Based on the provided image, we can extract the following information:

1. The Problem Setup: The table shows the relationship between time spent studying ($x$, in minutes) and exam scores ($y$, out of 100). A linear regression equation $y = mx + b$ is needed to predict scores and study times.

2. Equation: The regression equation provided in the image is: $y = 0.086x + 71$

Solving the Questions:

1. Predicting the Score:

For a student studying 1 hour 30 minutes ($90$ minutes), substitute $x = 90$ into the regression equation: $y = 0.086(90) + 71$ $y = 7.74 + 71 = 78.74 \approx 79$ The predicted score is 79 points.

2. Time Required to Score At Least 80:

Set $y = 80$ and solve for $x$: $80 = 0.086x + 71$ $80 - 71 = 0.086x$ $9 = 0.086x$ $x = \frac{9}{0.086} \approx 104.65$ Rounded up, the time is approximately 105 minutes.

Let me know if you'd like a breakdown or further explanation for these calculations!

Further Learning:

Here are some related questions to deepen your understanding:

1. What does the slope ($m = 0.086$) represent in the context of this regression?
2. How would the equation change if the intercept ($b = 71$) were adjusted to account for a different baseline score?
3. How does increasing study time impact scores based on the current slope?
4. Could a quadratic model fit this data better? Why or why not?
5. What are potential limitations of using this linear regression model for predicting scores?

Tip: Always verify your regression equation by calculating the residuals (differences between actual and predicted $y$-values) to check the model's accuracy.