91Ó°ÊÓ

Hormone replacement therapy (HRT) is thought to increase the risk of breast cancer. The accompanying data on \(x=\) percent of women using HRT and \(y=\) breast cancer incidence (cases per 100,000 women) for a region in Germany for 5 years appeared in the paper. The authors of the paper used a simple linear regression model to describe the relationship between HRT use and breast cancer incidence. \begin{tabular}{cc} & Breast Cancer \\ HRT Use & Incidence \\ \hline \(46.30\) & \(103.30\) \\ \(40.60\) & \(105.00\) \\ \(39.50\) & \(100.00\) \\ \(36.60\) & \(93.80\) \\ \(30.00\) & \(83.50\) \\ \hline \end{tabular} a. What is the equation of the estimated regression line? b. What is the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use? c. What would you predict the breast cancer incidence to be in a year when HRT use was \(40 \%\) ? d. Should you use this regression model to predict breast cancer incidence for a year when HRT use was \(20 \%\) ? Explain. e. Calculate and interpret the value of \(r^{2}\). f. Calculate and interpret the value of \(s_{e}\).

Step by step solution

01

Create scatterplot and calculate correlation

First of all, create a scatterplot of the data points to visualize the relationship. Then, calculate the correlation coefficient (\(r\)) to determine the strength and direction of the linear relationship between HRT use and breast cancer incidence.

02

Calculate the slope and y-intercept

Using the formula for the slope \((b = r * \frac{S_{y}}{S_{x}})\) and the formula for the y-intercept \((a = \bar{y} - b\bar{x})\), calculate the slope and y-intercept of the regression line.

03

Estimate the Regression Line

The equation of the estimated regression line is \(y = a + bx\). Substitute the calculated slope (b) and y-intercept (a) into this equation.

04

Interpret the slope

The slope (b) represents the estimated average change in breast cancer incidence associated with a 1 percent increase in HRT use. In simpler terms, it indicates how much the predicted value of Y (breast cancer incidence) changes for each one-unit change in X (HRT use).

05

Predict a value based on the regression line

To predict breast cancer incidence in a year when HRT use was 40%, substitute \(x = 40\) into the equation and calculate the value for \(y\).

06

Evaluate the appropriateness of the model

Discuss whether this regression model should be used to predict a value outside the scope of the original data. Examine the range of HRT usage in the given data and compare it with the value of HRT use one wants to predict (20%).

07

Calculate and interpret \(r^{2}\)

Calculate \(r^{2}\) by squaring the correlation coefficient. This is the coefficient of determination that measures the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

08

Calculate and interpret the standard error of the estimate \(s_{e}\)

Compute the standard error of the estimate (\(s_{e}\)) using the formula: \(s_{e} = \sqrt{\frac{1}{n-2} \sum (y - \hat{y})^2}\) where \(\hat{y}\) are the estimated values of \(y\). \(s_{e}\) provides a measure of the dispersion of observed around the predicted values.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Correlation Coefficient

The correlation coefficient, often represented as \( r \), is a vital statistical measure. It shows the strength and direction of a linear relationship between two variables. Here, variables are HRT use and breast cancer incidence. This coefficient ranges from -1 to 1. A value close to 1 indicates a strong positive linear relationship, meaning as HRT use increases, breast cancer incidence also rises. Similarly, a value close to -1 suggests a strong negative relationship. Meanwhile, a value near 0 implies no linear relationship.

To calculate \( r \), data points need to be either manually computed or input into a statistical software tool. In this exercise, a scatterplot helps visualize the data, while the correlation coefficient gives a numeric value to the linear relationship's strength and direction.

Regression Line Equation

The regression line equation is at the heart of simple linear regression analysis. It is written as \( y = a + bx \), where:

• \( y \) is the predicted value of the dependent variable (breast cancer incidence).
• \( a \) is the y-intercept, representing the expected value of \( y \) when \( x \) is zero.
• \( b \) is the slope of the line, indicating the change in \( y \) for each one-unit change in \( x \).
• \( x \) is the independent variable (percentage of HRT use).

Calculate \( b \) using the formula \( b = r \cdot \frac{S_{y}}{S_{x}} \). Here, \( S_{y} \) and \( S_{x} \) are the standard deviations of \( y \) and \( x \), respectively.

Next, find \( a \) with the equation \( a = \bar{y} - b\bar{x} \). Substitute \( a \) and \( b \) into the regression line equation to predict outcomes.

Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), is a key concept in assessing the goodness-of-fit of a linear regression model. It represents the proportion of variance in the dependent variable that is predictable from the independent variable.

Calculated by squaring the correlation coefficient \( r \), \( r^2 \) varies between 0 and 1. An \( r^2 \) of 1 implies that the regression model perfectly predicts the dependent variable's variance, while an \( r^2 \) of 0 suggests no explanatory power at all.

In this context, knowing \( r^2 \) allows insight into how well HRT usage explains the variance in breast cancer incidence rates. Higher \( r^2 \) indicates stronger predictability, making the model more reliable and useful for credible predictions.

Standard Error of Estimate

Standard error of estimate \( s_{e} \) is an essential measure in regression analysis. It tells us how spread out the observed data points are around the predicted values from the regression line.

Calculating \( s_{e} \) involves the formula \( s_{e} = \sqrt{\frac{1}{n-2} \sum (y - \hat{y})^2} \), where \( \hat{y} \) represents the predicted values and \( n \) is the number of data points. This formula shows how tightly data points cluster around the regression line.

Lower standard error means that the data points are close to the regression line, indicating a good fit of the model. Conversely, a higher standard error suggests that the model may not perfectly fit the data, making its predictions less precise. Understanding \( s_{e} \) is vital for evaluating the accuracy of a regression model.