There are many measurements of the human body that are positively correlated. For example, the length of one's forearm (measured from elbow to wrist) is approximately the same length as the foot (measured from heel to toe). They are positively correlated because, as one measurement increases, so does the other measurement.
You will discover through this project whether a human's arm span (measured across the body with the arms extended) is correlated to his height.
You will need to collect data from 11 people, which will give you 12 data points including your own personal data. You will turn in and answer questions regarding only one scatter plot if doing the project alone.
Part One: Measurements
Measure your own height and arm span (from finger-tip to finger-tip) in inches. You will likely need some help from a parent, guardian, or sibling to get accurate measurements. Record your measurements on the "Data Record" document. Use the "Data Record" to help you complete Part Two of this project.
Measure 11 additional people, and record their arm spans and heights in inches.
Part Two: Representation of Data with Plots
Using graphing software of your choice, create a scatter plot of your data. Predict the line of best fit, and sketch it on your graph.
Copy and paste your scatter plot into a word processing document.
Part Three: The Line of Best Fit
Include your scatter plot and the answers to the following questions in your word processing document.
1.Which variable did you plot on the x-axis, and which variable did you plot on the y-axis? Explain why you assigned the variables in that way.
2.Write the equation of the line of best fit using the slope-intercept formula y=mx+b
. Show all your work, including the points used to determine the slope and how the equation was determined.
3.What does the slope of the line represent within the context of your graph? What does the y-intercept represent?
4.Test the residuals of two other points to determine how well the line of best fit models the data.
5.Use the line of best fit to help you to describe the data correlation.
6.Using the line of best fit that you found in Part Three, Question 2, approximate how tall is a person whose arm span is 66 inches?
7.According to your line of best fit, what is the arm span of a 74-inch-tall person?
To complete the project, you will need to follow these steps:
Part One: Measurements
1. Measure your own height and arm span in inches.
2. Record your measurements on the "Data Record" document.
3. Measure 11 additional people and record their arm spans and heights in inches.
Part Two: Representation of Data with Plots
1. Use graphing software of your choice to create a scatter plot of your data.
2. Plot the arm span on the x-axis and the height on the y-axis.
- The x-axis usually represents the independent variable, which is the variable that is expected to cause changes in the other variable. In this case, arm span is expected to affect height.
- The y-axis usually represents the dependent variable, which is the variable that is expected to change in response to the independent variable. In this case, height is dependent on arm span.
Part Three: The Line of Best Fit
1. On your scatter plot, predict the line of best fit and sketch it.
2. Write the equation of the line of best fit using the slope-intercept formula.
- The slope-intercept formula is y = mx + b, where m represents the slope and b represents the y-intercept.
- Determine the slope by selecting two points on the line of best fit and calculating the change in y divided by the change in x between those points.
- Use one of the points on the line of best fit and the slope to calculate the y-intercept using the formula b = y - mx.
Answer the following questions and include them in your word processing document:
1. Explain why you assigned the variables in that way (x-axis and y-axis).
- The variables were assigned in that way because arm span is expected to be the independent variable affecting the dependent variable, which is height.
2. Write the equation of the line of best fit using the slope-intercept formula y = mx + b. Show all your work.
- Use the slope and y-intercept values you obtained from Part Three, Question 2, to plug into the equation y = mx + b and write the complete equation.
3. Explain what the slope and y-intercept represent.
- The slope represents the rate of change in height for each unit change in arm span. A positive slope indicates that as arm span increases, height also increases.
- The y-intercept represents the predicted value of height when arm span is zero. It signifies the height of a person with no arm span.
4. Test the residuals of two other points to determine how well the line of best fit models the data.
- Calculate the residual for each of the two points by subtracting the predicted height (based on the line of best fit equation) from the actual height. Compare the residuals to assess how well the line of best fit fits the data. Smaller residuals indicate a better fit.
5. Use the line of best fit to describe the data correlation.
- Based on the line of best fit, describe the relationship between arm span and height. Is it a positive or negative correlation? How strong is the correlation based on the slope?
6. Approximate the height of a person whose arm span is 66 inches using the line of best fit.
- Plug the arm span value of 66 inches into the equation of the line of best fit and solve for the corresponding height.
7. Determine the arm span of a person who is 74 inches tall using the line of best fit.
- Plug the height value of 74 inches into the equation of the line of best fit and solve for the corresponding arm span.