When is it appropriate to model data with a linear function? Give an example of real-world data that can be modeled with a linear function. Include the linear function and a sample of the data.
It is appropriate to model data with a linear function when there is a linear relationship between the independent variable (x) and the dependent variable (y). A linear relationship means that the data points roughly form a straight line on a scatter plot.
One example of real-world data that can be modeled with a linear function is the relationship between the number of hours studied and the corresponding test score achieved by students. Let's say a teacher collects data on the number of hours studied by a group of students and their respective test scores. The data is as follows:
Hours Studied (x): 1, 2, 3, 4, 5
Test Score (y): 65, 70, 75, 80, 85
To model this data with a linear function, let's assume the linear function is y = mx + b, where y represents the test score and x represents the hours studied. The goal is to find the values of m and b that best fit the data.
Using the first and last data points, we can calculate the slope (m) of the line:
m = (85 - 65) / (5 - 1) = 20 / 4 = 5
Now we can substitute the slope (m) and any data point (x, y) into the linear equation to solve for the y-intercept (b). Let's use the first data point:
65 = 5(1) + b
65 = 5 + b
b = 65 - 5 = 60
Therefore, the linear function that models this data is y = 5x + 60.
Now, we can use this linear function to make predictions for test scores based on the number of hours studied or analyze the approximate increase in test scores for each additional hour studied.
shorten answer
A linear function is appropriate for modeling data when there is a linear relationship between the independent and dependent variables. For example, the relationship between the number of hours studied and test scores can be modeled using a linear function. If a group of students' data shows the following:
Hours Studied: 1, 2, 3, 4, 5
Test Score: 65, 70, 75, 80, 85
The linear function that models this data is y = 5x + 60, where y represents test score and x represents hours studied. With this function, we can predict test scores based on hours studied or analyze the approximate increase in test scores for each additional hour studied.