing HYPOTHESIS

A hypothesis is a conjecture or statement that aims to explain certain phenomena. To seek for the answers to

queries, a researcher tries to find and present evidences then tests the resulting hypothesis using statistical

tools and analysis. In statistical analysis, assumptions are given in the form of a null hypothesis, the truth of

which will either be rejected or failed to be rejected (accepted) within a certain critical interval.

Components of a Formal Hypothesis Test

(a) Null Hypothesis (denoted by Ho) is a statement about a value of a population parameter (such as the

mean), and it must contain the condition of equality and must be written with the symbol =, , or . (b) Alternative Hypothesis / Research Hypothesis (denoted by H1) is the statement that must be true if the

null hypothesis is false and it must be written with the symbol , < or >. NULL AND ALTERNATIVE HYPOTHESES

You might legitimately ask, What does it really mean when researchers test hypothesis or perform tests of significance? The concept is actually simple and direct. We are trying to find out if two (or more) things are the same or if they are different. What actually are null and alternative hypotheses? The null hypothesis is that there is no difference between or among population means, variances or proportions. For now, remember that the key part of the definition is no difference. The hypothesis that is subjected to testing to determine whether its truth can be rejected or failed to be

rejected (accepted) is the null hypothesis (H0). This hypothesis states that there is no significant relationship or

no significant difference between two or more variables, or that one variable does not affect another variable.

In statistical research, the hypotheses should be written in null form.

Example: Suppose you want to know whether method A is more effective than method B in teaching high

school mathematics. The null hypothesis for this study will be one of the following:

Ho: There is no significant difference between effectiveness of method A and method B in

teaching high school mathematics. (AMETHOD = BMETHOD)

34 Ho: Method A is as effective as method B in teaching high school mathematics (AMETHOD = BMETHOD)

The other type of hypothesis is the alternative hypothesis (H1 or HA) that challenges the null hypothesis. The

alternative hypothesis is what is known as the research hypothesis. This hypothesis specifies that there is a

significant relationship or significant difference between two or more variables or that one variable affects

another variable. Sometimes the alternative hypothesis is referred to as the research hypothesis. The

alternative hypothesis or research hypothesis is what the researcher expects to find. This is why the research,

and hence the statistical analysis, is being done.

In the example above, the alternative hypothesis can be one of the following:

Non-Directional (Area inTwo-tails)

H1: There is a significant difference between the effectiveness of method A and method B in

teaching high school mathematics. (AMETHOD BMETHOD)

Directional (Area in Right-Tail)

H1: Method A is more effective than method B in teaching high school mathematics. (AMETHOD > BMETHOD)

Directional (Area in Left-Tail)

H1: Method A is less effective than method B in teaching high school mathematics. (AMETHOD < BMETHOD)

Examples:

Null Hypothesis (Ho) Non-Directional Alternative Hypothesis (H1)

(Research Hypothesis)

Directional Alternative Hypothesis (H1)

(Research Hypothesis) Europeans are no more or less obedient to authority than Americans

Europeans differ from Americans with respect to obedience to authority

Americans are more obedient to authority than Europeans

Christians have the same suicide rate as Non-Christians

Christians do not have the same suicide rate as Non-Christians

Christians have more suicide rates than Non-Christians

The mean age of gamblers in the Asia is 30 years old

The mean age of gamblers in the Asians not 30 years old

The mean age of gamblers in the Asia is below years old

The mean monthly salary of statistics professors is at least 60,000.

The mean monthly salary of statistics professors is different from 60,000.

The mean monthly salary of statistics professors is more than 60,000.

One-half of all internet users make on-line purchases

All internet users making on-line purchases is not one-half

Fewer than one-half of all Internet users make on-line purchases

The proportion of defective

computers is equal to 0.05.

The proportion of defective

computers is different from 0.05.

The proportion of defective

computers is less than 0.05.

Womens heights have a standard

deviation that is equal to 2.8 inches

which is the standard deviation for

mens heights.

Womens heights have a standard

deviation that is different from 2.8

inches which is the standard

deviation for mens heights.

Womens heights have a

standard deviation less than 2.8

inches which is the standard

deviation for mens heights.

