91Ó°ÊÓ

The table shows the heights (in inches) and weights (in pounds) of 14 college men. The scatterplot shows that the association is linear enough to proceed. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 68 & 205 \\ \hline 68 & 168 \\ \hline 74 & 230 \\ \hline 68 & 190 \\ \hline 67 & 185 \\ \hline 69 & 190 \\ \hline 68 & 165 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 70 & 200 \\ \hline 69 & 175 \\ \hline 72 & 210 \\ \hline 72 & 205 \\ \hline 72 & 185 \\ \hline 71 & 200 \\ \hline 73 & 195 \\ \hline \end{array} \end{aligned} $$ a. Find the equation for the regression line with weight (in pounds) as the response and height (in inches) as the predictor. Report the slope and the intercept of the regression line, and explain what they show. If the intercept is not appropriate to report, explain why. b. Find the correlation between weight (in pounds) and height (in inches). c. Find the coefficient of determination and interpret it. d. If you changed each height to centimeters by multiplying heights in inches by \(2.54\), what would the new correlation be? Explain. e. Find the equation with weight (in pounds) as the response and height (in inches) as the predictor, and interpret the slope. f. Summarize what you found: Does changing units change the correlation? Does changing units change the regression equation?

Step by step solution

01

Calculation of Regression Line

First, calculate the regression line of the data which is in the form \(y = mx + c\), where \(y\) is the weight (response variable), \(x\) is the height (predictor variable), \(m\) is the slope and \(c\) is the intercept. The formula is \(y=\bar{y}+m(x-\bar{x})\), where \(\bar{y}\) and \(\bar{x}\) are the means of the \(y\) and \(x\) values respectively. Importantly, \(m=r\left(\frac{s_y}{s_x}\right)\), where \(r\) is the correlation coefficient, \(s_y\) is the standard deviation of the \(y\) values, and \(s_x\) is the standard deviation of the \(x\) values.

02

Interpretation of Slope and Intercept

The slope represents how much the weight changes for each increase in height by 1 inch. The intercept represents the weight when the height is zero, which has no practical meaning in this context since people cannot have zero height.

03

Calculation of Correlation

The formula to calculate correlation is \(r=\frac{n\sum xy -( \sum x)( \sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}\) where \(x\) is height and \(y\) is weight, and \(n\) is the number of data points.

04

Calculation of Coefficient of Determination

Next, find the coefficient of determination which is the square of the correlation coefficient. It denotes the fraction of the variance in the weights that can be explained by the heights.

05

Changing Units

If height is converted to centimeters from inches by multiplying by 2.54, the correlation does not change because correlation is a unit-less measure of the strength of linear relationship between two variables.

06

Effect of Changing Units on Regression Line

If the units of the height are changed from inches to centimeters, the slope will change but the y-intercept and correlation will remain the same. This is because the slope is proportional to the unit change of x, which is height in this case, while the y-intercept and the correlation don't depend on the units of measurement.

07

Summary

Changing units does not change the correlation because it is unitless and measures only the strength of linear relationship between two variables. The slope of the regression line changes because it is influenced by the units of the predictor variable height. The y-intercept stays the same because it represents where the line crosses the y axis at x=0 and does not depend on the units of measurement, but it is not meaningful in this context because height cannot be zero.

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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, often represented by the symbol \( r \), is a crucial measure in regression analysis. It quantifies the degree of a linear relationship between two variables, such as height and weight in our example. The value of \( r \) ranges from -1 to 1.

If \( r = 1 \), it indicates a perfect positive linear relationship, meaning as one variable increases, the other variable increases at a consistent rate. Conversely, an \( r = -1 \) denotes a perfect negative linear relationship, indicating that as one variable increases, the other decreases consistently.

A value of \( r = 0 \) suggests no linear relationship between the variables. In practical contexts:

• A positive \( r \) indicates that as height increases, weight tends to increase.
• A negative \( r \) would suggest that as height increases, weight tends to decrease.
• The closer \( r \) is to 1 or -1, the stronger the linear relationship.

Importantly, the correlation coefficient does not change when the units of measurement are altered, as it is a dimensionless measure.

Coefficient of Determination

The coefficient of determination, denoted as \( R^2 \), provides insight into how well the regression line fits the data. It is calculated as the square of the correlation coefficient (\( r^2 \)).

The \( R^2 \) value tells us the proportion of variance in the dependent variable (weight, in this case) that is predictable from the independent variable (height).

• An \( R^2 \) of 0.9, for example, implies that 90% of the variance in weight can be explained by height.
• Conversely, an \( R^2 \) of 0.3 would mean only 30% of the weight variance is due to height.

Understanding \( R^2 \) is crucial as it provides a single number summary of the strength of the relationship, and helps in determining the effectiveness of the predictor variable.

Linear Relationship

A linear relationship describes a straight-line relationship between two quantitative variables. In regression analysis, this is expressed by the equation: \( y = mx + c \), where \( y \) represents the dependent variable (weight), \( x \) is the independent variable (height), \( m \) is the slope, and \( c \) is the y-intercept.

When examining a linear relationship:

• The slope \( m \) indicates how much \( y \) changes for a unit change in \( x \). For example, a slope of 2 implies that for each inch increase in height, weight increases by 2 pounds.
• The intercept \( c \), in the context of height and weight, is the expected weight when height is zero. Although this has no realistic interpretation in our example, it provides a starting point for the regression line.

A strong linear relationship is indicated by a high correlation coefficient and subsequently a substantial coefficient of determination, showing that the data points closely fit the regression line.

Predictor Variable

In the context of regression analysis, the predictor variable is the independent variable used to forecast the dependent variable. In our example, height is the predictor variable used to predict weight.

Key considerations for a predictor variable include its ability to accurately reflect changes in the dependent variable and the strength of its linear relationship with the dependent variable. The predictor variable's reliability is often measured by the correlation coefficient and the coefficient of determination.

• In linear regression, choosing the right predictor variable is paramount. A predictor variable with a high correlation to the dependent variable enhances the accuracy of predictions.
• However, the units of the predictor variable can affect the slope of the regression equation, though not the relationship's strength, as indicated by the correlation coefficient.

Understanding the role and effects of the predictor variable is essential for making reliable predictions and drawing valid conclusions from the regression analysis.