# Portfolio Assessment in Statistical Method in Educational Research Portfolio Assessment in Statistical Method in Educational Research. here is an example of how  this form of assessment is developed.

Question One (1)

a). What is educational statistics? Mention the various branches of statistics as a;

i). Discipline and

ii). As a measure and explain them, stating their characteristics and examples

b). State at least, three roles/importance of statistics to education

c). Mention at least three types of variables and explain them with clear examples

d). Define educational data? Mention the;

i). Levels of data and their implications respectively with adequate examples

ii). Categories of data

Portfolio Assessment in Statistical Method in Educational Research

Question Two (2)

a). What do you understand by measures of Central tendency? Briefly discuss what you understand by Kurtosis and Skewedness.

b). Mention the three major measures of central tendency,

i). State where each of them is most appropriate

c. State four (4) applications of measures of central tendency

d. Group the data below, and estimate:

i). The mean

ii). Mode and

iii). Median

2        5        4        7        6        9        10      7        20      30      12      22

17      16      25      18      15      10      21      28      3        4        8        6

7        14      10      11      29      19

Portfolio Assessment in Statistical Method in Educational Research

Question Three (3)

a). What is measure of variability? Mention at least three (3) measures of variability and discuss them

b). Using the information in question (2d), calculate the;

i). Range

ii). Semi-interquartile range and

iii). The standard deviation of the data

c. Mention 5 applications of measures of variability

Portfolio Assessment in Statistical Method in Educational Research

Question Four (4)

a). A professor in the Faculty of Education UNN conducted a study in order to determine the achievement of male and female students in RME 603. After computing the mean achievement of the students, it was shown that the male students outperformed the female students but an inferential statistical test further conducted revealed that there was no significant difference in their mean achievement.

What would you say is the reason behind the difference in the means of the students? Justify your choice.

b). Given these samples sizes (n = 100, 200, 400 and 800) compute the standard errors of mean using a standard deviation of 28. Compare the values and explain your observation(s) or implications

c.) Explain the concept of central limit theorem and states the various theorems explaining each of them respectively? Given that the population IQ mean is 100 with standard deviation of 15. If we sample 50 lecturers, what is the probability that the average IQ of the sample is greater than 105?

d.) In a normal distribution of 2000 scores assumed to be normally distributed with a population mean, and a standard deviation D = 20, use the properties of the normal distribution to estimate the following probabilities

I. P(N< 90)

II. P(N>130)

II. P(N between 90 and 115)

IV. States at least four (4) applications of the area under the normal distribution curve.

Portfolio Assessment in Statistical Method in Educational Research

Questions Five (5)

(a) What is correlational technique? Consider the scores obtained by 2018/2019 first year postgraduate students in RME 603 (X1) and 601 (X2) test respectively.

 S/No X1 X2 1 20 18 2 10 17 3 19 16 4 12 10 5 15 14 6 18 13 7 14 9 8 11 11 9 17 20 10 16 12

Using the information above, compute;

i). The Pearson Moment Correlation Coefficient and interpret the result

ii). Spearman Rank Correlation Coefficient and interpret your result

ii). Rank the scores in the table above and compute the Kendall`s tau correlation coefficient

(b) Consider the rankings of students by four independent Professors in the Faculty of Science Education UNN on their performance in PGC 601 as shown below:

 S/N P1 P2 P3 P4 1 1 3 1 2 2 5 4 2 8 3 6 6 5 1 4 3 1 4 7 5 2 5 8 5 6 7 3 7 3 7 8 7 6 4 8 4 2 3 6

i). Compute the Kendall Coefficient of Concordance and interpret the result

ii). Why do the values of Kendall Coefficient of Concordance range from 0 to 1 and never negative?

(c) A researcher in an attempt to determine the relationship between anxiety level and students achievement in NECO examination collected the following results as shown below

 S/No Anxiety Level Students Achievement 1 1 50 2 1 48 3 1 40 4 0 60 5 0 54 6 0 66 7 0 80 8 0 49 9 0 68 10 0 70

i). Compute the biserial for the data above

ii). Interpret your result

iii). Should the correlation be greater than justify why otherwise, under what condition can the correlation coefficient be greater than 1?