Test Statistic a statistic used to determine the relative position of the mean, variance or proportion in the hypothesized probability distribution of sample means. Test Statistic is a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis. The test statistic converts the sample statistic (such as the sample mean) to a score (such as the z score) with the assumption that the

35 null hypothesis is true. The test statistic can therefore be used to gauge whether the discrepancy between the sample and the claim is significant. Critical Region The region on the far end of the distribution. If only one end of the distribution, commonly termed the tail, is involved, the region is referred to as one-tailed test; if both ends are involved, the region is known as two-tailed test. When the computed test statistic (z, t, F, 2, etc.) falls in the critical region, reject the null hypothesis. The critical region is sometimes called the rejection region. The probability that a test statistic falls in the critical region is denoted by . The critical region is the set of all values of the test statistic that cause us to reject the null hypothesis. Nonrejection Region the region of the sampling distribution not included in ; that is, the region located under the middle portion of the curve. Whenever the test statistic falls in this region, the evidence does not permit the researcher to reject the null hypothesis. The implication is that the results falling in this region are not unexpected. The nonrejection region is denoted by (1 - ). Critical Value The number that divides the distribution (normal or skewed) into the region where the null hypothesis will be rejected and the region where the null hypothesis will fail to be rejected. A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis. The critical values depend on the nature of the null hypothesis, the relevant sampling distribution, and the significance level .

Test of Significance a procedure used to establish the validity of a claim by determining whether the test statistic falls in the critical region. If it does, the results are referred to as significant. This test is sometimes called the hypothesis test.

The significance level (denoted by ) is the probability that the test statistic will fall in the critical region

when the null hypothesis is actually true. If the test statistic falls in the critical region, we will reject the null

hypothesis, so is the probability of making the mistake of rejecting the null hypothesis when it is true. The

common level of significances are 10%, 5% and 1% but the most preferred in educational/

psychological/sociological research is 5%.

To test the null hypothesis of no significance in the difference between the two variables, one must set the

level of significance first. This is the probability of committing a type I error (). A type I error is the probability

of rejecting the null hypothesis when in fact it is a true hypothesis. The probability of accepting a null hypothesis

when in fact it is a false hypothesis is called a type II error ().

36

DIRECTIONAL (One-Tailed) AND NON-DIRECTIONAL (Two-tailed) TESTS

In testing statistical hypotheses, you must always ask a key question: Am I interested in the deviation of one population mean from another population mean in one or both directions? The answer is usually implicit in the way Ho and H1 are stated. If you are interested in determining whether the mean of one data is significantly different from the mean of the other data, you should perform a two-tailed test, because the difference could either be negative or positive. If you are interested in whether the mean of one data is significantly larger or smaller than the other mean data, you should perform a one-tailed test. A one-tailed test is indicated for questions like: Is a new drug superior to a standard drug? Does the air pollution level exceed safe limits? Has the death rate been reduced for those who quit smoking? A two-tailed test is indicated for questions like: Is there a difference between cholesterol levels of men and women? Does the mean age of a group of volunteers differ from that of the general population? Notice the difference in the way these questions are worded. In a potential one-tailed test, you will see words like exceed, reduced, higher, lower, more, less, and better.

A test is called directional (area in one-tail) if the region of rejection lies on one extreme side of the

distribution (either left or right) and non-directional (area in two-tails) if the region of rejection is located on

both ends of the distribution.

Non-directional (Two-tailed) test:

The critical region is in the two extreme

regions (tails) under the curve.

37

Conclusions in Hypothesis Testing The original claim sometimes becomes the null hypothesis and at other times becomes the alternative hypothesis. The standard procedure of hypothesis testing requires that always test the null hypothesis and that initial conclusion will always be one of the following:

1.) Reject the null hypothesis 2.) Fail to reject the null hypothesis

ACCEPT vs. FAIL TO REJECT Some texts say accept the null hypothesis instead of fail to reject the null hypothesis. Whether to use the term accept or fail to reject, we should recognize that we are not proving the null hypothesis but merely saying that the sample evidence is not strong enough to warrant rejection of the null hypothesis. The term accept is somewhat misleading, because it seems to imply incorrectly that the null hypothesis has been proved. The phrase fail to reject says more correctly that the available evidence is not strong enough to warrant rejection of the null hypothesis.

