Question

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: \(\begin{array}{rrrrrrrrr}x & 79.4 & 81.8 & 81.8 & 82.3 & 83.7 & 84.3 & 84.3 & 85.2 \\ y & 4.3 & 4.6 & 4.8 & 4.7 & 5.0 & 4.9 & 4.4 & 5.0 \\ x & 87.0 & 87.3 & 87.7 & 88.1 & 88.1 & 88.6 & 89.0 & 89.5 \\ .1 & 6.1 & 4.7 & 57 & 5.7 & 5.2 & 5.5 & 5.0 & 5.3\end{array}\) \(87.3\) \(4.7\) \(87.7\) \(5.7\) \(\begin{array}{rrrr}1 & 88.1 & 88.6 & 89.0 \\ 7 & 5.2 & 5.5 & 5.0\end{array}\) \(x\) \(y\) \(6.1\) \(8.1\) \(5.7\) 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 \(1-\mathrm{cm}\) increase in chest circumference? with a \(10-\mathrm{cm}\) 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.

Short answer

The nature of the relationship depicted in the scatterplot depends on the distribution of points. Assuming a positive correlation, the least-squares line can be verified to be \(\hat{y}=-4.54 + 0.1123*x\) with the given summary quantities. On average, roughly 0.1123 change in vital capacity is associated with a 1-cm increase in chest circumference, and 1.123 with a 10-cm increase. The vital capacity predicted for a Tibetan native whose chest circumference is 85 cm would be approximately \(\hat{y}=-4.54 + 0.1123*85\). Lastly, vital capacity is not completely determined by chest circumference because of other unmeasured factors influencing vital capacity.

Step-by-step solution

Construct a Scatterplot

Firstly, the relationship between the vital capacity (y) and the chest circumference (x) needs to be examined visually by constructing a scatterplot using the provided data points. In the scatterplot, each point reflects a unique combination of chest circumference (x-coordinate) and the corresponding vital capacity (y-coordinate). The pattern of the scatterplot gives the nature of the relationship between the x and y.

Calculate the Least-Squares Line

The equation of the least-squares line can be verified using formula, which is \(\hat{y} = a + bx\), where `a` is the y-intercept and `b` is the slope. The y-intercept `a` is calculated by using the formula \(a = \bar{y} - b*\bar{x}\), and the slope `b` is calculated by using the formula \(b = (\sum{x*y} - n*\bar{x}*\bar{y}) / (\sum{x^2} - n*\bar{x}^2)\). By substituting the given summary statistics into these formulas, the values can be verified as provided. This line can be drawn on the scatterplot.

Interpret the Slope

The slope of the line, which is 0.1123, represents the average change in the y-value (vital capacity) for each 1-unit increase in the x-value (chest circumference). Thus, a 1-cm increase in chest circumference corresponds to an average increase of 0.1123 in vital capacity. For a 10-cm increase, you simply multiply the slope by 10, which gives you an average change in vital capacity of \(0.1123 * 10 = 1.123\).

Predict Vital Capacity

To predict the vital capacity for a Tibetan native whose chest circumference is 85 cm, plug in \(x = 85\) into the equation of the least-squares line, which gives \(\hat{y} = -4.54 + 0.1123*85\). Find the y-value to get the predicted vital capacity.

Determine Relationship

To answer whether vital capacity is completely determined by chest circumference, consider the scatterplot and the residual variations (the vertical deviations of the points from the line). If the residuals are small and randomly dispersed around zero, it could be said that chest circumference is a useful predictor. However, vital capacity can never be 'completely' determined by chest circumference, as there are always external factors and potential measurement errors influencing vital capacity.

Key concepts

Scatterplot Construction

To initiate an effective statistical analysis, one often begins with the creation of a scatterplot. This visual representation is the first step in exploring the potential relationship between two quantitative variables. In the context of our exercise, chest circumference (measured in centimeters) is displayed along the x-axis, and vital capacity (measured in liters) is displayed along the y-axis. Each pair of corresponding values forms a point on the graph.

When you create a scatterplot, you're seeking patterns, trends, and outliers which are critical in understanding the nature of the relationship between variables. The position and distribution of these points can suggest a variety of relationships—direct or inverse correlation, or no apparent correlation at all. In the given case, the scatterplot might indicate a trend where increases in chest circumference are associated with larger vital capacities, suggesting a positive correlation.

Least-Squares Regression

The heart of predictive analysis in linear regression is the least-squares line. This statistically determined line has the specific goal of minimizing the sum of squared differences (known as residuals) between the observed values and the values predicted by the line. Imagine you're trying to draw the best possible straight line through a scatter of points on a graph; the least-squares line does precisely that.

To calculate the least-squares regression line, we use the summary data provided. The slope, 'b', conveys how much y (vital capacity) is expected to increase when x (chest circumference) increases by one unit. The y-intercept, 'a', indicates the expected value of y when x is zero, which, in the context of biological data, can sometimes be an abstract concept as negative or zero values might not be feasible. By applying the calculated slope and y-intercept to the formula \(\hat{y} = a + bx\), we arrive at our predicted regression line, laying the foundation for further predictive analysis.

Correlation Between Variables

Identifying the correlation between variables is a significant step in understanding how strongly two variables are related. In statistical terms, correlation is quantified by the correlation coefficient, ranging from -1 to 1. A value close to 1 implies a strong positive correlation, meaning as one variable increases, the other tends to increase as well. Conversely, a value close to -1 signifies a strong negative correlation. A value near zero implies a weak or no linear relationship.

The slope of the least-squares line demonstrates the average increase in vital capacity corresponding to a one-unit increase in chest circumference, thereby providing insight into the nature of their relationship. The pattern shown by the scatterplot, when supplemented with the calculation of the correlation coefficient, will offer a more precise understanding of how closely the changes in chest circumference are related to changes in vital capacity.

Predictive Modeling

Predictive modeling is a statistical technique employed to forecast outcomes based on historical data. In our textbook exercise, the use of the least-squares regression line allows us to predict the vital capacity of a Tibetan native based on their chest circumference. By inputting the chest circumference measurement into the equation, we derive an estimated vital capacity. Yet, it is vital to acknowledge that predictive models are not infallible; they make assumptions based on the available data. The scatterplot and residual analysis will guide us in assessing the model's accuracy.

In predictive modeling, it's crucial to evaluate the confidence we have in our predictions and recognize the limitations. For instance, chest circumference might be a significant predictor of vital capacity, but it is not the sole determining factor. Biologically, many other variables could influence vital capacity, such as fitness level, altitude acclimatization, and health status, reminding us of the multifactorial nature of biological traits.