(d) Under what condition would you as a researcher use the phi coefficient and the tetrachoric correlation coefficient?

 Item 1 Item 2 0 1 1 25 40 0 15 20

i) Using the appropriate statistics from the table above, taking correct response to 1 and incorrect to be 0; find the relationship between items 1 and 2 assumed to be measuring a construct that is assumed to be normally distributed in the population on a physics test.

ii) Interpret your result

 X1 (0) X2 (1) Y1 (1) 4 5 Y2 (0) 6 4

i). Supposing the data above is on two nominal dichotomous variables, compute the Phi correlation coefficient.

(e) In an attempt to determine the CGPA of PG student in UNN from their RME 601 and RME 603 scores, a professor obtained the following information from 10 students

 Students RME 601 RME 603 CGPA 1 500 30 2.8 2 550 32 3.0 3 450 28 2.9 4 400 25 2.8 5 600 32 3.3 6 650 38 3.3 7 700 39 3.5 8 550 38 3.7 9 650 35 3.4 10 550 31 2.9

Use the information above to compute

1. The partial correlation and
2. Part correlation for the CGPA

(f) Given three variables, study time, understanding ability and academic achievement; X1, X2 and X3 respectively, where r12 = 0.6; r13 = 0.5 and r23 = 0.8. find the coefficient of multiple correlation for R1(23)

Portfolio Assessment in Statistical Method in Educational Research

Questions Six (6)

a). What is statistical inference?

i). Differentiate between casual inference and statistical inference.

ii). Differentiate estimation from Null Hypothesis Test of Significance (NHTS)

b. Define estimation. Mention three uses of estimation and state the various forms of estimation explain them each.

c. Given a sample size of 20 with standard error of 8.60 and a mean of 58.0. Estimate the confidence interval at 5% level of significance.

d. Given that a certain variable has a standard deviation of 10, and we want to estimate the population mean at 95% confidence, what is the sample size required to achieve a margin of error of 3?

e). The scores of Science Education PG students in RME 603 are normally distributed with a standard deviation of 8. If a random sample of 30 students has an average score of 68, find the 95% confidence interval for the population mean of all the students’ scores.

Question Seven (7)

a). What do you understand by hypothesis?

i). State and explain the various types of statistical hypothesis.

ii). State two approaches of decision rule in hypothesis testing and state their rules respectively

b). A researcher carrying out a study to determine the influence of gender on students’ achievement in RME 604 stated a null hypothesis to guide his study at 5% level of significance. At the end of his investigation, the researcher found out that that there was a significant difference in the performances of male and female students in RME 604, but he failed to reject the null hypothesis. What type of error has the researcher committed and why?

i). In not more than four lines each, explain the meaning of critical value and rejection region

ii). State the major steps in hypothesis testing

iii). Under what condition would you as a researcher use the z-test and the t-test? What is the difference between the two tests statistics?

c). In a study to investigate the influence of location on students` achievement in Geography, the following scores of students were obtained from a Geography Achievement test administered on a sample of 28 SS 2 students made up of 14 urban and 14 rural as shown in the table below

Urban                                                        Rural

20      23      16      17      19               13      17      20      12      8

22      25      19      28      16               12      20      12      11      19

21      18      15      12                         10      18      20      16

Using the information above, compute the mean and standard deviation

i). State the hypothesis

ii). Set the criterion for rejecting the hypothesis

iii). Choose the appropriate statistics to be used and compute the test statistics

iv). State the decision rule

v). Draw the inference

Question Eight (8)

a). Mention and briefly explain three major assumptions of ANOVA

b). Why is it necessary and under what condition is it advisable to use ANOVA?

c). A researcher in an attempt to compare the mean achievement of eight (8) different groups in a Physics test, decided to carry out a multiple t-test instead of ANOVA.

i). Determine the numbers of test that the research will need to compute

ii). What will happen to the level of significance if it was set at 5% level of significance?

d). Consider the table below

 S/No Method X1 Method X2 Method X3 1 18 18 26 2 22 14 27 3 18 15 18 4 23 14 22 5 19 19 23 6 24 21 19 7 20 17 27 8 21 17 26

Using the information above:

i). State the null hypothesis

ii). Set the criterion for rejecting the null hypothesis

iii). Choose the appropriate statistics to be used and compute the test statistics

iv). What proportion of the students’ performance is attributable to the teaching methods

v). Why did you choose the statistics you employed?

