91Ó°ÊÓ

Body measurements, Part III. Exercise 8.13 introduces data on shoulder girth and height of a group of individuals. The mean shoulder girth is \(107.20 \mathrm{~cm}\) with a standard deviation of \(10.37 \mathrm{~cm}\). The mean height is \(171.14 \mathrm{~cm}\) with a standard deviation of \(9.41 \mathrm{~cm} .\) The correlation between height and shoulder girth is 0.67 (a) Write the equation of the regression line for predicting height. (b) Interpret the slope and the intercept in this context. (c) Calculate \(R^{2}\) of the regression line for predicting height from shoulder girth, and interpret it in the context of the application. (d) A randomly selected student from your class has a shoulder girth of \(100 \mathrm{~cm}\). Predict the height of this student using the model. (e) The student from part (d) is \(160 \mathrm{~cm}\) tall. Calculate the residual, and explain what this residual means. (f) A one year old has a shoulder girth of \(56 \mathrm{~cm}\). Would it be appropriate to use this linear model to predict the height of this child?

Step by step solution

01

Calculate slope (b) of the regression line

The slope of the regression line, denoted as \( b \), is calculated using the formula \( b = r \frac{s_y}{s_x} \), where \( r \) is the correlation coefficient, \( s_y \) is the standard deviation of height, and \( s_x \) is the standard deviation of shoulder girth. With the provided data, this becomes \( b = 0.67 \cdot \frac{9.41}{10.37} = 0.608 \).

02

Calculate intercept (a) of the regression line

The intercept \( a \) is calculated using the formula \( a = \bar{y} - b\bar{x} \), where \( \bar{y} \) is the mean height and \( \bar{x} \) is the mean shoulder girth. Therefore, \( a = 171.14 - 0.608 \cdot 107.20 = 105.00 \).

03

Formulate the regression equation

The regression equation for predicting height \( \, \hat{y} \, \) from shoulder girth \( \, x \, \) is \( \, \hat{y} = a + bx \, \). With \( a = 105.00 \) and \( b = 0.608 \), the equation is \( \hat{y} = 105.00 + 0.608x \).

04

Interpret slope and intercept

The slope 0.608 indicates that for each additional cm in shoulder girth, the height increases by approximately 0.608 cm. The intercept 105.00 represents the estimated height when shoulder girth is 0 cm, but it's often not meaningful in this context.

05

Calculate \( R^2 \) value

\( R^2 \) is the square of the correlation coefficient, so \( R^2 = 0.67^2 = 0.4489 \). This means 44.89% of the variation in height can be explained by the shoulder girth.

06

Predict the height for a 100 cm shoulder girth

Using the regression equation \( \hat{y} = 105.00 + 0.608 \cdot 100 \), predict the height: \( \hat{y} = 105.00 + 60.8 = 165.8 \mathrm{~cm} \).

07

Calculate the residual for the student

The residual is the difference between the observed height and the predicted height. For a height of 160 cm, the residual is \( 160 - 165.8 = -5.8 \mathrm{~cm} \). This indicates the model overestimates the actual height by 5.8 cm.

08

Evaluate model suitability for a one-year-old

Using this model to predict the height of a one-year-old with a shoulder girth of 56 cm is inappropriate because the model is based on data from older individuals, making extrapolation unreliable.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient is a statistical measure used to express the strength of a linear relationship between two variables. In this exercise, the correlation coefficient, denoted as \( r \), was found to be 0.67. This value ranges between -1 and 1, where values closer to -1 or 1 indicate a stronger linear relationship.
For example, if \( r \) equals 1, it suggests a perfect positive linear relationship, whereas if \( r \) equals -1, the relationship is perfectly negative.
The positive value of 0.67 in our exercise indicates a moderate to strong positive relationship between shoulder girth and height: as shoulder girth increases, height tends to increase as well.
This information is critical in forming a basis for the linear regression model that seeks to predict one variable based on another.

Linear Regression Model

A linear regression model is used to predict values based on an established relationship between two variables. Here, we're predicting height based on shoulder girth. The model is expressed as \( \hat{y} = a + bx \), where \( \hat{y} \) is the predicted value, \( a \) is the intercept, and \( b \) is the slope.

• The slope, calculated as 0.608, signifies how much height is expected to change with every additional cm of shoulder girth.
• The intercept, calculated as 105.00, would be the height if the shoulder girth were zero; however, this isn't meaningful for our context of predicting human height.

The resulting regression equation \( \hat{y} = 105.00 + 0.608x \) allows us to estimate the height for any given shoulder girth in the dataset range, which helps in understanding trends and making scientific predictions.

Residual Calculation

Residual calculation is a method used to measure the difference between observed and predicted values in a regression analysis. It helps us understand how well the model predicts the actual data.
In this specific case, if a student's observed height is 160 cm and their predicted height using our model is 165.8 cm, the residual is \( 160 - 165.8 = -5.8 \) cm.
A negative residual indicates that the model overpredicts the height, meaning the actual height was less than what was predicted. Conversely, a positive residual would indicate underprediction. Calculating residuals for multiple data points can help identify any patterns that might suggest modifications to the model for better accuracy.

Standard Deviation

Standard deviation is a statistic that measures the dispersion or spread of a set of values relative to its mean. In the context of this exercise, we have two standard deviations: that of shoulder girth (10.37 cm) and height (9.41 cm).

• A smaller standard deviation denotes that the values are closer to the mean, suggesting less variability.
• A larger standard deviation indicates more spread out values, suggesting more diversity in the dataset.

Understanding standard deviation is crucial for calculating the slope \( b \), as it appears in the formula \( b = r \frac{s_y}{s_x} \), where \( s_y \) and \( s_x \) are the standard deviations of height and shoulder girth, respectively. It helps explain how scattered or clustered the data points are, influencing the reliability and accuracy of the regression model.