Fundamental Concepts In Inferential Statistics1

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

BRIEF CONTENTS

1. Fundamental concepts in inferential statistics 1

2. Descriptive statistics 52

3. Comparison of two means with z-test and t-test 117

4. Comparison of means with analysis of variance (ANOVA) 164

5. Comparison of means with two-way analysis of variance 199

6. Hypothesis testing with analysis of covariance 229

7. Correlation 247

8. Simple regression 284

9. Information and communication technology in data analyses:

SPSS in action 300

10. Correlation and regression analyses, using SPSS 354

11. Execution of ANOVA, two-way ANOVA and ANCOVA

with SPSS 398

12. Nonparametric statistics with SPSS 433

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

DETAILED CONTENTS

CHAPTER 1: FUNDAMENTAL CONCEPTS IN INFERENTIAL STATISTICS

Inferential Statistics 1

Population 4

Sample 5

Qualities of a good sample 6

Merits of sampling 7

Nature of population elements: 7

Cost reduction: 8

Accuracy of results: 8

Data collection speed: 8

Sampling Techniques 8

Probability sampling techniques 9

Simple Random Sampling Technique 9

Example of Simple Random Sampling 10

Stratified Random Sampling Technique 10

Advantages of stratified random sampling 11

Systematic Sampling Technique 12

Cluster Random Sampling Technique 13

Nonprobability Sampling Procedures 14

Convenience Sampling Technique 15

Purposive Judgment Sampling 15

Purposive Quota Sampling 16

Sample Size Suitability 16

Hypothesis 21

Uses of Hypothesis in Research 22

Characteristics of Hypothesis 26

Types of Hypothesis 30

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

The Research Hypothesis 30

The Null Hypothesis 31

Steps for Testing Hypothesis 33

Level of Significance 34

Decision Errors 37

Type I Error 39

Type II Error 40

Balancing of Type I error and Type II error 41

Central Limit Theorem 42

The Normal Curve 45

One-Tailed and Two-Tailed Tests 47

One-Tailed Test of Significance 48

Two-Tailed Test of Significance 49

CHAPTER 2: DESCRIPTIVE STATISTICS

Scales of Measurement 53

Nominal Scale 53

Ordinal Scale 54

Interval Scale 55

Ratio Scale 56

Organization of Research Data 57

Grouped frequency distribution 61

Organization of Nominal Scale Data 63

Graphic Presentation of Data 70

Histogram 71

Frequency polygon 72

Cumulative frequency curve 74

Characteristics of Distribution 75

Normal Curve 75

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

Positively Skewed Curve 75

Negatively Skewed Curve 76

Leptokurtic Curve 77

Platykurtic Curve 77

Bimodal Curve 78

Pie Chart 79

Bar Chart 80

Measures of Central Tendency 81

The Mean 82

Midpoint Method of Finding Mean 83

Assumed Mean Method of Finding Mean 84

The Mode 86

The Median 86

Measures of Variability 89

Standard Deviation 91

Raw Scores Method 91

Deviation from the Mean Method 95

Grouped Frequency Distribution Method 98

Variance 99

Range 102

Semi-Interquartile Range 103

Measures of Relative Position 106

The Z-score 107

T-Score 108

Stanine Scores 111

Percentile Ranks 113

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

CHAPTER 3: COMPARISON OF TWO MEANS WITH Z-TEST AND T-TEST

Distribution of means 117

The Z-Test for Difference Between One Sample Mean And 117

Population Mean When the Population Standard Deviation Is Known 121

The T-Test for Difference between One Sample Mean and

Population Mean When the Population Standard Deviation is Unknown 125

The T-Test for Difference Between Means from Two Independent

Samples 131

Corrected Sum of Squares Method for Computation of Two

Independent Samples T-Test 138

Unbiased Variance Method for Computation of Two Independent

Samples T-Test 140

Summary of Results for Research Articles 143

The Z-Test for Difference Between Two Independent Samples 144

The T-Test for Dependent Samples 150

Difference Scores Standard Deviation Method 152

Direct Difference Scores Method 156

Standard Error of the Difference Between Two Dependent Means Method 158

CHAPTER 4: COMPARISON OF MEANS WITH ANALYSIS OF VARIANCE

(ANOVA)

