8210 wk2 assignment

Assignment Tutorials

Posted on: Thursday, May 26, 2022 10:30:46 AM EDT

Please consider the following as a general guide of what is expected within all assignments:

- Title Page [see Walden University Template for formatting].

- Introduction [required]: When drafting a formal, scholarly or academic paper alway start with an introduction. The introduction immediately orients any audience to the paper's purpose. IE: The following is a selection of articles on fair hiring practices. Each source will be annotated to inform an audience of the general focus and scope of the sources.......

- Articles [if a bibliography] each article or research source is listed by formal reference. Immediately thereafter, the author of the bibliography gives a concise overview of the article and/or reference's content and purpose [one paragraph]. The bibliography continues with what the writer has gleamed from the source as relevant to the paper's purpose [see introduction above].

OR

-Content [assignments other than annotated bibliographies] using APA formatted headings, hold the hand of your audience assisting through topical transitions allowing them to follow and anticipate.

- Summary [required]: having considered a topic, looking at varied sources to learn about the topic synthesize the sources into a few summary statements. Continuing with the example: The selection and review of articles on fair hiring practices makes evident some of the most common errors to avoid are.....OR...a couple of statements threading together what has been learned by your scholastic engagement of the content.

PRONOUNS: Note in the above instructions not a single demonstrative or personal pronouns appears.Overuse of pronouns is considered a potential "...affront to clarity an can exclude a passive audience (APA 7.0)."

APA Tutorial: Do not lose valuable points in grading by excluding core elements or avoiding headings necessary to facilitate audience access or by including pronouns limiting and/or prohibiting audience access. Please focus upon the misuse/overuse of pronouns considered an "affront" upon clarity with academic/scientific/formal writing. Pronouns potentially exclude your audience and unnecessarily conceal critical content. Take a look at an example;

WRONG: This information was prepared to make clear that those critical polices of the agency you must follow when hiring somebody..

RIGHT: The brief, informational brochure presents the mandatory policies of the Federal Office of Discrimination when engaging hiring processes.

Posted by: John Billings

Posted to: RSCH-8210D-2/RSCH-8210C-2-Quantitative Reasoning-2022-Summer-QTR-Term-wks-1-thru-11-(05/30/2022-08/14/2022)-PT27

Re-authenticate SPSS

Posted on: Thursday, May 26, 2022 10:27:51 AM EDT

Attention Students,

SPSS is a software package that provides students with statistical analysis, modeling, predictive, and survey research tools used in many of our courses and advanced research activities. Students can download the latest version of SPSS from the web free of charge. Walden is pleased to be able to continue offering this important resource free of charge for the duration of your program; and now the process for receiving SPSS is easier than ever.

Please note that all students will need to enter an updated access code on September 30, 2013 to re-authenticate their SPSS license. Read the instructions about updating an expired license. If you are in need of the latest version SPSS, version 20.0, you may download it by following the installation instructions.

Do I need to download the latest version?

· Yes, if you have an older version and would like the current version.

· Yes, if you had SPSS but your license has since expired and you need to continue using it.

· Yes, if you are beginning your first course where you will be using SPSS.

· If you have any questions about this change or need assistance downloading SPSS, please contact Walden's Student Support Team from your myWalden portal by choosing the Support tab and clicking "Click to Chat." You can also call 1-800-WALDENU (925-3368) or e-mail [email protected]. Please include your name, student ID number, and degree program with any correspondence.

· Posted by: John Billings

· Posted to: RSCH-8210D-2/RSCH-8210C-2-Quantitative Reasoning-2022-Summer-QTR-Term-wks-1-thru-11-(05/30/2022-

This is an example paper from my instructor if you are having problems with the software but hopefully not

Guide to Research Statistics:

A Monograph for Use with ED: 8900 Courses

Written by: John W. Billings, Ed. D.

