1. Explanatory variable: Time of day (measured in hours from 7 AM). Response variable: Average waiting time (in minutes).
2. Scatterplot: Plot points with x-axis as time (7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17) and y-axis as average waiting time (1, 3, 5, 8, 12, 18, 25, 30, 28, 20, 10). Connect points with line segments.
3. Yes, there is a clear outlier at 5 PM (x=17, y=10). It deviates significantly from the increasing trend before it and the decreasing trend after it.
4. The scatterplot shows a nonlinear relationship. Waiting time increases from 7 AM to 2 PM, peaks around 2 PM, then decreases through 5 PM. The relationship is not linear; it has a curved shape resembling an inverted U or a parabola.
5. The range is 1 to 30 minutes.
6. At 1:30 PM (x=13.5), interpolate between (13,25) and (14,30). Estimate ≈ 27.5 minutes.
7. The point (13,25) means that at 1 PM, the average waiting time was 25 minutes.
8. The direction is positive from 7 AM to 2 PM (waiting time increases with time), then negative from 2 PM to 5 PM (waiting time decreases with time). The form is curved (nonlinear). The strength is strong for the overall pattern, but the outlier at 5 PM weakens the fit slightly.
9. r² = 0.92. Interpretation: 92% of the variation in average waiting time can be explained by the time of day using the linear regression model.
10. The predicted average waiting time at 7 AM (x=7) is 1 minute. At 5 PM (x=17), it is 10 minutes.
11. Difference: 10 - 1 = 9 minutes.
12. If actual waiting time is 15 minutes at 1 PM (x=13), predicted is 25 minutes. Residual = 15 - 25 = -10 minutes. This means the actual waiting time was 10 minutes less than predicted.
13. For 15-minute wait: Solve 15 = 2.5*(time) - 10 → time = 10 hours (10 AM). For 20-minute wait: Solve 20 = 2.5*(time) - 10 → time = 12 hours (12 PM).
14. The two problems are different because one asks for the time given a waiting time (inverse problem), while the other asks for the waiting time given a time (direct prediction). The first requires solving the equation for x, while the second substitutes x into the equation.
15. The last two problems answer different questions: one finds input for a given output, the other finds output for a given input. They use the same model but solve for different variables.
Parent Tip: Review the logic above to help your child master the concept of linear regression worksheet.