Question Nine (9)

a). Mention the assumptions of ANCOVA

b). State and explain two purposes for ANCOVA

c). Mention three (3) importance of ANCOVA

d). Mention six (6) Post Hoc comparison procedures, classify them and illustrate where they are most appropriate

e). Using the information in question (8d) above, choose the appropriate procedure and compute the Post Hoc. Justify your reason(s) for choosing the procedure

Question Ten (10)

a). State three reasons why we have to use simple linear regression instead of correlation

b). What do you understand by;

i). Criterion variable?

ii). Predictor variable?

iii)Regression Model?

c). State the assumptions and major steps in the computation of simple linear regression analysis

d). The scores obtained by 10 candidates in a physics examination and the number of times they attended summer lesson in a month are as follows:

 S/N Study habit  (X) Invigilators` Rating (Y) 1. 18 28 2. 23 21 3. 30 40 4. 14 43 5. 25 42 6. 20 34 7. 15 25 8. 12 26 9. 10 30 10. 11 32

Using the information in the table above, estimate the:

i). Regression model

ii). Standard error

iii). Variance error

iv). Coefficient of determination and state its implication

v). Coefficient of alienation and state its implications.

Memory Matrix Development and Application

PORTFOLIO ASSESSMENT ANSWERS TO VARIOUS QUESTIONS

Question One

1. What is educational statistics? Mention the various branches of statistics as a;
2. Discipline and
3. As a measure and explain them, stating their characteristics and examples
4. State at least, three roles/importance of statistics to education
5. Mention at least three types of variables and explain them with clear examples
6. Define educational data? Mention the;
7. Levels of data and their implications respectively with adequate examples
8. Categories of data

Answers to Question One (1)

• Educational statistics simply refers to the application of statistics to the field of education. It is concerned with applied statistics which involve the application of already derived or generated formulas and equations (Applied Statistics) to educational data to achieve a more meaningful organization, analysis and interpretation of educational data. Hence, the ability to compute basic mathematical operations while taking into account the knowledge of these derived formulas is what is needed. However, the abilities to read tables, graphs and as well interpret basic results are necessary.

In conclusion, it is simply the extensive and useful applications of statistics to education in order to clearly understand and describe human characteristics, changes in human behaviours and drawing of sound inferences from educational data.

1. Branches of statistics as a discipline are
2. Pure Statistics: This branch of statistics which is also called theoretical statistics is concerned with the derivation and study of the mathematical properties of the various statistics (derived measures).
3. Applied Statistics: Applied statistics here refers to the application of these statistics in solving human problems. There is no rigorous computation here since it is concerned with the use of statistical measures and techniques in solving practical problems.
4. Branches of statistics as a measure are:
5. Descriptive statistics: This comprises those techniques for expressing, describing and summarizing data. Descriptive statistics is concerned with the presentation and summarization of data. In essence, it can only give information about the sets of data or groups in which those information are collected from. It cannot give information beyond that. Examples are pie chart, measures of central tendency, measures of variability, correlation etc.

Characteristics of descriptive statistics are

1. They are used in the description of data
2. They are used for presenting and summarizing information gotten from a set of data
3. They cannot be used to draw inference or conclusions on a set of data/information
4. They can give information on where a set of data is gotten from
5. They include charts, correlation, variability etc.
6. Inferential statistics: This statistical measure allows one to draw conclusions, generalization, inferences about population characteristics (parameters) based on sample characteristics (statistics). It is used in order for us to understand the entire population based on the information gotten from the sample.

Inferential statistics is classified into two:

1. Parametric inferential statistics: These are those statistical techniques that make use of strict assumptions out of a given distribution or population or scores. These assumptions includes; normality assumption, homogeneity of variances, continuous data etc. Examples of these statistics are: t-test, ANOVA, ANCOVA, z-test etc.
2. Non-parametric statistics: These are those statistical techniques that make no assumption about a distribution. Hence, they are called distribution-free statistics. Examples are the F-test, Chi-square, Kruskal Wallis, Mann-Whitney test etc.