Introduction:

The methods of statistical calculation, based on data, are valuable resources within the field of applied research. If not used regularly, however, these methods are not easily retained. This means, when used sporadically, these methods can be cumbersome and errors imminent. Additionally, if not readily used, the products of statistical methods are not understood, and their value compromised. This guide is written with three instructional goals in mind. First, the guide is written by linking easily understood and applicable scenarios to certain statistical methods. Second, the document is not time specific. It is written to be a perpetual reorientation for practitioners within their respective fields. Third, the document clarifies the meaning of the products of statistical analysis in order that numbers can become useful in discerning courses of action. Whether attending a special education annual reviewing, analyzing mandated test scores or reading system reports, one only need to scan this material to be immediately reoriented to, and understand, general statistical analyses.

Sampling:

In the most, simple of terms, sampling refers to the means by which subjects are selected for use within research exercises. A population is the total number of subjects with at least one, common and relevant characteristic. For example, “...all fourth, grade students within a local school district.” would express a population. A sample is a portion of that total population or, “…35 of the 128 fourth graders within a local school district.” For a sample to have practical value, it should be representative of the total population. By practical, the writer suggests that the intent of a proposed research project is to eventually generalize results upon the larger population; the representative sample should reflect accordingly. Generally, if 11% of the fourth graders have individualized education plans [special education students] then the sample should have a similar, representative percentage. If two-thirds of a population is female, then the sample should reflect that gender disparity. The reader should note that these are general guidelines for representative samples. The specific nature of a proposed study might alter guidelines. For example, a curricular initiative designed to assist gifted students might intentionally exclude special education students from its sample. A study on premature menstruation

among inner city females would exclude all males. Evaluation of a sample’s representative integrity typically takes place after a selection of an initial sample has been made, though this is not an inflexible rule. Breaking a total population into representative subgroups prior to selection, such as males, Hispanics and so forth, can compromise a researcher’s attempts at random sampling.

The most frequent error in educational statistics is the presumption of random sampling. A random sample indicates that all members of an original population had equal opportunity for selection. Reflect upon the following scenario. There are 128 fourth graders within a local school district listed in alphabetical order. The researcher decides to “randomly” choose every fourth student for a research sample. This is not random. The predetermined order of the student names upon the initial list, and the determinant of every fourth compromises randomness. In this design, each student has a one-in-four chance of selection. If the same students’ names were placed on individual pieces of paper and placed within a basket, shaken and blindly pulled, then every student has an equal chance of selection. That would be random. There are reasons, within certain studies, for the selection of a sample not to be random. In sound statistical method, however, the means of sampling must be carefully considered and then amply expressed to a potential audience for their collective consideration.

As to the size of a sample, the presiding rule is that as a sample increases, the more likely it is to be representative. Mathematically, a researcher must understand that eventually one is called to defend the significance of a change within a sample. In statistics, degrees of significance are not subjective. A sample of ten is treated and there is certain, measurable change in one member. Juxtapose that to a sample of one thousand, who upon being exposed to another treatment, reveals ninety students have changed. Knowing nothing else, what treatment might you be predisposed to choose?

Categorizing Statistical Processes with Research Proposals

Early in the design process, a researcher must become sensitive to the two general categories, and two general uses into which statistics are sorted.

Inferential Statistics use representative samples to make generalizations about larger populations.

Descriptive Statistics yield information about very specific populations. For example, within a comprehensive high school, a researcher might isolate advanced placement students for specialized study. The results of this study would not be implied upon all students without further analysis.

Parametric Designs are used with interval and ratio scales only and when the scores of the sample groups are normally distributed, variances within the sample groups compared are alike and the subjects within the sample groups are randomly chosen. A t-test is the most common parametric measure. The majority of statistical tests used within the field of education are classified as parametric. For example, an investigation of how a student’s IQ scores [interval scale] compare to a peer’s is a parametric measure.

Nonparametric Designs are often restricted to nominal or ordinal scales, though interval and ratio scales that do not meet the assumptions of parametric design may be used. Non-parametric designs are easier to compute. The Chi-Square test is an example of non-parametric statistics. Typically, when a researcher is comparing responses upon a survey coded “1” for girls and “2” for boys, calculations such as determining a simple average does not make practical sense. This is the type of situation in which non-parametric measures are applied.

Scales:

Quantitative data comes in various forms. Within applied research, the designer can often decide in which format data is made available or decide which data to use based on its format. Below is a summation of the forms in which quantitative data are typically available.

