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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. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} & \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} & \begin{array}{c} \text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 68 & 205 & 70 & 200 \\ \hline 68 & 168 & 69 & 175 \\ \hline 74 & 230 & 72 & 210 \\ \hline 68 & 190 & 72 & 205 \\ \hline 67 & 185 & 72 & 185 \\ \hline 69 & 190 & 71 & 200 \\ \hline 68 & 165 & 73 & 195 \\ \hline \end{array} $$ 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 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 \(\mathrm{cm}\) ) as the predictor, and interpret the slope. f. Summarize what you found: Does changing units change the correlation? Does changing units change the regression cquation?

Step by step solution

01

Find the Regression Line's Equation

To find the equation for the regression line, first calculate the means of the height and weight variables. Then, calculate the deviations from the mean for each value of height and weight. Compute the slopes by dividing the sum of products of deviations by the square of the sum of deviations of height. Using the slope and mean values, calculate the intercept of the regression line.

02

Explanation of Regression Line's parameters

The slope shows how much, on average, the weight changes for each one-inch increase in height, while the intercept shows the expected weight for individuals with zero height. However, it's inappropriate to report an intercept that makes no sense in the context of your data, like a zero-height student.

03

Find the Correlation between Weight and Height

Correlation between the weight and height can be found by dividing the sum of the products of the deviations of height and weight by the product of their standard deviations.

04

Compute the Coefficient of Determination

The coefficient of determination can be found by squaring the correlation coefficient. It demonstrates the proportion of variance in the dependent variable that can be predicted from the independent variable.

05

Evaluate the Effect of Changing Units

The correlation is unitless and does not change even if the units of height are changed from inches to centimeters.

06

Find the Regression Equation after Changing Units

When changing units, the slope value will change depending on the magnitude of the unit conversion. The slope will be smaller than when using inches because 1 cm change represents lesser change in physical terms than 1 inch.

07

Summarise the Findings

Changing units does not change the correlation value as it is unitless. On the other hand, the regression equation does get affected by a unit change because the slope depends on the scale of measurement. Both these analytical measurements are crucial in making statistical predictions, understanding trends and relationships between variables.

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Key Concepts

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

Linear Regression

Linear regression is a staple of statistical analysis, particularly when one is interested in understanding the relationship between two continuous variables. Simply put, it helps model the linear relationship between a dependent variable, which is what we're trying to predict, and one or more independent variables, the predictors.

In the exercise given, height is the independent variable (predictor) and weight is the dependent variable (response). The regression line's equation is symbolized by \( y = mx + b \), where \( m \) represents the slope, and \( b \) stands for the intercept. This equation shows how weight changes, on average, with each inch of height. The slope \( m \) is particularly important because it points to the rate of change in weight for each additional inch of height. However, we must be cautious with the intercept \( b \) as it can suggest an unrealistic weight for a hypothetical person with zero height.

Correlation Coefficient

The correlation coefficient, often denoted by \( r \), measures the strength and direction of the linear relationship between two variables. Its value ranges from -1 to 1, where \( r = 1 \) signifies a perfect positive linear relationship, \( r = -1 \) indicates a perfect negative linear relationship, and \( r = 0 \) implies no linear relationship at all.

In our case with height and weight, a positive correlation coefficient would mean that generally, as height increases, weight increases too. The closer the value of \( r \) is to 1, the stronger the linear relationship between the variables. Calculating \( r \) involves scaling the covariance by the product of the variables' standard deviations, which inherently makes the correlation coefficient unitless and thus immune to changes in the units of measurement.

Coefficient of Determination

The coefficient of determination, represented by \( r^2 \), provides insight into how much of the variability in the dependent variable can be explained by the independent variable in a linear regression model. It is the square of the correlation coefficient, and it ranges from 0 to 1.

A \( r^2 \) value of 1 means the model explains all the variability in the response data around its mean, while a value of 0 indicates no explanatory power at all. When we interpret the coefficient of determination in the context of the height and weight example, we can say that a certain percentage of the variation in weight is predictable from the height. This is a key statistic in regression analysis as it gives a clear, quantifiable measure of how well our linear model fits the data.

Statistical Prediction

Statistical prediction involves using a model, like the linear regression model, to estimate an unknown outcome. With the regression model established between height and weight, we can use someone's height to predict their weight. This is especially powerful in fields ranging from healthcare to sports analytics, wherever numerical predictions are needed.

Nonetheless, predictions are only as good as the model's accuracy and the data's integrity. Predictions based on a model with a high coefficient of determination are typically more reliable because a greater variance is explained by the predictor, thereby reducing potential prediction error.

Unit Conversion in Data Analysis

Unit conversion in data analysis ensures comparability across datasets and can impact the interpretation of the coefficients in a regression model. Although unit conversion does not affect the correlation coefficient, which is dimensionless, it directly influences the slope in a regression equation.

When the height from inches is converted to centimeters, the slope will represent the change in weight per centimeter of height, rather than per inch. The numerical value of the slope will adjust to accommodate the new units, but the underlying relationship remains constant. Therefore, it's crucial to understand the units used for each variable to avoid misunderstandings when interpreting the results of the analysis. When units are changed, as highlighted in our exercise, the regression equation must be recalculated while the correlation remains intact.