Final answer:
The question focuses on calculating the regression line and correlation coefficient between GPA and studying hours. It also involves testing for the existence of a relationship between the two variables and predicting GPA for a given value of study hours.
Step-by-step explanation:
To address the student's question, we need to run a linear regression analysis for which the dependent variable is the GPA and the independent variable is the Hours of study. The regression line represents the relationship between GPA (dependent variable) and Hours spent studying (independent variable).
Here is a step-by-step explanation:
- Identify the dependent variable (½GPA½) and independent variable (½Hours½).
- Draw a scatter plot to visually inspect the relationship between GPA and Hours.
- Use statistical software or a calculator capable of regression analysis to calculate the least-squares regression line, which will have the form ½ŷ = a + bx½, where ½ŷ½ is the predicted GPA, ½a½ is the y-intercept, and ½b½ is the slope of the line.
- Calculate the correlation coefficient (½r½) to measure the strength and direction of the linear relationship between the two variables. If |r| is close to 1, there is a strong linear relationship.
- To test the claim of a relationship, perform a hypothesis test for regression analysis, providing the test statistic (such as the t-statistic) and the P-value. If the P-value is less than the chosen significance level (commonly 0.05), reject the null hypothesis and conclude that there is a significant relationship.
- Predict the GPA for a student who averages zero hours per week studying by using the y-intercept (½a½) of the regression line, which represents the predicted GPA when the independent variable is zero.
Remember, the significance of the correlation coefficient and whether the regression line is a good fit for the data can only be determined after performing these calculations.