NOMINAL SCALES: A number is assigned to a member of a sample as a mere label or categorization. For example, the number worn by an athlete on a team is an example of a nominal scale. A sports information director might then record the frequencies of certain athletic events to a player by shirt number. A query such as how many assists do guards have in relation to forwards in basketball could be derived by looking at the frequencies with which assists are assigned to the shirt numbers associated with those positions.

ORDINAL SCALES: When thinking about ordinal scales, think about smallest to biggest or oldest to youngest. A couple of educational examples might be a study of weight loss in which students are rank ordered: from largest to smallest or listing student test scores from 100 to the lowest score within the class.

INTERVAL SCALES: Similar to ordinal scales, only this time data are recorded in predetermined intervals between points on the scale. Many criterion-referenced tests place students within performance intervals such as “needs improvement” or “proficient.” Intervals within this scale do not have an absolute zero. For example, a student with an IQ of 100 is not 80% as smart as a student who has an IQ of 125.

RATIO SCALES: A ratio scale is an interval scale with an absolute zero. For example, a child that weighs 120 pounds is twice as heavy as one who weighs 60 pounds.

Measures of Central Tendency

When using statistical measures, there are certain tools that ensure controls from statistical anomalies. Perhaps better stated, these tools assume predictable patterns within data and readily reveal anomalies that alert statisticians accordingly. Regardless of what is being measured, the larger a given sample the more evident patterns become. By looking at these patterns, and individual variations from these patterns, the more powerful one’s statistical analyses become. As these next few sections unfold, consider a certain scenario. You are a parent of a high school student and you have just discovered your child’s American History test amongst some books. The score is 78. But what does that score mean? What is its value? To begin to answer that question consider the calculated set of statistical tools called the Measures of Central Tendency. These include mean, median and mode.

Mean: Mean is more typically called an average amongst statistical novices. Remember when you were a student and an academic quarter was coming to a close and you would calculate your average in anticipation of a grade? To determine the mean of a set of scores, one simply adds all of the raw scores and divides that sum of the scores by the number of total scores.

Median: To determine the median, one simply orders the raw scores from the lowest to the highest and then, determines by counting, the middle score. The median is the score at which half the test scores are above and half are below.

Mode: The mode of a set of scores is the individual test score that occurs most frequently. In smaller samples there might not be a distinct mode. The larger a sample gets, the more evident the modes tend to be, and on some occasions, there can even be more than one.

Application of Measures of Central Tendency:

The following (Table 1.1) is a list of sophomore test scores assigned students within an American History course. The test asked students one hundred questions and awarded one point for each correct answer. A perfect raw score would be 100.

Table 1.1 100 86 71 56

98 86 71 55

97 84 70 53

94 82 69 51

93 80 68 50

92 78 67 50

90 76 66 48

89 74 65 41

87 72 61 40

87 71 59 38

∑= 2865 mean: 2865⁄40 = 71.625 median: 71 mode: 71

This list was purposely designed to represent a predictable pattern or what is called within statistical analysis as a normal distribution of scores. A normal distribution of test scores suggests that given a random sample of students, their attending scores will distribute across a predictable, predetermined range. Additionally, as this or any sample becomes larger, the scores will distribute in certain, subtle, and predictable range allotments. Below (Table 1.2) is a representation of a normal distribution of scores referred to as a Bell Curve. The Bell Curve dates to 18th C France and visually plots data by respective frequencies.

Table 1.2

.13% 2.19% 13.59% 34.13% 34.13% 13.59% 2.19% .13%

Graph: The Normal Curve is a bell-shaped curve

68.26%

95.44%

99.74%

Assume that Axis OY is the number of students who obtained a given grade on the American History Test. Assume that Axis OX is the actual raw score of any one student. At the extreme right of Axis OX [designated by a black dot] is the student score of 100. Likewise, at the extreme left of this axis would be a student score of 0. At the bottom of Axis OY [designated by the same black dot] is the frequency with which the score 100 occurred within our working sample of “sophomore test scores assigned students within an American History course.” We know that the frequency of a raw test score of 100 in this sample is one.

