Statistical Analysis of Educational Factors Affecting Student Performance: A Case Study Approach How do various educational factors influence student performance, and what statistical methods can be used to analyze these factors? (Word count: 800 - 1000 words) Case Study 1: An educational statistics professor is examining his grade book to determine the relationship and differences among a variety of assignment scores. He is examining these scores to determine if his course is effective. Every semester, the professor gives his students a pretest and a posttest to determine if they improved in their understanding of the course concepts. Thus, the first question he examines is as follows: Is there a difference in the number of points students earned on their educational statistics pretests and posttests? Here the null hypothesis is 𝐻 ௢ : There is no difference in the number of points students earned on their educational statistics pre - tests and post - tests. vs the alternative hypothesis 𝐻 ଵ : There is difference in the number of points students earned on their educational statistics pre - tests and post - tests. The main assumptions are as follows: 1) The number of points earned by the students in the pre - test ( 𝑋 ଵ , 𝑋 ଶ , … , 𝑋 ௡ ) and that in the post - test i.i.d ( 𝑌 ଵ , 𝑌 ଶ , … , 𝑌 ௡ ), where n is the sample size, when paired {( 𝑋 ଵ , 𝑌 ଵ ) , ( 𝑋 ଶ , 𝑌 ଶ ) , … , ( 𝑋 ௡ , 𝑌 ௡ ) } follow Bivariate Normal Distribution with mean ( 𝜇 ଵ , 𝜇 ଶ ) and variance covariance matrix Σ . 2) The sample must consist of 2 related groups, here they are the scores of the pre - test prior to the course and a post - test after the course. 3) There are no significant outliers in the differences between the two related groups. Thus in this case 𝐻 ௢ : 𝜇 ଵ = 𝜇 ଶ vs 𝐻 ଵ : 𝜇 ଵ ≠ 𝜇 ଶ The test used is a paired sample t - test. Let 𝑑 ௜ = 𝑋 ௜ − 𝑌 ௜ Then 𝑑 ̅ = 𝑋 ത − 𝑌 ത ~ N (0, 𝜎 ଶ ) where 𝜎 ଶ can be derived from Σ . Let S= ∑ ( 𝑑 ೔ − 𝑑 ത ) మ ೙ ೔ = భ ௡ − ଵ Thus under null , 𝑑 ത ೄ √ ೙ follows t distribution with n - 1 degrees of freedom. We reject null at 0.05 level of significance if the value of | 𝑑 ത ೄ √ ೙ |> 𝑡 ( ௡ − ଵ ) , 0 . 0ఱ మ In this case, The following output was received from SPSS: T - TEST PAIRS=postest WITH pretest (PAIRED) /CRITERIA=CI(.9500) /MISSING=ANALYSIS. T - Test Notes Output Created 07 - Mar - 2013 02:18:56 Comments Input Data C: \ Users \ Abhishek \ Downloa ds \ zFinal (3).sav Active Dataset DataSet1 Filter <none> Weight <none> Split File <none> N of Rows in Working Data File 105 Missing Value Handling Definition of Missing User defined missing values are treated as missing.