Assumptions that Underlie ANOVA 165

Normal Distribution of the Population 165

Independence of Observations 165

Homoscedasticity 166

Example of ANOVA for Hypothesis Testing 166

Raw-Scores Method of ANOVA Computation 168

The F-Ratio 168

Mean square 168

Degrees of freedom 169

Sources of variation 170

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

Summary of ANOVA Equations 172

Sources of variation 173

Degrees of freedom 174

Mean squares 175

The F-ratio 175

Summary of ANOVA Results 175

Deviation from Grand Mean Method of Computing ANOVA 177

The F-ratio 177

Mean squares 177

Degrees of freedom 178

Sources of variation 178

Degrees of freedom 181

Mean squares 181

The F-ratio 182

Multiple Comparisons 183

Post Hoc or a Posteriori Tests 183

Fisher’s Protected T-Tests (ť) 184

Fisher’s LSD 185

Application of Fisher’s LSD: 189

Scheffé’s Test (Fs) 190

Tukey’s HSD (q) 196

CHAPTER 5: COMPARISON OF MEANS WITH TWO-WAY ANALYSIS OF

VARIANCE

Advantages of Two-Way ANOVA 199

Greater generalizability of results 200

Determination of Interaction Effect 200

Two-way ANOVA is more economical 201

Example of Two-Way ANOVA for Hypothesis Testing 201

Raw-Scores Method of Computing Two-Way ANOVA 204

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

The F ratios in two-way ANOVA 204

Mean squares in two-way ANOVA 206

Degrees of freedom in two-way ANOVA 208

Sources of variation in two-way ANOVA 208

Summary of two-way ANOVA equations 212

Calculation of Sources of Variation in Two-Way ANOVA with

Raw-Scores Method 213

Calculation of degrees of freedom in Two-Way ANOVA 216

Calculation of Two-way ANOVA Mean Squares 216

Calculation of F ratios in two-way ANOVA 217

Deviation from Grand Mean Method of Calculating Two-Way ANOVA 220

The F ratios 221

Sources of variation 221

The mean squares 222

Degrees of freedom 223

CHAPTER 6: HYPOTHESIS TESTING WITH ANALYSIS OF COVARIANCE

Uses of Analysis of Covariance 229

Determination of change 230

Elimination of nuisance variables effects 230

ANCOVA reduces the error term 230

ANCOVA adjusts the group means 231

ANCOVA uses randomization 231

Assumptions that Underlie ANCOVA 232

Valid and reliable measurement of the covariate 232

Linearity of the dependent variable and covariate relationship

Homogeneity of regression 233

Example of Analysis of Covariance 233

The F ratio of ANCOVA 236

Adjusted mean squares 236

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

Adjusted degrees of freedom 236

Sources of Variation 237

Sum of products of X and Y 238

Summary of ANCOVA equations 240

Calculation of the ANCOVA 241

CHAPTER 7: CORRELATION

Facts about Correlation 248

Tests of Relationships and Tests of Differences 250

Correlation Statistical Techniques 251

Pearson Product Moment Correlation Coefficient 252

Standard Scores Method 252

Example of Correlation, Using Standard Scores Method 253

Testing Significance of Correlation Coefficient with t Distribution 256

Sum of Squared Deviations Method of Calculating Correlation 258

Raw Scores Method of Calculating Correlation 260

Population Standard Deviation Method 264

Scatterplot 267

Partial Correlation 271

Partial Correlation as Coefficient of Correlation between Residuals of

Variables X and Y on a Third Variable 271

Conventional Technique for Computation of Partial Correlation 275

Multiple Correlation Coefficient 276

Multiple Correlation and Hypothesis Testing 279

Importance of Multiple Correlation 281

CHAPTER 8: SIMPLE REGRESSION

Regression and Correlation Compared 284

Raw Scores Regression Coefficient Method 285

Example of Regression 286

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

Regression Slope and Regression Intercept 289

Standardized Regression Coefficient 291

Regression Analysis in Hypothesis Testing 295

CHAPTER 9: INFORMATION AND COMMUNICATION TECHNOLOGY IN DATA

ANALYSES: SPSS IN ACTION

Execution of Descriptive Statistics with SPSS 307

T-Test Analyses with SPSS 328

Execution of Independent Samples T-Test, Using SPSS 328

Application of SPSS in Correlated-Samples T Test 341

Single Sample T Test with SPSS 349

CHAPTER 10: CORRELATION AND REGRESSION ANALYSES, USING SPSS

Execution of Pearson Product Moment Correlation, Using SPSS 355

Partial Correlation with the Use of SPSS 364

Multiple Correlation Execution with SPSS 373

Execution of Simple Regression with SPSS 378

Multiple Regression Execution with SPSS 386

Enter Method of Executing Multiple Regression with SPSS 387

Stepwise Method of Executing Multiple Regression with SPSS 392

CHAPTER 11: EXECUTION OF ANOVA, TWO-WAY ANOVA AND ANCOVA

WITH SPSS

Application of SPSS in Independent Samples ANOVA 399

Application of SPSS in Repeated Measures ANOVA 406

Two-Way ANOVA Execution with SPSS 415

Execution of ANCOVA with SPSS 424

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

CHAPTER 12: NONPARAMETRIC STATISTICS WITH SPSS

Wilcoxon Test for Two Related Samples 434

Sign Test for Two Related Samples 437

Mann-Whitney U-Test 439

Kruskal-Wallis K Independent Samples Test 443

Friedman K Related Samples Test 447

Spearman Correlation Coefficient with SPSS 450

Chi-Square, Using SPSS 458

___Published by European Centre for Research Training and Development UK (www.eajournals.org)