Math Problem Statement

Karen is studying the relationship between the time spent exercising per day and the time spent outside per day and has collected the data shown in the table. The line of best fit for the data is ^y=0.16x+45.5. Assume the line of best fit is significant and there is a strong linear relationship between the variables. Exercising (Minutes)  Time Spent Outside (Minutes) 705790601106213067

According to the line of best fit, what would be the predicted number of minutes spent outside for someone who spent 70 minutes exercising? Round your answer to two decimal places, as needed.

Provide your answer below:

Solution

The equation of the line of best fit is given by:

y^=0.16x+45.5\hat{y} = 0.16x + 45.5

Where:

  • y^\hat{y} represents the predicted time spent outside (in minutes).
  • xx represents the time spent exercising (in minutes).

We are asked to predict the time spent outside for someone who spends 70 minutes exercising. Therefore, we substitute x=70x = 70 into the equation:

y^=0.16(70)+45.5\hat{y} = 0.16(70) + 45.5

Now, calculate the result:

y^=11.2+45.5=56.7\hat{y} = 11.2 + 45.5 = 56.7

Thus, the predicted time spent outside is 56.7 minutes.

Would you like further explanation on how to use the equation of best fit or any other details?

Here are some related questions:

  1. How can you find the equation of the line of best fit if the data is provided in a table?
  2. How do you interpret the slope and y-intercept of the line of best fit in real-world terms?
  3. What is the significance of rounding the predicted value in regression analysis?
  4. How can you calculate residuals for a linear regression model?
  5. What is the difference between interpolation and extrapolation in regression?

Tip: Always ensure that the units of your input and output variables align, especially when dealing with real-world applications!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Best Fit Line
Prediction using Linear Equations

Formulas

y = mx + b (Equation of a line)

Theorems

Linear Relationship and Best Fit Line

Suitable Grade Level

Grades 9-11