Question

Leonardo da Vinci (1452-1519) drew a sketch of a man, }\end{array}\( indicating that a person's armspan (measuring across the back with your arms outstretched to make a "T") is roughly equal to the person's height. To test this claim, we measured eight people with the following results: $$ \begin{array}{l|clll} \text { Person } & 1 & 2 & 3 & 4 \\ \hline \text { Armspan (inches) } & 68 & 62.25 & 65 & 69.5 \\ \text { Height (inches) } & 69 & 62 & 65 & 70 \\ \text { Person } & 5 & 6 & 7 & 8 \\ \hline \text { Armspan (inches) } & 68 & 69 & 62 & 60.25 \\ \text { Height (inches) } & 67 & 67 & 63 & 62 \end{array} $$ a. Draw a scatterplot for armspan and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables. b. If da Vinci is correct, and a person's armspan is roughly the same as the person's height, what should the slope of the regression line be? c. Calculate the regression line for predicting height based on a person's armspan. Does the value of the slope \)b$ confirm your conclusions in part b? d. If a person has an armspan of 62 inches, what would you predict the person's height to be?

Short answer

Explain your reasoning using the steps provided in the solution.

Step-by-step solution

Draw a scatterplot

To draw a scatterplot, we will plot armspan on the x-axis (horizontal) and height on the y-axis (vertical). Also, use the same scale for both axes.

Describe the relationship

After plotting the points, look for any patterns or trends in the scatterplot. It's important to describe the direction, form, and strength of the relationship.

Determine the expected slope

If da Vinci's claim is correct, then the slope of the regression line should be 1, as the armspan should be roughly equal to the height.

Calculate the regression line

To calculate the regression line, we need to find the values for the slope \(b\) and the y-intercept \(a\) using the following formulas: \(b = \frac{n\cdot\sum(xy)-\sum x\cdot\sum y}{n\cdot\sum(x^2)-(\sum x)^2}\) \(a = \bar{y} - b\bar{x}\) where \(\sum x\), \(\sum y\), and \(\sum(xy)\) are sums of the respective values, \(\bar{x}\) and \(\bar{y}\) are the averages, and \(n\) is the number of data points.

Compare the calculated slope to the expected slope

Compare the calculated value of the slope \(b\) to the expected value from Step 3. If the calculated value is close to 1, it confirms da Vinci's claim.

Make a prediction

Using the calculated regression line, predict the height for a person with an armspan of 62 inches. To do this, plug the given armspan value into the regression equation and calculate the predicted height.

Key concepts

Scatterplot

A scatterplot is a type of graph used to visually display the relationship between two numerical variables. In our exercise, we plot the armspan and height measurements of eight individuals, with armspan on the x-axis and height on the y-axis. Both axes use the same scale to reflect equality in the units of measure, facilitating easy comparison.
By plotting each person's data as a point on this graph, we aim to observe patterns and trends.
Typical trends to look for in a scatterplot include:

• Direction: Do the points seem to rise or fall? A positive direction (upward trend) suggests that as one variable increases, so does the other.
• Form: Is it linear (forms a straight line) or non-linear? A linear pattern suggests a consistent rate of increase (or decrease).
• Strength: How closely do the points follow their pattern? Strong relationships have points close to a line or curve.

For da Vinci's claim, we hope to see points aligning close to a straight line that suggests equal armspan and height, indicating a strong positive linear relationship where armspan increases as height does.

Slope Calculation

The slope of a regression line in a scatterplot represents the relationship's rate of change between two variables. It's a key element in linear regression analysis, helping us predict one variable based on another.
The slope (denoted as \( b \)) is calculated using the formula:
\[ b = \frac{n \cdot \sum(xy) - \sum x \cdot \sum y}{n \cdot \sum(x^2) - (\sum x)^2} \]

This formula requires:

• The number of data points \( n \)
• The sum of products of paired scores \( \sum(xy) \)
• The sum of each variable \( \sum x \) and \( \sum y \)
• The sum of squared values of the independent variable \( \sum(x^2) \)

A slope of 1 would perfectly confirm Da Vinci's assertion of equality in armspan and height, as it directly implies no difference per unit increase or decrease. Calculating and comparing this value allows us to explore how closely reality matches the historical claim.

Da Vinci's Claim

Leonardo da Vinci, apart from his artistic prowess, made various anatomical and scientific observations. One such claim suggests that a person's armspan is approximately equal to their height. This intriguing idea provides us an opportunity to apply regression analysis to test its validity with empirical data.

To examine da Vinci's assertion, we utilize linear regression—a statistical method for drawing a line through a scatterplot that best fits the data points. Ideally, if armspan and height are indeed identical, the scatterplot should show points lying close around a line with a slope of 1.
But what does that mean? Simply, for every inch of increase in armspan, there would be an equal inch increase in height.
Explorations like this one allow us to weigh the intuitive wisdom of historical figures like da Vinci against data-driven analysis, learning about both the human body and the power of statistics in making predictions.

Predicting Height

Once the regression line is established, it can be used to predict one variable when the other is known. In this context, we determine height when the armspan is given, using the regression equation:
\[ y = a + bx \]
where \( y \) is the predicted height, \( a \) is the y-intercept, \( b \) is the slope, and \( x \) is the given armspan.

By substituting a specific armspan into the equation, we can compute an estimated height. For example, given an armspan of 62 inches, we would substitute 62 into the equation as \( x \), solving for \( y \).
With this process, regression models offer a powerful tool for predicting future data points by leveraging observed patterns. The practical application extends beyond human height predictions, aiding in numerous scientific, economic, and engineering fields. It teaches us how statistical predictions can inform our understanding of consistent patterns in nature and human characteristics.