The table shows hours studied and test scores for five students. Plot a scatter plot, describe the correlation, draw a line of best fit, find the regression equation, predict a score, and explain regression analysis.

1.2.1 Scatter plot points: (2, 50), (6, 65), (8, 85), (10, 90). 1.2.2 The scatter plot shows a strong positive linear correlation. As hours studied increase, test scores increase nearly linearly with little scatter around an upward trend. 1.2.3 The intuitive line of best fit passes close to all points with a positive slope approximately 5. It roughly follows from near (2, 50) through (6, 65) and (10, 90), balancing distances above and below. 1.2.4 Equation of the regression line y = mx + c. Step 1: Data and sums table. |c|c|c|c| x & y & x^2 & xy \\ 2 & 50 & 4 & 100 \\ 6 & 65 & 36 & 390 \\ 8 & 85 & 64 & 680 \\ 10 & 90 & 100 & 900 \\ x = 26 & y = 290 & x^2 = 204 & xy = 2070 \\ Step 2: Slope formula m = (n xy - ( x)( y))/(n x^2) - ( x)^2 where n=4. Substitute values: m = (4 × 2070 - 26 × 290)/(4 × 204 - 26^2) = (8280 - 7540)/(816 - 676) = (740)/(140) = (37)/(7). Step 3: Intercept formula c = ( y - m x)/(n). Substitute: m x = (37)/(7) × 26 = (37 × 26)/(7) = (962)/(7), y - m x = 290 - (962)/(7) = (2030 - 962)/(7) = (1068)/(7), c = (1068/7)/(4) = (1068)/(28) = (267)/(7). Regression line: y = (37)/(7)x + (267)/(7) or approximately y = 5.29x + 38.14. y = (37)/(7)x + (267)/(7) 1.2.5 For x=7 hours studied, y = (37 × 7 + 267)/(7) = (259 + 267)/(7) = (526)/(7) ≈ 75.14. Predicted test score: 75. 75 1.2.6 Regression analysis finds the line of best fit that minimizes the sum of squared residuals (differences between observed and predicted y-values). This provides the most accurate linear model for the data, enabling reliable predictions for x-values within the observed range (e.g., 2 to 10 hours).