91Ó°ÊÓ

Use appropriate multiple regression software of your choice and enter the data. Note that the data are also available for download at the Companion Sites for this text. Medical: Blood Pressure The systolic blood pressure of individuals is thought to be related to both age and weight. For a random sample of 11 men, the following data were obtained: $$\begin{array}{ccc|ccc} \hline \begin{array}{c} \text { Systolic } \\ \text { Blood Pressure } \end{array} & \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} & \begin{array}{c} \text { Systolic } \\ \text { Blood Pressure } \end{array} & \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline x_{1} & x_{2} & x_{3} & x_{1} & x_{2} & x_{3} \\ \hline 132 & 52 & 173 & 137 & 54 & 188 \\ 143 & 59 & 184 & 149 & 61 & 188 \\ 153 & 67 & 194 & 159 & 65 & 207 \\ 162 & 73 & 211 & 128 & 46 & 167 \\ 154 & 64 & 196 & 166 & 72 & 217 \\ 168 & 74 & 220 & & & \\ \hline \end{array}$$ (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, which variable has the greatest spread of data values? Which variable has the smallest spread of data values relative to its mean? (b) For each pair of variables, generate the sample correlation coefficient \(r\) Compute the corresponding coefficient of determination \(r^{2}\). Which variable (other than \(x_{1}\) ) has the greatest influence (by itself) on \(x_{1} ?\) Would you say that both variables \(x_{2}\) and \(x_{3}\) show a strong influence on \(x_{1}\) ? Explain your answer. What percent of the variation in \(x_{1}\) can be explained by the corresponding variation in \(x_{2}\) ? Answer the same question for \(x_{3}\) (c) Perform a regression analysis with \(x_{1}\) as the response variable. Use \(x_{2}\) and \(x_{3}\) as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in \(x_{1}\) can be explained by the corresponding variations in \(x_{2}\) and \(x_{3}\) taken together? (d) Look at the coefficients of the regression equation. Write out the regression equation. Explain how each coefficient can be thought of as a slope. If age were held fixed, but a person put on 10 pounds, what would you expect for the corresponding change in systolic blood pressure? If a person kept the same weight but got 10 years older, what would you expect for the corresponding change in systolic blood pressure? (e) Test each coefficient to determine if it is zero or not zero. Use level of significance \(5 \% .\) Why would the outcome of each test help us determine whether or not a given variable should be used in the regression model? (f) Find a \(90 \%\) confidence interval for each coefficient. (g) Suppose Michael is 68 years old and weighs 192 pounds. Predict his systolic blood pressure, and find a \(90 \%\) confidence range for your prediction (if your software produces prediction intervals).

Step by step solution

01

Calculate Summary Statistics

For each variable, calculate the mean and standard deviation. Let's denote the systolic blood pressure as \(x_1\), age as \(x_2\), and weight as \(x_3\). Use statistical software or a calculator to find these values. Once you have the mean \( \bar{x} \) and standard deviation \( s \), calculate the coefficient of variation (CV) as \( CV = \frac{s}{\bar{x}} \times 100\% \). Identify which variable has the greatest and smallest spread relative to its mean.

02

Compute the Correlation Coefficient

Use statistical software to calculate the sample correlation coefficient \( r \) for each pair \((x_1, x_2)\), \((x_1, x_3)\), and \((x_2, x_3)\). Calculate the coefficient of determination \( r^2 \) as \( r^2 = r^2 \times 100\% \) to express the influence of one variable on another. Analyze which variable \(x_2\) or \(x_3\) has a stronger influence on \(x_1\) by comparing the \( r^2 \) values.

03

Perform Multiple Regression Analysis

Set up a multiple regression with systolic blood pressure \(x_1\) as the dependent variable and both age \(x_2\) and weight \(x_3\) as independent variables. Use software to find the regression coefficients and the overall coefficient of multiple determination \( R^2 \). This \( R^2 \) will show the percentage of variance in \(x_1\) explained by \(x_2\) and \(x_3\) together.

04

Construct the Regression Equation

Based on the multiple regression output, write the equation as \( x_1 = \beta_0 + \beta_1 x_2 + \beta_2 x_3 \), where \( \beta_0 \) is the intercept, and \( \beta_1 \) and \( \beta_2 \) are the coefficients for age and weight, respectively. Interpret the coefficients as the change in systolic blood pressure per unit increase in age or weight. Predict changes in \(x_1\) with fixed \(x_2\) and changing \(x_3\), and vice versa.

05

Test the Significance of Coefficients

Perform hypothesis tests on \( \beta_1 \) and \( \beta_2 \) to see if they are significantly different from zero, using a 5% significance level. Significant p-values (less than 0.05) suggest the corresponding variable is influential. Insignificant p-values suggest the variable could be omitted from the model.

06

Confidence Intervals of Coefficients

Calculate the 90% confidence intervals for each coefficient \( \beta_1 \) and \( \beta_2 \). This interval estimates where the true population parameter lies, providing insight into the reliability of the coefficients.

07

Predict Systolic Pressure and Confidence Interval

Using the regression model, substitute Michael's age (68) and weight (192) into the equation to predict his systolic blood pressure. Also, generate a 90% prediction interval if the software provides this feature, which gives a range where Michael's systolic pressure is likely to fall.

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Key Concepts

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

Systolic Blood Pressure

Systolic Blood Pressure is an essential measurement in understanding cardiovascular health. It represents the pressure in your arteries when your heart beats. Key factors influencing systolic blood pressure include age and weight.
Understanding the changes in systolic pressure helps doctors monitor conditions like hypertension. In analyzing data, especially in multiple regression, systolic blood pressure is often used as a dependent variable.
This is because it can be influenced by other factors like age and weight, which are used as independent variables. By conducting a regression analysis, you can assess how age and weight contribute to blood pressure changes, helping to predict future health risks.

Correlation Coefficient

The correlation coefficient, often denoted as \(r\), measures the strength and direction of a linear relationship between two variables. Ranging from -1 to 1, values close to 1 indicate a strong positive relationship, while values near -1 imply a strong negative relationship.
In our exercise, the correlation coefficient helps determine how strongly age and weight are connected to systolic blood pressure.
A correlation analysis within multiple regression helps establish whether these independent variables are worthy predictors of changes in blood pressure. Knowing the correlation also aids in understanding whether increasing age or weight is more influential in modifying systolic pressure.

Coefficient of Determination

The Coefficient of Determination, represented as \(R^2\), reveals the proportion of variance in a dependent variable explained by the independent variables in a regression model.
The value of \(R^2\) ranges between 0 and 1, where 0 indicates that the model explains none of the variability and 1 indicates it explains all the variability in the data. In multiple regression, \(R^2\) reflects how well age and weight together predict systolic blood pressure.

• An \(R^2\) close to 1 means age and weight are good predictors of blood pressure changes.
• An \(R^2\) closer to 0 suggests a weak predictive capability of the model.

This metric is crucial because it informs the reliability of the predictions made by the regression equation.

Confidence Interval

A Confidence Interval offers a range within which we can expect the true population parameter to lie, based on sample data. In regression analysis, confidence intervals are calculated for regression coefficients, providing insights into their accuracy and reliability.
A 90% confidence interval implies that if we were to repeat the sampling process many times, 90% of the calculated intervals would contain the true parameter.

• Wide intervals indicate less precision and suggest a need for more data.
• Narrow intervals imply more precise estimates and stronger conclusions.

In our exercise, understanding the confidence intervals aids in determining the reliability of predictions regarding systolic blood pressure based on age and weight.