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 80, 2 comma 70, 2 comma 80, 2 comma 90, 3 comma 80, 3 comma 100, 4 comma 90, and 4 comma 98, and a line of fit drawn passing through the points 0 comma 70 and 1 comma 75

Find the slope of the line of fit and explain its meaning in the context of the data.

80; a student who studies for 0 hours is predicted to earn 80% on the test
70; a student who studies for 0 hours is predicted to earn 70% on the test
10; for each additional hour a student studies, their grade is predicted to increase by 10% on the test
5; for each additional hour a student studies, their grade is predicted to increase by 5% on the test

Answer 1

To find the slope of the line of fit, we use the two points through which the line passes: (0, 70) and (1, 75).

The formula for the slope \( m \) of a line given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting in the values from the two points:

\[ m = \frac{75 - 70}{1 - 0} = \frac{5}{1} = 5 \]

Thus, the slope of the line of fit is \( 5 \).

Meaning in context: This slope indicates that for each additional hour a student studies, their predicted grade on the test is expected to increase by 5%. This suggests a positive relationship between study time and test scores within the context of this data collection.