91Ó°ÊÓ

The paper "Increased Vital and Total Lung Capacities in Tibetan Compared to Han Residents of Lhasa" (American Journal of Physical Anthropology [1991]:341-351) included a scatterplot of vital capacity (y) versus chest circumference \((x)\) for a sample of 16 Tibetan natives, from which the following data were read: $$ \begin{array}{rrrrrrrrr} x & 79.4 & 81.8 & 81.8 & 82.3 & 83.7 & 84.3 & 84.3 & 85.2 \\ y & 4.3 & 4.6 & 4.8 & 4.7 & 5.0 & 4.9 & 4.4 & 5.0 \\ x & 87.0 & 87.3 & 87.7 & 88.1 & 88.1 & 88.6 & 89.0 & 89.5 \\ y & 6.1 & 4.7 & 5.7 & 5.7 & 5.2 & 5.5 & 5.0 & 5.3 \end{array} $$ a. Construct a scatterplot. What does it suggest about the nature of the relationship between \(x\) and \(y\) ? b. The summary quantities are $$ \sum x=1368.1 \quad \sum y=80.9 $$ \(\sum x y=6933.48 \quad \sum x^{2}=117,123.85 \quad \sum y^{2}=412.81\) Verify that the equation of the least-squares line is \(\hat{y}=\) \(-4.54+0.1123 x\), and draw this line on your scatterplot. c. On average, roughly what change in vital capacity is associated with a 1 -cm increase in chest circumference? with a 10 -cm increase? d. What vital capacity would you predict for a Tibetan native whose chest circumference is \(85 \mathrm{~cm} ?\) e. Is vital capacity completely determined by chest circumference? Explain.

Step by step solution

01

Scatterplot Construction

Plot chest circumference (x) on the horizontal axis and vital capacity (y) on the vertical axis. The nature of the relationship can be determined by the pattern of the points plotted.

02

Constructing Least-squares line

To calculate the least-square line, one can ascertain the values of its slope (b1) and y-intercept (b0) as follows: b1 = (n(∑xy) - ∑x∑y) / (n∑x^2 - (∑x)^2) and b0 = (∑y - b1∑x) / n where n represents the number of data points (16 in this case). Substituting the provided summary quantities, the equation of the least-squares line is obtained. This line can then be added to the scatterplot.

03

Interpret the slope of the line

The coefficient of x in the least-squares line equation (-4.54 + 0.1123x) represents the change in vital capacity associated with a 1 cm increase in chest circumference. Multiply this coefficient by 10 to compute the change associated with a 10 cm increase.

04

Prediction of Vital Capacity

To predict the vital capacity for a chest circumference of 85 cm, substitute x = 85 in the least-squares line equation.

05

Interpretation of relationship

Examine the scatterplot to infer whether vital capacity is completely determined by chest circumference. If all the data points are on or very close to the line, one could infer a strong relationship, but not complete dependence, as there is always some room for variation due to other influencing factors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least-squares line

Understanding the least-squares line is essential when analyzing the relationship between two variables. In the context of chest circumference (\(x\text{ cm}\)) and vital capacity (\(y\text{ L}\)), we want to find the line that best fits the scatter of data points. This line minimizes the sum of the squares of the differences (errors) between the observed values (\(y\text{ of each data point}\)) and the predicted values (\(y\text{ on the line}\)).

Mathematically, this line is represented by the equation \(\hat{y} = b_0 + b_1x\), where \(b_0\) is the y-intercept and \(b_1\) is the slope showing how much \(y\) changes for each one-unit increase in \(x\). From the exercise, we have \(\hat{y} = -4.54 + 0.1123x\). This means that for every additional centimeter in chest circumference, the vital capacity increases by about 0.1123 liters, indicating a positive relationship. Learning how to compute and interpret this line provides profound insights into the nature of the relationship and is vital for accurate predictions and conclusions.

Vital capacity

Vital capacity refers to the maximum amount of air a person can expel from their lungs after maximum inhalation. It is an important measure of lung and respiratory health. In our specific exercise, we measure vital capacity in liters (\(y\text{ L}\)) and look at how it relates to another variable—chest circumference.

Vital capacity can vary widely based on a range of factors, including age, sex, height, and ethnicity. In the study of Tibetan natives, knowing that chest circumference can be one of the physical correlates to vital capacity provides useful health insights and also serves to underline that statistical relationships do not imply direct causation but rather an association influenced by multiple variables.

Chest circumference analysis

When analyzing chest circumference (\(x\text{ cm}\)), we’re looking at the measurement around the thorax, which can be an indicator of lung capacity. In the provided research, the correlation between chest circumference and vital capacity in Tibetan natives is explored through scientific investigation and statistical methods.

Chest circumference can serve as a physical indicator of health and growth. However, this analysis should always be interpreted with caution. While there may be an observable pattern or trend suggested by data (as depicted in a scatterplot), it’s crucial to remember that correlation does not necessarily imply causation. In other words, while increased chest circumference might indicate a higher vital capacity, it’s not the sole determinant, and other factors should also be considered.

Statistical data interpretation

Statistical data interpretation involves inferring conclusions or making predictions based on data analysis. In our textbook example, this includes predicting vital capacity from chest circumference, understanding the meaning of the slope of our least-squares line, and considering the extent to which chest circumference explains variations in vital capacity.

Interpreting statistical data also demands a degree of skepticism, particularly regarding how much one variable can determine another. The scatterplot from the textbook exercise opens up an opportunity to discuss how closely the data aligns with the least-squares line. If the scatter is tightly clustered around the line, it suggests a strong relationship. However, outliers or a wide scatter would indicate other factors at play. The real-world interpretation would also involve understanding the limitations of the study, considering confounding variables, and recognizing the scope of predictions based on the model.