Scatter Plot, Linear Regression, and R-Value
The following chart is a scatter plot illustrating the relationship between the heights of eight 10-year old boys in inches and weight in pounds. The scatter plot shows the changes in weight as height changes. According to the chart, there is a random distribution of the heights and weights of the selected sample. The records do not reflect any uniform pattern that shows a direct relationship between the variables. The trend indicates that height is not a good predictor of weight for any selected sample.
Figure 1: Scatter plot for the height (inches) and weight (pounds) of 8 random 10-year old boys
According to the scatter plot in Figure 1 above, the regression line for the model is y = -0.592x + 103.19. The regression line gives a negative slope of -0.592 and an intercept of 103.19. The negative slope suggests a negative relationship between height and weight variables for the selected sample. It indicates that an increase in height results in a general decrease in the weight of the sample. The negative relationship between the variables is also an indication that tall people have low weights compared to the shorter people. This implies that height could be an appropriate predictor of weight. The findings are consistent with the general belief that shorter people are heavier than taller people.
The scatter plot shows an R-squared value of 0.2129. As a measure of goodness-of-fit, the value indicates a 21.29% variance in the independent variables as explained by the dependent variable. It suggests that the regression model does not properly fit the data. This could be linked to the randomness of the data such that there is no specific pattern in explaining the relationship between the variables. Most of the data points fall outside the regression line, hence a high variation. The R-value is obtained from the square-root of R-squared value. According to the analysis, the R-value of the regression model is -0.4614. The negative value is linked to the negative slope exhibited in the regression line and equation.
The R-value is a correlation coefficient that measures the strength and direction of the relationship between two variables. In the regression model between heights (inches) and weight (pounds), the negative R-value of -0.4614 indicates a negative linear relationship between the variables. It shows that weight will decrease as height increases. Since the R-value is less than -0.5, we conclude that a weak linear relationship between the variables. The weak relationship implies that the height variable cannot be a good predictor of weight from a sample population. Therefore, there is a weak negative correlation between height and weight variables.
Scenario for Application
The concept of correlation and regression is applicable in predicting a dependent variable using an independent variable. In a school setting, for instance, one would use the regression model to test the relationship between student scores and study hours per semester. As a student, I would use the concept to predict my scores based on the average study hours. Since the regression model shows the type of relationship and degree of association between two or more variables, I could apply it in establishing the linkage between performance and study hours. It could also be applied in predicting the weight of a group of students based on their heights.