Characteristics of inferential statistics

1. They are used in the drawing of generalization/conclusion to a larger population from a given sample
2. They allow for a deeper understanding about a population from the information gotten from the sample
3. Some of the inferential statistics make assumption (ANOVA, t-test, etc.) while others make no assumption about the distribution (Chi-square, f-test, Kruskal Wallis etc.).
• Importance (at least three, 3) of statistics to education are:
1. Statistics has contributed to the development of educational and behavioural science research by providing a more efficient way of handling data and dealing with complex educational problems.
2. The application of inferential statistics has widen the understanding of educational effects
3. The knowledge of educational statistics has made it possible for teachers to be able to select and use appropriate standardized instruments (or tests) in evaluating instructional outcomes
4. The knowledge of educational statistics helps the teacher to report the results of investigations of common classroom phenomena in acceptable language.
• Types of variables (at least three, 3) in education with examples are:
1. Moderator variables: These are a third variable that affects the strength of the relationship between a dependent and an independent variable. It is an independent variable that can either be categorical of continuous (i.e. qualitative or quantitative). Examples of categorical or qualitative moderator`s variables are; gender, location, race, socio-economic status etc. while the examples of quantitative variables include; age, weight, height etc., a typical illustration is in the study, Gender and Test Anxiety as Predictors of Examination Malpractice; the moderator factor there is gender (male and female).
2. Extraneous variables: These are independent variables that environmental factors which can have effect on the dependent variables(s) but whose effects are not controlled. Extraneous variables are capable of invalidating a study or findings and make it difficult for the researcher to know whether the result of the findings was actually produced by the manipulation of the independent variables or treatment or it was caused by another random factor. Examples could be noise, atmospheric temperature etc., a typical illustration is in a study involving the use of a new teaching methods verses the conventional method; if the experimental group and the control group are being taught by different teachers, then the teacher factor becomes the extraneous variable because it will be very difficult to tell if the outcome is because of the new teaching method or because the teacher who taught was better. To control this, the same teacher is expected to teach the both groups in order to eliminate this extraneous variable.
3. Dependent variables: These are variables in which the manipulation of the dependent variables affects. They can be explained as the variation or the changes which the researcher observes and which can be attributed or attributable to the independent variable. An illustration is in a study involving; effects of peer tutoring on the academic achievement of senior secondary schools students in Chemistry. The dependent variable here is the academic achievement of senior secondary school students.
4. Independent variable: These are variables whose effects the researcher is interested in. it is the variable that the changes in of variation in the study is attributable to. Hence, its manipulation affects the dependent variable. Independent variables can either be manipulable (the researcher can assign value to subjects at different levels) or non-manipulable independent variables (subjects are already assigned to the level prior to the study). An illustration is in a study involving; the effects of observational technique on students interest and achievement in Physics among secondary school students. The independent variable here is the observational technique because its manipulation will determine the achievements and interest of students in physics.
• Educational data can be referred to as those measurable or observable characteristics of educational variables in educational researches or investigations.

The levels of educational measurement and their implications

1. Nominal data
2. Ordinal data
• Interval data
1. Ratio data

Nominal Data: Data at this level of measurement is concerned with classification, categorization or naming of variables. In this level of measurement, we simply imply that one thing is different from the others but it does not mean that it is greater or lesser than the other. It is the least refined level of measurement since no mathematical operation can be conducted here. Only simple count can be possible at this level of measurement and the use of numeric values here can only be for identification or coding. For instance, male can be coded as 1 and female coded as 2 but it does not imply that female is greater than male or male is lesser than female.

Examplesinclude: Gender (male and female), location (urban and rural), countries (Nigeria, Ghana, Canada …), political party (APC, PDP, CAN, UNPP….), colours (red, black, yellow, blue…..) etc.

