Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

The table shows the percents \(P\) of women in different age groups (in years) who have been married at least once. (Source: U.S. Census Bureau) $$\begin{array}{|c|c|}\hline \text { Age group } & \text { Percent, } P\\\\\hline 18-24 & 14.6 \\\25-29 & 49.0 \\\30-34 & 70.3 \\\35-39 & 79.9 \\\40-44 & 85.0 \\\45-49 & 87.0 \\\50-54 & 89.5 \\\55-59 & 91.1 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a logistic model for the data. Let \(x\) represent the midpoint of the age group. (b) Use the graphing utility to graph the model with the original data. How closely does the model represent the data?

Step-by-Step Solution

Answer

A logistic model for the data is created with the form \(P = \frac{1}{1 + e^{-(a + bX)}}\) where \(a\) and \(b\) are obtained using a graphing utility. The model is then graphed with the original data using the same utility to verify the goodness of fit. The r-squared value, calculated from the model, will indicate how closely the model represents the data.

Step 1: Midpoint Calculation

The first step is calculating the midpoint of each age group. For example, the midpoint of the range 18 - 24 years is \(x = \frac {18 + 24}{2} = 21\). This should be done for all age groups.

Step 2: Logistic Regression Calculation

Logistic regression calculations are complex and cannot be done manually. The general formula for logistic regression is \(P = \frac{1}{1 + e^{-(a + bX)}}\). Here, \(P\) is the probability of occurrence of an event (In this case, the percentage of women married), \(a\) is the logistic regression coefficient, \(b\) is the slope of the line curve, \(X\) is the independent variable (In this case, the midpoint age). The values of \(a\) and \(b\) have to be calculated using a graphing utility by inputting the midpoints of the age groups and corresponding percentages of women who have been married.

Step 3: Graphing the Model

After obtaining the regression parameters \(a\) and \(b\), a logistic regression graph can be plotted using a graphing utility. The X-axis represents age and the Y-axis the percentage of women who are married.

Step 4: Model Verification

Finally, using the graphing utility's statistical features, analyze how closely the model represents the data. This can be done by analyzing the graph behaviour and calculating the r-squared (coefficient of determination) value for the model. The closer the r-squared value is to 1, the better our model fits the data.

Key Concepts

Midpoint Calculation

Understanding how to calculate the midpoint is essential in many statistical analyses, including logistic regression. The midpoint, as the name suggests, is the central point of a data range. It is calculated by adding the lower and upper limits of the range and then dividing by two. For example, if we're considering an age group of 18-24 years, the midpoint would be calculated as
\[ x = \frac{18 + 24}{2} = 21 \.\] Such calculations allow researchers and statisticians to represent a range of values by a single, central number. It simplifies the dataset and provides a precise point for further calculations. For logistic regression, the midpoint acts as the independent variable around which the model will curve.

Logistic Regression Analysis

Logistic regression is a powerful statistical technique used for modeling the probability of a binary outcome – like pass or fail, win or lose, healthy or sick. It's particularly useful for understanding the relationship between a categorical dependent variable and one or more independent variables. The calculation of logistic regression involves finding the best fit for the logistic function, which is an S-shaped curve. The equation is of the form \[ P = \frac{1}{1 + e^{-(a + bX)}} \] where \(P\) represents the probability, \(e\) is the base of the natural logarithm, \(a\) is the intercept of the regression equation, \(b\) is the coefficient that describes the slope of the curve, and \(X\) is the independent variable (such as the midpoint of the age groups in the exercise). The coefficients \(a\) and \(b\) must be estimated from the data, and this is typically done using maximum-likelihood estimation, a process often facilitated by statistical software.

Graphing Utility Usage

A graphing utility, such as a calculator or software program, is an invaluable tool when it comes to statistical analysis. These utilities assist in plotting data points and regression curves, which helps to visually interpret the relationships between variables. When it comes to logistic regression, plotting the S-curved model onto the graph alongside actual data points aids in visual comparison between the predicted values by the model and the real-world data. This visual tool complements statistical measures and is especially helpful in conveying complex concepts in simpler visual formats, facilitating easier understanding. Users can also adjust scale and range of axes to focus on specific data intervals for a more detailed analysis.

Statistical Model Verification

Verifying a statistical model is the final and a crucial step in the logistic regression analysis process. It involves assessing how well the model fits the actual data. One of the most common ways to do this is by evaluating the r-squared value, also known as the coefficient of determination. This statistic reveals the proportion of variability in the dependent variable that can be explained by the logistic model. An r-squared value close to 1 indicates a high level of correlation and thus a strong model. In logistic regression, alternative measures like the deviance R-squared or the McFadden's R-squared might instead be used due to the binary nature of the data. This process of verification not only validates the logistic model but also ensures that the predictions made by the model can be relied upon, ultimately supporting sound decision-making based on the analysis.