Continue to assume that Axis OY is the number of students who obtained a given grade on the American History Test and that Axis OX is the actual raw score of any one student. At the center of Axis OX [designated by a white dot] is the student score of 71. At the top of Axis OY [designated by the same white dot] is the frequency with which the score 71 occurred within our working sample of “sophomore test scores assigned students within an American History course.” We know that the frequency of a raw test score of 71 within this sample is three. You now have a working knowledge of how data can be visually distributed upon a dual axis graph and how to interpret this data.

Standard Deviations:

Continuing to look at Table 1.2, you will see below the graph a box labeled standard deviations and the number values of -3 through 3 evenly distributed across the graph through vertical lines. You will also note that there are representative percentages within the parameters of those vertical lines. Simple addition reveals that 68.26% of all grades,

within a normal distribution of scores fall within the standard deviations of –1 and 1. These two standard deviations, and the combined scores within, are often referred to as the average population.

Historically, this is where the assignment of letter grades within American Academia was born. The range of grades between SD –1 and 1 represents the ‘C’ student. If you look at the table again, you will see that the last vertical bar to the right, so designating the Standard Deviation of 3, represents the ‘A’ students. For many, many years, the Bell Curve sorted populations into labeled grading variances. The predictable debate that unfolded attacked the presumption that only .13% of a normative distribution represented superior performance while a full two thirds represented average. In our dedicated scenario of the American History tests, the student with a raw score of 100 would get a letter score of ‘A’. More controversial is that the student who received a grade of 98, within this designed sample of History students would be awarded a ‘B’. In an ASCD publication called The Truth about Testing, James Popham, Ph. D., a patriarch of standardized testing exercises a debate over the misuse of standardized testing within unfolding American Educational Reform Movements. It is a valued read due to Popham’s contribution to the original designs of standardized tests and the predominance of mandated testing in public education.

Standard Deviations sort scores so that researchers can identify and analyze degrees of statistical significance amongst subjects within samples. Namely, if you had a student that performed at a level of two deviations below the average [SD –2], then one understands that this student was outperformed by 95.57% of peers. This is a degree of statistical significance.

While software packages [e.g., SPSS, 2005] readily calculate standard deviations, it is important to understand the basic method for generating them. To solve for standard deviation one must first solve for the sum-of-differences. In our scenario, the average American History test score is determined to be 71.625. If calculating SD by hand, the researcher would then take each, individual test scores and subtract it from the established mean. [e.g., 100-71.625=28.375] These products, or differences from the mean, would then be added and their total divided by the total number of test scores. The result is the standard deviation.

Variance

Variance is calculated immediately after standard deviation by simply taking the SD and squaring it. The variance is the average distances from the mean score in squared units. It is necessary to know how to calculate the variance as it is a key element in calculating future statistical products such as a t-test.

Application of Standard Deviation and Variance:

To check your knowledge of how to calculate standard deviation and variance, the following simple list of scores and calculation products is offered:

Test Scores: 11 8 5 2

10 7 4

9 6 3

åX = 65.00 Mean = 6.5 VAR = 8.24 SD = 2.87

Z-scores:

Continuing with our high school scenario, a teacher of Algebra II wants to know if a student within his class performs better in Algebra or Physics. The teacher then decides to compare students’ performance on his classroom test and on the test in Physics. There is a problem. These are two different tests with likely two different means and standard deviations. In order to compare these two sets of tests, we need to recalculate them into the same scale. This conversion of scores, in order that they might be further analyzed is to produce a product called standard scores.

To create standard scores we will convert them, first, to z scores. Z scores change a raw score to a notation that indicates how many standard deviations that original score is from its mean within its original group. For example, we have a raw score on the American History test of 74, a mean of 71 and a standard deviation of 3. From this we know that (71+3=74) or that a score of 74 is a z score of +1.

If we convert all scores, between two sets of tests, to z scores, we have converted the scores to a scale where the mean is now 0 and the standard deviation is 1. To calculate a z score, use the formula: raw score mean/standard deviation.