Ordinal Data: Data at this level of measurement, in addition to having the properties of nominal data, there is a presence or the property of “ordering or ranking”. This implies that variables or elements of a group can be ordered i.e. arranged in an ascending or descending order. As a result, we not only say that one thing is different from the other, we can also say that one thing is greater than or lesser than the others. In essence, two things are equal or different.

Examples: positions of students in a class; 1st, 2nd, 3rd, etc.; grades, A, B, C…; Shoe sizes; size 4, size 10, size 15 etc.

It is important to know that the conclusions from this level of measurement is limited since it is not possible to ascertain by how much one thing is greater or lesser than the other. In other words, equal interval does not correspond to equal attributes measured. i.e., we cannot categorically state that the difference between the performance of the persons assigned 1st and 2nd in a class is equal to the performance of those assigned 8th and the 9th respectively.

Interval Data: These levels of data, in addition to having the property of “ordinal data” also possess the property of equal-interval. This implies that the difference between two similar points on a measuring scale is equal. Hence, we cannot only say that one thing is greater or lesser than the other, but we can also be assertive as to by how much or what extent it is greater or lesser. Therefore, we can say that equal interval corresponds to equal attributes measured, that is, the difference between a score of 20 and 30 is equal to the difference between the score of 40 and 50. However, there is absence of “absolute zero”. Any zero here is arbitrary or relative i.e. a school of zero does not imply total lack of attribute measured. Mathematical operations such as addition (+) and subtraction (-) are possible here

Examples: temperature scale: 0oc, 10oc, 15oc, 20oc etc.; days in the calendar, 1, 2, 3 ,4, 5 etc., students’ scores; 10, 20, 30, 40, etc.; ages; 20years, 30 years, 40 years, 60 years etc.

Ration Data: Data at this level is the most refined data. In addition to having all the properties of interval scale, there is a presence of absolute zero. Hence, there is a true origin, and this is where measurement begins. In addition, we can say that something is twice or half the size of value of the other. Therefore, the presence of zero (0) in measurement here implies a complete absence or lack of such an attribute. This measurement is however, often used in the physical sciences, the only kind of such measurement in education is the deviation score of students which will always sum up to zero (0). Mathematical operations such as addition (+), subtraction (-), multiplication (x) and division () are all possible here.

Examples: Length, 0m, 2m, 4m, 10m etc., weight, 0N, 10N, 20N, 30N etc.; Temperature in Kelvin scale, 0K, 10K, 100K, 273K etc. Note that a weight of 0N implies total lack of weight.

• Categories of data
1. Discrete Data: These are data which measure only specific values. In the category of data, there are no intermediate values between any two such values. They are discontinuous data since they are only determined by counting and cannot be easily measured. This category of data is comprised of the nominal and ordinal level of measurement.

Examples: colours, number of professors in UNN, no of students in RME 603 class, etc.

1. Continuous Data: these are data for which any value is possible within any defined range of values. These data cannot be determined by counting by through higher measurement operations. As a result, between any two values, there are uncountable numbers of intermediate values. This category of data is comprised of the interval and ration level of measurement.

Examples: heights, weight, test score and age etc.

Question Two

1. What do you understand by measures of Central tendency? Briefly discuss what you understand by Kurtosis and Skewedness.
2. Mention the three major measures of central tendency,
3. State where each of them is most appropriate
1. State four (4) applications of measures of central tendency
2. Group the data below, and estimate:
3. The mean
4. Mode and
• Median

2        5        4        7        6        9        10      7        20      30      12      22

17      16      25      18      15      10      21      28      3        4        8        6

7        14      10      11      29      19

Answers to Question Two

• Measure of central tendency is a statistical technique that tries to locate the center of a given distribution. It target is to local the focus or center of that distribution such that the estimate(s) from the distribution will be used to describe the characteristic of that distribution. There are three (3) basic measure of central tendency. They include; Mode, Mean and Median.

Measure of skewedness is a characteristic of a set of data that describe the shape of such a data. It is the measure of lack of symmetry or the extent of departure from symmetry of a given distribution. It can either be positive (when the mode is less than the median which in turn is less than the mean) or negative (when the mean is less than the median which in turn is less than the mode). Below are a negatively skewed and a positively skewed distribution.

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