Based on the image, here are the key points of the statistical analysis problem:

Observed Data

• Variables: Number of lovers (independent variable) and GPA (dependent variable).
• Correlation coefficient (r): -0.80 (rounded to 2 decimal places).
• Null hypothesis $H_0$: $\rho = 0$ (no correlation).
• Alternative hypothesis $H_1$: $\rho \neq 0$ (correlation exists).
• p-value: 0.0296 (rounded to 4 decimal places).

Questions and Interpretation

1. Hypothesis Test Conclusion (α = 0.05):

   • The p-value (0.0296) is less than α = 0.05. This means we reject the null hypothesis.
   • Correct answer: "There is statistically significant evidence to conclude that there is a correlation between the number of lovers students have had in their lifetimes and their GPA. Thus, the regression line is useful."

2. Coefficient of Determination $r^2$:

   • $r^2 = (-0.80)^2 = 0.64$.
   • This indicates that 64% of the variation in GPA can be explained by the number of lovers.

Would you like further clarification on any part of the problem? Below are related questions that might expand your understanding:

1. How is the correlation coefficient $r$ calculated from raw data?
2. What does the p-value represent in hypothesis testing?
3. How is the coefficient of determination ($r^2$) interpreted in context?
4. What would happen if the p-value was higher than the significance level?
5. Why is it important to consider context when interpreting correlation results?

Tip: Always examine the scatterplot of the data when interpreting correlation to ensure no unusual patterns (like nonlinearity) that might mislead conclusions.