5. Two different tests were designed to measure understanding of a topic. The two tests were given to ten students with the following results:
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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: $\overline{x}=78$, $\overline{y}=77.8$

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: $a=\overline{y}-b\overline{x}=77.8-1.3039(78)=15.7884$

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