TESTING HYPOTHESIS

The following are suggested steps when testing the truth of a hypothesis

1. Formulate the null hypothesis (Ho) and the alternative hypothesis (H1)

2. Set the desired level of significance ()

3. Determine the appropriate test statistic to be used in testing the null hypothesis

4. Compute for the value of the statistic to be used

5. Find the critical value (tabular value) from a table

6. Compare the computed value to the tabular value and state the decision rule:

If the absolute computed value is greater than the tabulated value (tabled value), reject the null

hypothesis.

7. Make a conclusion and interpret the result in a non-technical manner.

Directional (Right-tailed) test:

The critical region is in the extreme right

region (tail) under the curve.

Directional (Left-tailed test):

The critical region is in the extreme left

region (tail) under the curve.

38

Activity No. 5

Null and Alternative Hypotheses

The following are claims about a phenomenon. Identify whether each hypothesis stated as null or alternative. If it is an alternative, further identify whether the alternative hypothesis is one-tailed (directional) or two-tailed (non-directional) test?

1. The mean amount of Coke in cans is at least 12 ounces.

2. Salaries among women business analysts have a standard deviation greater than 126,000.00

3. More than 50% of gun owners favour stricter gun laws.

4. Nasal congestion occurs at a higher rate among drug users than those who do not use drug.

5. Proportion of drinkers among convicted arsonists is greater than the proportion of drinkers convicted

of fraud.

6. Ages of faculty cars vary less than the ages of student cars.

7. The treatment and placebo groups have the same mean.

8. Men and women have different mean height.

9. Obsessive-compulsive patients and healthy persons have the same mean brain volume.

10. There is no difference between the mean for obsessive-compulsive patients and the mean for healthy

persons.

11. The mean amount of carbon monoxide in filtered cigarettes is equal to the mean amount of carbon

monoxide for non-filtered cigarettes.

12. Dyspepsia occurs at a higher rate among drug users than those who do not use drug.

13. There is a difference between the pre-training and post-training mean weights.

14. Women with a college degree have incomes with a higher mean than women with a high school

diploma.

15. Waiting times for the single line have lower standard deviation than the waiting times for any one of

several lines.

16. Dozenol tablets are more soluble after being stored for one year than before storage.

17. Percentage of women ticketed for speeding is less than the percentage of men.

18. The average number of sold paracetamol tablets is more than 100 per day.

19. There is a significance difference in the scores of the engineering and computer science students in a

mathematics quiz administered by their professor.

20. There is no significant difference between the mean heights of the two groups of trees planted with

two different types of soil.

1. 8. 15. 2. 9. 16. 3. 10. 17. 4. 11. 18. 5. 12. 19. 6. 13. 20 7. 14.

39

PARAMETRIC VERSUS NON-PARAMETRIC STATISTICS

Parametric statistics require quantitative dependent variables and are usually applied when these variables are measured on either interval or ratio characteristics. Statistical techniques that involve analysis of means, variances and sums of squares are under parametric statistics. Parametric statistics require assumptions about the distribution of scores within the population of interest. The nonparametric statistics focus on differences between distributions of scores and that can be used to analyze quantitative variables that are measured on an ordinal or even nominal level. Nonparametric statistics do not require many of the assumptions about distributional properties of scores that parametric statistics rely on.

