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Exercise 10 (page 160 ) presented data on the lean body mass and resting metabolic rate for 12 women who were subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate, in calories burned per 24 hours, is the rate at which the body consumes energy. Here are the data again. $$\begin{array}{llllllll}\hline \text { Mass: } & 36.1 & 54.6 & 48.5 & 42.0 & 50.6 & 42.0 & 40.3 & 33.1 & 42.4 & 34.5 & 51.1 & 41.2 \\\\\text { Rate: } & 995 & 1425 & 1396 & 1418 & 1502 & 1256 & 1189 & 9131124 & 1052 & 1347 & 1204 \\\\\hline\end{array}$$ (a) Use your calculator to help make a scatterplot. (b) Use your calculator's regression function to find the equation of the least-squares regression line. Add this line to your scatterplot from part (a). (c) Explain in words what the slope of the regression line tells us. (d) Calculate and interpret the residual for the woman who had a lean body mass of \(50.6 \mathrm{~kg}\) and a metabolic rate of 1502 .

Step by step solution

01

Create a Scatterplot

Input the data into your graphing calculator or software capable of generating a scatterplot. Lean body mass values (x-values) are entered in one list, and metabolic rate values (y-values) go in another list. Plot these points on a graph to form your scatterplot. This visual representation helps identify any potential linear relationship between the variables.

02

Find the Least-Squares Regression Line

Using the regression function on your calculator, find the equation of the line of best fit for the data. This line minimizes the sum of the squared differences between the observed and predicted values. The regression function will provide you with a line in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Compute this and add it to your scatterplot for reference.

03

Interpret the Slope

The slope of the regression line \( m \) indicates the change in the metabolic rate (calories per 24 hours) for each 1 kg increase in lean body mass. Explain this relationship quantitatively with respect to the slope value obtained in Step 2.

04

Calculate the Residual

The residual is the difference between the observed value of the metabolic rate and the value predicted by the regression line for the woman with a lean body mass of 50.6 kg. Use the regression equation from Step 2 to calculate the predicted metabolic rate for 50.6 kg. Then, subtract this predicted value from the actual metabolic rate of 1502 calories.

05

Interpret the Residual

The residual shows how much the actual metabolic rate of the woman deviates from what the prediction suggests. A positive residual means the actual rate is higher than predicted, while a negative one indicates the opposite. Use this information to comment on the accuracy of the regression model for this particular data point.

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Key Concepts

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

Scatterplot

A scatterplot is a useful visual tool for examining the relationship between two numerical variables. In this exercise, we have lean body mass as the independent variable (x-axis) and the resting metabolic rate as the dependent variable (y-axis). By plotting the data for these 12 women, each point represents a combination of mass and metabolic rate.

This visualization helps us quickly see any patterns or trends, such as whether heavier lean body mass is associated with higher metabolic rates. Identifying a linear or non-linear trend is crucial, as it guides us in deciding which statistical methods, like regression analysis, may be appropriate.

• A well-created scatterplot will help predict and model relationships.
• Look for clustering of points or trends indicating a linear relationship.
• If data aligns closely to a straight line, a linear relationship is likely present.

Creating a scatterplot is the foundational step in data analysis before moving on to more complex calculations.

Residual Analysis

Residual analysis involves examining the difference between observed and predicted values of a dependent variable. This helps assess the accuracy of a regression model.

Residuals are calculated for each data point. It is the difference between an actual data point and the corresponding predicted value given by the regression line. A residual can inform us about how well our linear model fits the data.

• Calculate the predicted rate using the regression line equation.
• Subtract this predicted value from the actual observed metabolic rate to obtain the residual.

If the residual is positive, it suggests that the actual metabolic rate is higher than what the model predicted. Conversely, if it’s negative, the actual rate is lower. Analyzing residuals helps identify patterns or inconsistencies in data that the model may not account for, suggesting areas for potential refinement of the model.

Slope Interpretation

Interpreting the slope of a regression line is a crucial aspect of understanding the data relationship. In our context, the slope represents the rate of change in the metabolic rate per unit increase in lean body mass.

For instance, if the slope of the line is 30, it implies that for every 1 kg increase in lean body mass, the metabolic rate increases by 30 calories per 24 hours. The slope provides a quantitative measure of how strongly the two variables are related.

Understanding the slope is vital for making predictions and understanding the dynamics of the data set. A steeper slope means a stronger relationship, and a flat slope indicates a weaker one. This information can be valuable when considering physical implications or practical applications of the data.

Data Visualization

Data visualization is an essential tool for understanding and interpreting complex datasets. By visually representing data, we can easily grasp relationships, trends, and outliers that might not be obvious from raw data.

When working with data like lean body mass and metabolic rate, it's helpful to augment our scatterplot and regression analysis with additional visual tools like trend lines and box plots, if needed. These can provide deeper insights into data distribution and variability.

• Enhance scatterplots with regression lines to show predicted trends.
• Use visualizations to spot outliers, which can affect accuracy.
• Consider supplementary plots to provide context, such as histograms of individual variables.

Effective data visualization supports better decision-making and communication of findings, making complex quantitative relationships more accessible to varied audiences.