5. Two different tests were designed to measure understanding of a topic. The two tests were given to ten students with the following results:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline Test \( x \) & 75 & 78 & 88 & 92 & 95 & 67 & 58 & 72 & 74 & 81 \\
\hline Test \( y \) & 81 & 73 & 85 & 88 & 89 & 73 & 66 & 75 & 70 & 78 \\
\hline
\end{tabular}
a) Write an equation for the least squares regression line.
b) Interpret the slope and \( y \)-intercept.
c) Calculate the residual for the student that scored 92 on test \( x \).

Step 1 ：Calculate means: \( \bar{x} = 78 \), \( \bar{y} = 77.8 \)

Step 2 ：Calculate sums: \( \sum_{i=1}^{10} x_i = 780 \), \( \sum_{i=1}^{10} y_i = 778 \), \( \sum_{i=1}^{10} x_i y_i = 61010 \), \( \sum_{i=1}^{10} x_i^2 = 59784 \)

Step 3 ：Calculate slope: \( b = \frac{10(61010)-780(778)}{10(59784)-780^2} = 1.3039 \)

Step 4 ：Calculate y-intercept: \( a = \bar{y} - b\bar{x} = 77.8 - 1.3039(78) = 15.7884 \)

Step 5 ：Least squares regression line: y = 1.3039x + 15.7884

Step 6 ：Slope interpretation: For every 1 point increase in Test \( x \), Test \( y \) is expected to increase by 1.3039 points

Step 7 ：y-intercept interpretation: If a student scored 0 on Test \( x \), their predicted score on Test \( y \) would be 15.7884

Step 8 ：Predicted \( y \) value for student with 92 on Test \( x \): \( y = 1.3039(92) + 15.7884 = 135.2453 \)

Step 9 ：Residual: Observed \( y \) - Predicted \( y \) = 88 - 135.2453 = -3.0789