BETWEEN-VERSUS WITHIN-SUBJECTS DESIGNS

Experiment 1 Consider an experiment where the investigator is interested in the relationship between two variables: type of drug and learning. The investigator wants to know whether two drugs A and B, differentially affect performance on a learning task. Fifty participants are randomly assigned to one of two conditions. In the first condition, 25 participants are administered drug A and then read a list of 15 words. They are asked to recall as many words as possible. A learning score is derived by counting the number of words correctly recalled (scores can range from 0 to 15). In the second condition, a different 25 participants read the same list of 15 words and respond to the same recall task after being administered drug B. The relative effects of the drugs on learning are determined by comparing the responses of the two groups. In this experiment, the investigator is studying the relationship between two variables: (1) type of drug and (2) learning as measured on a recall task. Type of drug is the independent variable and the learning measure is the dependent variable. The independent variable is set up so that participants who received drug A did not received drug B and those who received drug B did not received drug A, that is the two groups included different individuals. A variable of this type is known as a between-subjects variable because the values of the variables are split up between participants instead of occurring completely within the same individuals. Research designs that involve between-subjects independent variables are referred to as between-subjects designs or independent groups designs. Experiment 2 Consider a similar experiment that is conducted in a slightly different approach. A group of 25 participants are administered drug A and then given a learning task. One month later, the same 25 participants return to the experiment and are given the learning task after being administered drug B. The performance of these participants under the influence of drug B is then compared with their earlier performance under the influence of drug A. In this experiment, the 25 participants or subjects who received drug A also received drug B, that is, the same individuals participated in both conditions. A variable of this type is known as a within-subjects variable. Research designs that involve within-subjects independent variable are referred to as within-subjects designs or correlated groups designs or repeated measures designs.

SELECTION OF STATISTICAL TEST The importance of the selection of a statistical test rests on distinguishing between qualitative and quantitative variables and between within-subjects and between-subjects designs. The requires steps are: (1) identify the independent and dependent variables, (2) classify each as being qualitative or quantitative, (3) classify the independent variable as being between-subjects or within-subjects in nature and (4) note the number of levels that each variable has

40

INFERENCES ABOUT TWO MEANS

Two samples are independent if the sample values selected from one population are not related to or somehow paired with the sample values selected from the other population. If the values in one sample are related to the values in the other sample, the samples are dependent. Such samples are often referred to as matched pairs, or paired samples.

START

PAIRED t-TEST

t TEST

Pool the sample

variances

CASE 2

t TEST

Do not pool the

sample variances

CASE 3

Dependent

(Matched)

Samples

NO

Independent

Samples?

YES

Equal

Population

Variances? NOYES

INFERENCES ABOUT TWO MEANS:

Independent Samples Assumptions

1.) The two samples are independent. 2.) The two samples are simple random samples selected from normally distributed populations.

When these conditions are satisfied, use one of the three different procedures corresponding to the following cases: Case 1: The values of both population variances are known (In reality, this case seldom occurs)

Case 2: The two populations have equal variances (That is 22

2

1 )

Case 3: The two populations have unequal variances (That is 22

2

1 )

Case 1: Both Population Variances Are Known In reality Case 1 almost never occurs. Finding population variances typically requires that we know all of the values of both populations, and we can therefore find the values of their population means so there is no need to make inferences about their means.

41 Remember: ( )

Null Hypothesis (Ho): There is no significant difference between two population means 1 = 2 Alternative Hypothesis (H1): There is a significant difference between two population means 1 2 (Non-directional or two-tailed test) 1 < 2 (Directional: Right-tailed Test)

1 > 2 (Directional: Left-Tailed Test)

Notation for parameters and statistics when considering two populations

Choosing Between Cases 2 and 3: Preliminary F test approach: Apply the F test to test the null hypothesis that 12 = 22. Use the conclusion of the test as follows:

Use case 2 if F 2.50 and conclude that the two groups have equal variances Use case 3 if F > 2.50 and conclude that the two groups have different or unequal variances CASE 2: Equal Population Variances: Pool the Two Sample Variances Hypothesis Test: t-Test for two population means (assume equal variances)

21

212

21

2

2

1

2

21

2

2

1

2

2121 )()()()(

nn

nns

xx

n

s

n

s

xx

n

s

n

s

xxt

p

pppp

where pooled variance: )2(

)1()1(

21

2

22

2

112

nn

snsnsp

degrees of freedom: df = n1 + n2 2

42

43 Example: Independent simple random samples of 35 faculty members in private institutions and 30 faculty members in public institutions yielded the data on annual income in thousands of dollars in the following table. At the 5% significance level, do the data provide sufficient evidence to conclude that mean salaries for faculty in private and public institutions differ?

Private Institutions Public Institutions

19.881 x

s1 = 26.21 n1 = 35

18.732 x

s2 = 23.95 n2 = 30

Solution:

Step 1: State the null and alternative hypotheses Ho: Mean salaries for faculty in private and public institutions does not differ (1 = 2) H1: Mean salaries for faculty in private and public institutions differ (1 2) where 1 and 2 are the mean salaries of all faculty in private and public