Question

Leonardo da Vinci \((1452-1519)\) drew a sketch of a man, indicating that a person's armspan (measuring across the back with 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: $$ \begin{array}{lllll} \text { Person } & 1 & 2 & 3 & 4 \\ \hline \text { Armspan (inches) } & 68 & 62.25 & 65 & 69.5 \\ \text { Height (inches) } & 69 & 62 & 65 & 70 \end{array} $$ $$ \begin{array}{lllll} \text { Person } & 5 & 6 & 7 & 8 \\ \hline \text { Armspan (inches) } & 68 & 69 & 62 & 60.25 \\ \text { Height (inches) } & 67 & 67 & 63 & 62 \end{array} $$ 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. Calculate the correlation coefficient relating armspan and height. c. If you were to calculate the regression line for predicting height based on a person's armspan, how would you estimate the slope of this line? d. Find the regression line relating armspan to a person's height. e. If a person has an armspan of 62 inches, what would you predict the person's height to be?

Short answer

Answer: From the scatterplot, we can infer that there is a strong positive relationship between armspan and height, as indicated by the correlation coefficient of 0.873. The predicted height for someone with a 62-inch armspan is approximately 57.01 inches.

Step-by-step solution

Creating a scatterplot

Plot each person's armspan (x-value) and height (y-value) on a graph. Label the x-axis "Armspan (inches)" and the y-axis "Height (inches)". Use the same scale for both axes.

Statistical values

Compute the mean and standard deviation for both armspan and height. Mean of armspan: \(\bar{x}=(68+62.25+65+69.5+68+69+62+60.25)/8 = 65.5\) Mean of height: \(\bar{y}=(69+62+65+70+67+67+63+62)/8 = 65.625\) Standard deviation of armspan: \(\sigma_x=3.608\) Standard deviation of height: \(\sigma_y=3.45\)

Calculating the correlation coefficient

Use the formula for the correlation coefficient \(r\), which is: \(r = \frac{1}{n-1}\sum_{i=1}^n\frac{x_i-\bar{x}_{}}{\sigma_{x}}\frac{y_i-\bar{y}_{}}{\sigma_{y}}\) where \(x_i\) and \(y_i\) are the individual values of armspan and height, \(\bar{x}\) and \(\bar{y}\) are the means calculated in step 2, and \(\sigma_x\) and \(\sigma_y\) are the standard deviations calculated in step 2. The correlation coefficient is 0.873, indicating a strong positive relationship between armspan and height.

Estimate the slope of the regression line

The estimate of the slope of the regression line, \(b_1\) can be calculated as: \(b_1 = r\frac{\sigma_y}{\sigma_x} \) where \(r\) is the correlation coefficient, and \(\sigma_x\) and \(\sigma_y\) are the standard deviations of armspan and height. From step 3 and 4, \(b_1= 0.873\times\frac{3.45}{3.608} = 0.945\)

Finding the regression line equation

The regression line has the form \(y = b_0+b_1x\). From step 2, we have the mean of armspan \(\bar{x}=65.5\) and the mean of height \(\bar{y}=65.625\). From step 4, we have the slope, \(b_1=0.945\). We can now find the intercept, \(b_0 = \bar{y}-b_1\bar{x}\): \(b_0 = 65.625-0.945\times 65.5 = \text{-1.28}\) The regression line equation is: \(y=-1.28+ 0.945x\)

Predicting height for an armspan

Using the regression line equation from step 5, we can predict a person's height for an armspan of \(62\) inches: \(y= -1.28+0.945\times62 = 57.01\) So, we would predict a person with an armspan of \(62\) inches to have a height of approximately \(57.01\) inches.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical method used to investigate the relationship between two or more variables. In the context of this exercise, we're particularly interested in understanding how a person's armspan relates to their height. This involves finding a regression line, which is essentially a line of best fit through data points. This line helps to predict the value of the dependent variable (height) based on the independent variable (armspan).

To find this line, we need to calculate two main components: the slope (\(b_1\)) and the intercept (\(b_0\)). The slope tells us how much the height is expected to increase for each one-unit increase in armspan. The intercept indicates the expected height when the armspan is zero, though this might not be directly interpretable in all contexts. Together, these form the equation of the regression line: \(y = b_0 + b_1x\).

Regression analysis is crucial because it not only shows the strength of the relationship but also allows for future predictions and understanding trends in data. In our example, this analysis revealed that as armspan increases, height tends to increase as well, signifying a positive relationship.

Scatterplot

A scatterplot is a type of graph used in statistical analysis to visually display the relationship between two quantitative variables. In our exercise, the scatterplot illustrates the relationship between armspan and height across eight different people. Each point on the scatterplot represents one person's armspan and the corresponding height. This type of plot is extremely useful because it can immediately give us a visual sense of any correlation between the variables.

When creating a scatterplot for this exercise, both axes must use the same scale to accurately represent the data. By looking at the distribution and pattern of points, you can determine whether the variables have a positive, negative, or no correlation at all.

For instance, in this exercise, the scatterplot would show a pattern that suggests a positive correlation. This means as one variable (armspan) increases, the other variable (height) tends to increase as well. The scatterplot can also show us any outliers or unusual observations, which might need further investigation.

Armspan and Height Relationship

The relationship between armspan and height has intrigued scientists and artists alike for centuries, famously highlighted by Leonardo da Vinci. In statistical terms, this relationship can be explored using correlation and regression analysis.

The correlation coefficient (denoted as \(r\)) quantifies the strength and direction of this relationship. A strong positive correlation coefficient like 0.873, as calculated in our exercise, indicates that taller individuals tend to have a larger armspan. This aligns well with da Vinci's hypothesis but is grounded in statistical analysis rather than artistic interpretation.

Understanding the correlation between armspan and height is useful in various fields such as anthropometry, fashion design, and even sports sciences, where body proportions might influence clothing sizes or athletic performance. This relationship provides a fascinating insight into human body proportions, often reflecting a proportionality principle that can be observed across numerous populations.