Application of Z Scores:

SUBJECT TEST SCORE TEST MEAN TEST SD

Algebra II 74 71 3

Physics 68 56 6

Z score [Algebra II] = 74-71/3 = 3/3 = +1.00

Z score [Physics] = 68-56/6 = 12/6 = +2.00

These calculations indicate that a +2.00 in Physics is farther above the now standard mean performance than the +1.00 in Algebra II. The conclusion can be drawn that this student performs better within physics. Of course, this is a very simplified example. Within this scenario, a responsible teacher researcher would also look at the reliability of each test and other instructional variables such as style of test, means of instruction, class size and so forth.

T Scores:

Z Scores can generate negative numbers, decimals, and a value of 0. As these products might make future calculations more difficult, some researchers convert z scores into t scores or whole numbers. This is a very simple process. You simply multiple a z score by 10 and then add the constant value of 50.

Application of T Scores

Taking our two z scores from the Algebra II verse Physics scenario from above, the conversion would be as follows:

Algebra II Z score +1.00 1.00(10) +50 = 10+50 = 60

Physics Z score +2.00 2.00(10) +50 = 20+50 = 70

Percentile Rank:

Within psychological testing and some educational testing, researchers might desire to designate a score in relation to an entire population. A percentile rank so indicates that relationship. In any given distribution it might be recorded that a student falls at the 70th percentile. This would indicate that 70% of students within a sample scored below this student’s score while 30% scored above.

Stanines:

Originating from the phrase “a standard nine”, this is a scale that presumes exactly nine, equally distant points across a distribution of scores. The mean of a stanine scale is always 5 and the standard deviation is 2.

Application of Standard Deviations, Percentile Ranks, and Stanines

All variations of the distribution of these scores across a Bell Curve, can lead to an understanding through visual comparison. To this end, look at Table 1.3. On this table, the various measures are juxtaposed with the assistance of vertical markers. This table is an excellent tool for assisting statistical amateurs in a quick understanding of these measures’ relationships. It is also very useful for orienting parents within special education TEAM meetings.

TABLE 1.3

As Distributed Across a Bell Curve [Normal Distribution]

SD –3 -2 -1 0 1 2 3

% of cases (SD) .13 2.19 13.59 34.13 34.13 13.59 2.19 .13

Percentiles 2 16 50 84 98

Stanines 1 2 3 4 5 6 7 8 9

% in stanine 4 7 12 17 20 17 12 7 4

Correlation:

In statistics, correlations seek to calculate relationships between variables. Within applied research, correlations are most often used when a researcher uses two or more measures on the same sample. For example, a researcher is looking at emergent literacy with a group of third grade students. The school system has already adopted a published and validated reading assessment. Simultaneously, the state mandates an additional reading test each spring. The researcher, attempting to affect an increase in student achievement, within the specified area of reading comprehension, implements a research-based intervention. The researcher then wishes to correlate the school system’s internal measure to the state’s external measure in an attempt to verify that the intervention is responsible for student gains. The researcher seeks to define the relationship between these known variables.

Application of Correlation

Correlation is reported to a researcher within a range of –1.00 to 1.00. In a correlation of –1.00 a researcher cites a perfect, negative correlation. That is, one variable adversely affects another with certain consistency. A correlation of 0 suggests there is no measurable correlation between two identified variables. A correlation of +1.00 indicates a strong or positive correlation: one significantly influences change in another. The degree of correlation is what assists a researcher in making final assumptions or acting with established certainty [there are published tables that identify size of population or sample and interpret the expected levels of significance of correlation accordingly in virtually every statistics or algebra textbook. These published tables help a researcher to juxtapose the size of a sample with various levels of probability].

PROBABILITY: statistics discern probability and NOT certainty. We ultimately cannot prove anything! Rather, we narrate to our audience certain controls and parameters around of applied methods and then make statistical findings based on the probability that within replicated situations similar findings will reoccur.

LEVELS OF SIGNIFICANCE: establish when a product of statistical calculation is a number of certain legitimacy a number worthy of our interest. In most educational venues a product at a level of probability of 5% (p≤ .05) or less is deemed statistically significant.

The Pearson r [Pearson Product-Moment Coefficient].

