The following table shows the heights and weights of some...

The following table shows the heights and weights of some people. The scatterplot shows that the association is linear enough to proceed. $$ \begin{array}{|c|c|} \hline \text { Height (inches) } & \text { Weight (pounds) } \\ \hline 60 & 105 \\ \hline 66 & 140 \\ \hline 72 & 185 \\ \hline 70 & 145 \\ \hline 63 & 120 \\ \hline \end{array} $$ a. Calculate the correlation, and find and report the equation of the regression line, using height as the predictor and weight as the response. b. Change the height to centimeters by multiplying each height in inches by $2.54$. Find the weight in kilograms by dividing the weight in pounds by $2.205 .$ Retain at least six digits in each number so there will be no errors due to rounding. c. Report the correlation between height in centimeters and weight in kilograms, and compare it with the correlation between the height in inches and weight in pounds. d. Find the equation of the regression line for predicting weight from height, using height in centimeters and weight in kilograms. Is the equation for weight in pounds and height in inches the same as or different from the equation for weight in kilograms and height in centimeters?

The following table shows the heights and weights of some people. The scatterplot shows that the association is linear enough to proceed.
$$
\begin{array}{|c|c|}
\hline \text { Height (inches) } & \text { Weight (pounds) } \\
\hline 60 & 105 \\
\hline 66 & 140 \\
\hline 72 & 185 \\
\hline 70 & 145 \\
\hline 63 & 120 \\
\hline
\end{array}
$$
a. Calculate the correlation, and find and report the equation of the regression line, using height as the predictor and weight as the response.
b. Change the height to centimeters by multiplying each height in inches by $2.54$. Find the weight in kilograms by dividing the weight in pounds by $2.205 .$ Retain at least six digits in each number so there will be no errors due to rounding.
c. Report the correlation between height in centimeters and weight in kilograms, and compare it with the correlation between the height in inches and weight in pounds.
d. Find the equation of the regression line for predicting weight from height, using height in centimeters and weight in kilograms. Is the equation for weight in pounds and height in inches the same as or different from the equation for weight in kilograms and height in centimeters?

Key Concepts

Scatterplot Analysis

A scatterplot is a graphical representation that plots pairs of numerical data, one variable along each axis. It is an essential tool for visualizing the relationship between two variables, assessing the linearity of that relationship, and identifying potential outliers. The scatterplot is often the first step in exploratory data analysis before undertaking detailed modeling or regression analysis.

Linear Transformation and Unit Conversion

Linear transformations, such as converting measurement units (e.g., converting inches to centimeters or pounds to kilograms), affect the scale and intercept of the regression line but do not change the correlation coefficient because it is scale invariant. Understanding how these changes affect the regression equation is important when comparing models or interpreting statistical results under different measurement systems.

Least Squares Regression

Least squares regression is a method used to determine the best-fitting line through a set of data by minimizing the sum of the squared differences between observed values and the values predicted by the line. The resulting regression line provides a mathematical model that can predict the response variable based on the predictor variable. This approach forms the backbone of predictive modeling in statistics.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two continuous variables. It is represented by a value between -1 and 1, with values close to these extremes indicating strong linear association, and a value near 0 indicating little to no linear relationship. This concept is crucial for determining whether a linear model is appropriate for the given data set.

Transcript

00:01 So this scatter plot shows the relationship between height, which is being measured or used the variable h and weight, which is being represented by the variable w, of 12 men.

00:14 And the data is plotted as h comma w.

00:17 So instead of what is normally x, comma, y, we want to know that h is matching up with x and w is matching up with y.

00:26 So then we're asked to draw a line through the points 68, 155, and 73, 195.

00:33 So 68 would be 65, 66, 67, 68, and then 155 would be right here.

00:41 And then 73, 195, 71, 72, 73, and then 195 would be here.

00:49 So i'm going to go ahead and do that.

00:51 So one thing that's just worth noting is that neither of these two points, we're actually points of data of the 12 men, but they're just the two points that are going to give us the best line to model this data.

01:06 And so when we find our line of best fit or create a linear regression, we don't have to use data points within the set.

01:13 So now we're going to find the equation of this line in slope intercept form.

01:18 And so normally slope intercept form is y equals mx plus b, but remember we're not using y and x.

01:24 We're using w and h.

01:28 And so to find the slope between these two points, i might use the subtraction fraction fraction.

01:36 So m equals y2 minus y1 over x2 minus x1.

01:41 And so i would do 195 minus 155 over 73 minus 68, which would give me 40 over 5 for a slope of 8.

01:55 Now to find the intercept, i would choose one of my two points, so i might use actually the first one just because the number is a little bit smaller...

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