Question

Body Mass Index A study using body mass index (BMI) - an index of obesity-as a function OS1204 of income (\$ thousands) reported the following data for California in \(2016 .\) $$\begin{array}{l|cccccc}\text { Income } & 15 & 20.5 & 30 & 40 & 60 & 75 \\\\\hline \text { BMI } & 31.2 & 29.3 & 27.4 & 27.3 & 26.8 & 20.0\end{array}$$ a. If the researcher thinks that BMI is a function of income, which of the two variables is the independent variable \(x\) and which is the dependent variable \(y ?\) b. Find the least-squares line relating BMI to income. c. Construct the ANOVA table for the linear regression.

Short answer

Answer: The slope of the least-squares line is -0.2739, and the y-intercept is 28.5786. In this context, the slope represents the decrease in BMI for every one-unit increase in income, while the y-intercept represents the estimated BMI when the income is 0.

Step-by-step solution

Identify the dependent and independent variables

The independent variable is the one that doesn't rely on the value of another variable. In this exercise, the researcher thinks that BMI is a function of income. Therefore, the independent variable, x, is the income, and the dependent variable, y, is the BMI.

Calculate the necessary sums for the least-squares line

To find the least-squares line, we need to calculate the sums of x, y, xy, and x^2. Using the provided data: ∑x = 15 + 20.5 + 30 + 40 + 60 + 75 = 240.5 ∑y = 31.2 + 29.3 + 27.4 + 27.3 + 26.8 + 20 = 162 ∑xy = (15 x 31.2) + (20.5 x 29.3) + (30 x 27.4) + (40 x 27.3) + (60 x 26.8) + (75 x 20) = 8517 ∑x^2 = 15^2 + 20.5^2 + 30^2 + 40^2 + 60^2 + 75^2 = 13278.25 There are n = 6 data points.

Calculate the slope and the y-intercept of the least-squares line

Now, we can calculate the slope (m) and the y-intercept (b) of the least-squares line using the formulas: m = (n * ∑xy - ∑x * ∑y) / (n * ∑x^2 - (∑x)^2) m = (6 * 8517 - 240.5 * 162) / (6 * 13278.25 - (240.5)^2) = -0.2739 b = (∑y - m * ∑x) / n b = (162 - (-0.2739) * 240.5) / 6 = 28.5786 Thus, the least-squares line equation is: y = -0.2739x + 28.5786

Calculate the sums needed for the ANOVA table

For the ANOVA table, we need to calculate the sums of squares. Let's start with the total sum of squares: SST = ∑(y - y_mean)^2 First, find the mean of y: y_mean = ∑y / n = 162 / 6 = 27 Now, calculate SST: SST = (31.2 - 27)^2 + (29.3 - 27)^2 + (27.4 - 27)^2 + (27.3 - 27)^2 + (26.8 - 27)^2 + (20 - 27)^2 = 178.55 Next, find the sum of squares due to regression: SSR = m^2 * (∑x^2) / (n * ∑x^2 - (∑x)^2) SSR = (-0.2739)^2 * 13278.25 / (6 * 13278.25 - (240.5)^2) = 134.2827 Finally, calculate the sum of squares due to error: SSE = SST - SSR = 178.55 - 134.2827 = 44.2673

Complete the ANOVA table for linear regression

Now we can construct the ANOVA table: | Source | Sum of Squares (SS) | d.f. | Mean Square (MS) | | -------- | ------------------------- | -----| -----------------| | Regression (R) | SSR = 134.2827 | 1 | MS_R = SSR/1 | | Error (E) | SSE = 44.2673 | 4 | MS_E = SSE/4 | | Total (T) | SST = 178.55 | 5 | - | This table shows that the total variation (SST) in the BMI data can be attributed to the explanatory power of the linear regression (SSR) and the unexplained variation (SSE).

Key concepts

Body Mass Index (BMI)

Body Mass Index (BMI) is a numerical calculation that evaluates a person's body weight in relation to their height. It's a widely used tool to approximate a healthy body weight, taking into account variations in height. The formula to calculate BMI is given by \( BMI = \frac{weight (kg)}{height (m)^2} \).

It helps in categorizing individuals into various weight categories, such as underweight, normal weight, overweight, and obese. These categories can provide a general idea about health risks related to body weight, although it's essential to note that BMI has its limitations. For example, it may not accurately represent the muscle-to-fat ratio in different individuals, which means that muscular individuals may be classified as overweight when they are indeed healthy.

In the context of the exercise, the use of BMI as a dependent variable indicates that researchers are looking at how different levels of income (\textdollar thousands) could potentially influence a person's BMI. The study might investigate whether lower income correlates with higher BMI, which could hint at socioeconomic factors affecting health.

Least-Squares Line

The least-squares line, also known as the line of best fit, is crucial in linear regression analysis. It represents the best approximation of the set of data points by minimizing the sum of the squares of the vertical distances (residuals) of the points from the line.

When constructing a least-squares line, we use mathematical formulas to determine the slope (m) and y-intercept (b) of the line. The slope represents the rate of change of the dependent variable (in this case BMI) per unit change in the independent variable (income). The y-intercept is the value of the dependent variable when the independent variable is zero. The least-squares line for the given data in our exercise could suggest how income is likely to affect BMI.

The equation found in the exercise, \( y = -0.2739x + 28.5786 \), signifies that the BMI decreases overall as the income increases. The negative slope \( m = -0.2739 \) indicates this inverse relationship, and the y-intercept \( b = 28.5786 \) provides a starting point for BMI at an income level of zero. Understanding this equation can help us to predict BMI based on income and evaluate the strength of the relationship.

ANOVA Table

ANOVA stands for Analysis of Variance, and the ANOVA table is a foundational element in statistics that summarizes the components of variation in the data from a regression analysis. The table is made up of columns that show the source of variance, the sum of squares (SS), degrees of freedom (d.f.), and mean squares (MS).

The ANOVA table breaks down the total variability in the dependent variable (SST) into two parts: variability that can be explained by the linear model (SSR) and variability that cannot (SSE). The 'Regression' row quantifies the variation explained by the model, whereas the 'Error' row captures the random variation or noise.

The degrees of freedom associated with the error reflect the number of independent pieces of information that went into calculating the sum of squares due to error. In our exercise, we had 6 data points, so with one estimated parameter (the slope), we have 6 - 1 - 1 = 4 d.f. for the error.

Lastly, the mean squares are calculated by dividing the sum of squares by their respective degrees of freedom. These values are used to compute test statistics that can assess whether the regression model offers a significant fit to the data. In simple terms, a larger MS for Regression compared to Error indicates a more substantial impact of the independent variable on the dependent variable—here, showing the role of income as a predictor of BMI.