A Pearson r looks at the relationship between two sets of data to determine if there is a discernable correlation. For example, the National Reading Association has long held that there is no direct correlation between a student’s expanding vocabulary and an increase in reading comprehension. To test this theory, a local school system might take a grade level’s standardized vocabulary sub-test and juxtapose it to the same grade level’s reading comprehension sub-test. One would suppose, if the theory holds, that the distribution of individual students’ vocabulary sub-tests do not match up with that of the same students’ corresponding reading comprehension scores.

Before a researcher attempts a Pearson r certain criteria must be met.

The data must be in interval or ratio scales. Scales are readily defined within a variety of statistical texts. In applied research, one usually identifies early evidence of a problem. Quantitative evidence is reported in various scales. The majority of data in education is reported in intervals.

The data has a linear relationship. A linear relationship is data that begins with a negative relationship that evolves into a positive one. As a means of a common example, look at the relationship between age and weight.

If the data to be processed does not meet these two criteria, then the reader should consider the Spearmen Rank-Order Correlation. In the way of an example, a researcher might take a junior classes’ Grade –Point-Averages and place them in rank order: highest to lowest. Next, the researcher would place students’ frequency of behavior referrals in rank order: most frequent to least frequent. Performing a Spearman Rank-Order Correlation the researcher would test a theory as to whether there is a demonstrated correlation between a student’s poor GPA and increase in behavioral referrals.

Inferential Statistics

Often, within applied research models, the primary task is comparing one group to another. In a classic experimental model, a researcher identifies a control group with an experimental one. Avoiding the distraction of an ethical debate, consider a model where one-of-three 4th grade classroom teachers implements a research-based, instructional intervention. There appears to be a change. The question then becomes, can data from the affected group be compared to the remaining two classrooms to determine if the change is by mere chance or if the change is to a degree of statistical significance that the teacher can so declare: the new instruction increases student achievement. When the means of one group is compared to the means of another, the statistical calculation is known as a t-test.

T Test

On occasion, an applied researcher seeks to compare changes within separate groups to attempt to declare that an action, intervention or treatment implied upon one can be defended as the causative action of change from the other. This statistical model tries to determine if the changes in the means of separate groups is to a degree of certain statistical significance so as to eliminate critics that might suggest the change is one of mere chance or coincidence. When a researcher compares the means of one group to another, the researcher typically engages a T Test.

There are three, distinctly separate research models in which the mean of one group might be compared to another:

Independent Samples: Most often, an independent sample is when a researcher has two, similar groups from the same population. The researcher then establishes one of the groups as an experimental and the other as a control. In this way, the researcher makes a primary attempt to eliminate c

measurable variable: namely the treatment to which the experimental group is exposed. When comparing the means of these two groups one is performing a T Test of Independent Samples.

Single Sample: In a single sample T Test there is an identified sample and it is compared to the entire population from which it was initially selected. For example, amidst education reform initiatives, there is growing concern about academic achievement amongst vocational high school students when compared to the general population. If a researcher separated the scores of vocational high school students [a sample] from all high school students [the population] and compared their respective means, this would be a T Test of a Single Sample.

Paired Samples: Finally, especially within applied research, a student researcher’s powerbase might be limited to a certain, assigned classroom of students. Often, this situation limits the researcher to do a pre-test of a student sample, treat the sample, and then perform a post-test to measure potential change. When a researcher measures the means of the same group twice, that group can sometimes be referred to as either a dependent or correlated sample. The product of this exercise is called a T test of Paired Samples.

A t-test is calculated by:

t = μ1 – μ2

(n1 – 1) S12 + (n2 – 1)S22 ∙ (1/n1 + 1/n2)

n1 + n2 – 2

Where: μ1 = the mean of group 1

μ2 = the mean of group 2

S12 = the variance of group 1

S22 = the variance of group 2

n1 = the number of members of group 1

n2 = the number of members of group 2

Analysis of Variance [ANOVA]

If a researcher sets up a model that requires a comparison of means of three or more groups, there are two statistical options. The researcher can repeat the T Test as many times as is required. The problem with this is that there is a margin of error with the calculation of a t test and with each repeated calculation the error is compounded. So, a researcher can engage an ANOVA, which in simple language, calculates all the relevant T-Tests at once. The simplest form of an ANOVA is a One-Way ANOVA.

