Final answer:
In statistics, the coefficient of determination measures the strength of a linear relationship between variables. The independent variable is the manipulated or controlled variable, while the dependent variable is the measured or observed variable. Regression analysis can be used to find the line of best fit and the correlation coefficient to interpret the relationship.
Step-by-step explanation:
a. What are the independent and dependent variables?
The independent variable is the variable that is manipulated or controlled by the researcher. In this case, it could be the third exam score (x). The dependent variable is the variable being measured or observed. In this case, it could be the final exam score (y).
b. Draw a scatter plot.
A scatter plot is a graph that shows the relationship between two variables. The independent variable (third exam score) is plotted on the x-axis, and the dependent variable (final exam score) is plotted on the y-axis. Each data point represents a pair of values for the two variables. The scatter plot can help visualize if there is a linear relationship between the variables.
c. Use regression to find the line of best fit and the correlation coefficient.
In regression analysis, the line of best fit represents the relationship between the independent and dependent variables. This line is determined by minimizing the sum of the squared differences between the observed data points and the predicted values on the line. The correlation coefficient measures the strength and direction of the linear relationship between the variables.
d. Interpret the significance of the correlation coefficient.
The correlation coefficient, often denoted as 'r', ranges from -1 to +1. A value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. The significance of the correlation coefficient is determined through hypothesis testing. If the p-value (a measure of significance) is less than a predetermined significance level, typically 0.05, the correlation coefficient is considered significant and indicates a linear relationship between the variables.
e. Is there a linear relationship between the variables?
To determine if there is a linear relationship between the variables, we look at the scatter plot and the correlation coefficient. If the scatter plot shows a clear and consistent pattern, and the correlation coefficient is significantly different from zero, then we can conclude that there is a linear relationship between the variables.