Data was collected on the amount of time that a random sample of 8 students spent studying for a test and the grades they earned on the test. A scatter plot and line of fit were created for the data.

Question

scatter plot titled students' data, with points plotted at 1 comma 60, 2 comma 60, 2 comma 70, 2 comma 80, 3 comma 80, 3 comma 90, 4 comma 95, and 4 comma 100, and a line of fit drawn passing through the points 0 comma 40 and 2 comma 70

Find the y-intercept of the line of fit and explain its meaning in the context of the data.

30; for each additional hour a student studies, their grade is predicted to increase by 30% on the test
15; for each additional hour a student studies, their grade is predicted to increase by 15% on the test
70; a student who studies for 0 hours is predicted to earn 70% on the test
40; a student who studies for 0 hours is predicted to earn 40% on the test

Answer 1

To find the y-intercept of the line of fit that passes through the points \(0, 40\) and \(2, 70\), we can use the general form of the equation for a line, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. Calculate the slope (m): The slope can be calculated using the two points \( (0, 40) \) and \( (2, 70) \):

   \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{70 - 40}{2 - 0} = \frac{30}{2} = 15 \]

   So, the slope \(m\) is \(15\).

2. Find the y-intercept (b): We already have one of the points, \( (0, 40) \), which directly gives us the y-intercept:

   \[ b = 40 \]

This means that when \(x = 0\) (i.e., when a student has studied for 0 hours), the predicted value of \(y\) (the grade) is 40%.

Meaning in the context of the data: The y-intercept of 40 implies that a student who studies for 0 hours is predicted to earn 40% on the test.

Therefore, the correct answer is:

40; a student who studies for 0 hours is predicted to earn 40% on the test.