Like all of the prior statistical measures, doctoral students at Nova Southeastern University can access SPSS and key in data to generate necessary products. However, in light of the purpose of this monograph, the author offers the following scenario in order

to demonstrate an appropriate use. A Local health club has designed four weight loss programs. As the programs are implemented, the club owner wants to know which program might yield the greatest success for its clients. Once determined, the club owner hopes to open more sections and market that successful program. The programs are: diet workshops that teach better eating, a walking club that includes a diet, a weight- lifting program with public weigh-ins and a spinning class that observes the diet workshop as it spins. We propose, then, a null hypothesis which would be HO = μ1 = μ2 = μ3 = μ4.

The data to be collected will be quantitative. Each participant will record an initial weight and then individual weight loss will be monitored. Below (Table 1.4), the four programs are represented by name. In each group there are five participants. The score of each participant is the number of pounds lost. The following is the calculations for determining the F-ratio [ANOVA] for the data within Table 1.4.

Table 1.4

Program Diet Workshop Walking Weights Spinning

14 15 10 13

16 12 13 16

12 14 9 15

13 16 13 16

14 9 11 13

∑X 69 66 56 73 ∑XT = 264

∑X2 955 880 638 1067 ∑XT2 = 3540

n 5 5 5 5 NT = 20

µ 13.8 13.2 11.2 14.6 µT = 13.2

generated sums of squares:

SSW = ∑[∑X2 – (∑x)2/n ] SSW = 23.6

SSB = ∑[(∑x)2/n ] - (∑XT)2/NT SSB = 31.6

SST = ∑X2T - (∑XT)2/NT SST = 55.2

generated mean squares:

dfW = N-K = 16

dfB = K-1 = 3

dfT = N-1= 19

generated mean square with and mean square between:

MSW [mean square within] =SSW /N-K =23.6/16 =1.48

MSB[mean square between] =SSB /K-1 =31.6/3 =10.53

F-ratio = MSB /MSW = 10.53/1.48 = 7.14 p<.01

Therefore, because the product exceeds the level of probability, we conclude that there is a significant difference between at least one pair of the means from the various workshop offerings.

Chi Square for Surveys

When a researcher wants to analyze data from certain type of survey, a Chi Square is used. The calculation is used to analyze responses in numeric codes. For example, a survey asks a respondent to answer twenty questions. Each of the questions affords three responses. A ‘1’ indicates the desired answer of ‘yes’. A ‘2” indicates the desired answer of ‘no’. A ‘3’ indicates that the respondent ‘does not have enough information to respond.’

Two types of frequencies are recorded within this type of research model. The first set-of-frequencies are called observed. This is real or empirical information as offered by the respondents. The second set-of-frequencies are called expected. Expected outcomes are predicted or a result of one’s expertise. The end product of a Chi Square is to measure the differences between these two types of data to determine if respondents observed answer differently, to a degree of statistical significance, from what is expected.

Application of Chi Square

Though not the result of a survey, the flipping of a coin provides a good analogy for a Chi Square. If a random participant flips a coin one hundred times, what is the expected frequency? The answer would be 50 times the coin would reveal heads and 50 times the coin would reveal tails. Next a researcher asks one hundred high school students to flip a coin one hundred times, each, and record results. A Chi Square would take the students’ actual results [observed] and compare them to the 50/50 prediction [expected].

In the following formula the symbols represent:

X2 = chi square, O = observed frequencies, E = expected frequencies.

The formula for Chi Square is: X2 = ∑[(O-E)2/E] or Chi Square = The Sum of [Observed Frequencies – Expected Frequencies]2 Divided by the Expected Frequencies.

Considering an application of Chi Square within an educational setting, consider a study that looks at the relationship of students’ letter grades within an academic area and to students’ scores on a criterion-referenced test [CRT] in the same academic area. One would hypothesize that an ‘A’ would occur at a correlated frequency to the highest achievement interval on the CRT.

Summary

This monograph was written for the expressed purpose of providing a general overview of research statistics. It discussed, through practical analogies the purpose of performing such calculations. It provides direction on how to perform such statistical calculations. Finally, and most importantly, it helps an audience to understand the value of the products of these calculations