The Various Hypothesis of Regression - Assignment Example

The Null and the Alternative Hypothesis of Regression There is growing evidence to sustain advocates’ contention that “twenty-first century information and communications tools as well as more traditional computer-assisted instructional applications can positively influence student learning processes and outcomes” (Cradler, McNabb, Freeman and Burchett 46). This paper on regression analysis will utilize the variables involved in the aforementioned relation. For the first regression dataset, the relationship between academic performance in high school Algebra (mathgrade) and amount of exposure to technology in school (techschool). The following data set shows the academic performance in Algebra of 35 middle high school students represented by their grade in the subject as the dependent variable and amount of exposure to technology in school in terms of numbers of hours of exposure per week as the independent variable. Table 1. Dataset 1: academic performance in Algebra vs. amount of exposure to technology in school mathgrade techschool mathgrade techschool mathgrade techschool 80 4 82 4 90 10 88 7 78 2 68 2 87 6 93 10 71 2 82 5 70 2 76 4 84 4 80 4 71 4 77 2 83 5 79 5 91 7 70 2 75 5 75 2 90 7 85 5 91 10 83 5 80 4 77 2 84 4 71 2 69 2 81 4 89 10 71 4 85 5 The null hypothesis and the alternative hypothesis may be stated as: Null Hypothesis: There is no relationship between the amount of exposure to technology in school and academic performance in high school Algebra. Alternative Hypothesis: There is a relationship between the amount of exposure to technology in school and academic performance in high school Algebra. Following is a scatter plot of the two variables: Figure 1. Scatter plot: academic performance in high school Algebra vs. amount of exposure to technology in school The scatter plot generated using PASW 17 was not well-defined as to the correlation of the two variables. A test of linear correlation using two-tailed analysis and a level of significance of 0.01 also with the same software, however, revealed a correlation coefficient of 0.844 between the two variables. A correlation coefficient of 0.844 implies a very strong relationship between academic performance in Algebra and the amount of exposure to technology in school. This generalization was based on the interpretation table of correlation coefficients (De Vaus 259) shown in Appendix 1. Table 2 presents the PASW 17 output on the test of correlation. Table 2. Output of the test of correlation (two-way analysis; α=0.01) Grade in high school mathematics (Algebra) Number of hours per week of exposure to technology in school Grade in high school mathematics (Algebra) Pearson Correlation 1 .844** Sig. (2-tailed) .000 N 35 35 Number of hours per week of exposure to technology in school Pearson Correlation .844** 1 Sig. (2-tailed) .000 N 35 35 **. Correlation is significant at the 0.01 level (2-tailed). Table 2.1 to 2.3 feature the regression output generated from PASW 17, which includes the model summary, the ANOVA table and the Table 2.1. Model summary for regression analysis Model R R Square Adjusted R Square Std. Error of the Estimate 1 .844a .712 .703 3.95340 a. Predictors: (Constant), Number of hours per week of exposure to technology in school The model summary reveals information regarding the over-all model fit of the data set. In simple linear regression, R is actually the correlation between the the two variables mathgrade and techschool. It may be observed that the value of R (0.844) is the same as the correlation coefficient displayed in Table 1. Meanwhile, R Square represents the proportion of the variance in the dependent variable (academic performance in Algebra) which can be explained by the independent variable (amount of exposure to technology in school). Adjusted R square is an adjustment of the R square and generally considered to be a more accurate goodness-of-fit measure than R square, but is always smaller than R. The standard error of the estimate which is equal to 3.95340 is the standard deviation of the error term and the square root of the mean square for the residuals in the ANOVA table. It is a measure of the accuracy of the prediction from the regression analysis. Table 2.2. ANOVA table for regression analysis ANOVAb Model Sum of Squares df Mean Square F Sig. 1 Regression 1275.201 1 1275.201 81.590 .000a Residual 515.770 33 15.629 Total 1790.971 34 a. Predictors: (Constant), Number of hours per week of exposure to technology in school b. Dependent Variable: Grade in high school mathematics (Algebra) From Table 2.2. the sum of squares are associated with the sources of variance: (1) Regression quantifies the variance explained by the independent variable; and (2) Residual is a measure of the variance which is not explained by the independent variable. df stands for the degrees of freedom associate with the sources of variance. It may be observed that the F-statistic is very large at 81.590. The Sig or p-value is compared to the level of significance (α =0.05). The Sig value of .000 means that the p-value is less than .001, ( Read More