91Ó°ÊÓ

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?

Step by step solution

01

Calculate the sums

Calculate the sum of heights (\(X\)), weights (\(Y\)), products of each matched pair (\(XY\)) and squares for every value of \(X\) and \(Y\). For example, the sum of the heights \( \sum X = 60+66+72+70+63 = 331 \) inches.

02

Calculate the Means

Calculate the mean for heights (\( XÌ„ = \sum X / n = 331 / 5 = 66.2 \) inches) and weights (\( YÌ„ = \sum Y / n = 695 / 5 = 139 \) pounds) where n is the total number of data points.

03

Calculate Correlation Coefficient (r)

Calculate the correlation coefficient (r) using the formula \( r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n(\sum X^{2}) - (\sum X)^2][n(\sum Y^{2}) - (\sum Y)^2]}} \) Plug in the sums calculated from steps 1 and 2.

04

Calculate the slope and intercept

For the regression line, first calculate the slope (b1) using the formula \(b1 = \frac{\sum (X - XÌ„) (Y - YÌ„)}{\sum (X - XÌ„)^2}\) Then calculate the intercept (b0) using formula \( b0 = YÌ„ - b1 XÌ„\) which will give the equation of the regression line.

05

Convert the Units and Repeat Steps 1 to 4

Convert all heights into cm (inches * 2.54) and weights into kg (pounds / 2.205). For example, the first data point (60, 105) becomes approximately (152.4, 47.619). Repeat steps 1 to 4 for the converted data.

06

Compare the results

Finally compare the correlation coefficients and regression equations obtained when using height in inches and weight in pounds (parts a and c) with those obtained when using height in cm and weight in kg (parts b and d).

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Key Concepts

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

Linear Association

When we say the association between two variables is linear, it means that if one changes, the other tends to change predictably in a straight line when plotted on a graph. In our exercise, we are looking at how height and weight are related.
You might see that as height increases, weight also increases, which suggests a positive linear association.
This generally means that taller people tend to weigh more than shorter people.

• A linear association can be observed using a scatterplot.
• If the points approximate a straight line, the association is linear.
• In our exercise, the height represents the predictor variable and the weight is the response variable.

Understanding linear associations is crucial in statistics because it helps in making predictions based on data, and it sets the foundation for further analysis, like calculating the regression line.

Regression Line Equation

The regression line is like a best-fit line that models the relationship between our predictor and response variables. The equation of this line helps in predicting the response variable based on the predictor variable.
The general form of a regression line equation is: \[ Y = b_0 + b_1 X \]

• Here, \( b_1 \) represents the slope of the line.
• \( b_0 \) is the y-intercept, where the line crosses the y-axis.

This equation gives us the value of weight that corresponds to each given height in our dataset.

• The slope \( b_1 \) indicates how much weight changes for each inch of height.
• The intercept \( b_0 \) shows the predicted weight when height equals zero, even though it might not have practical real-world meaning in this context.

To find these values, we use our dataset to calculate the means and variances of our variables and apply them to regression formulas.

Correlation Coefficient

The correlation coefficient, often represented as \( r \), measures the strength and direction of the linear relationship between two variables.
The value of \( r \) ranges from -1 to +1:

• Values close to +1 indicate a strong positive linear relationship.
• Values close to -1 show a strong negative linear relationship.
• An \( r \) of 0 suggests no linear relationship.

In this exercise, calculating \( r \) for height and weight involves using the sums and means we calculated.
If \( r \) for height in inches and weight in pounds is similar to the \( r \) in centimeters and kilograms, it shows that the unit of measurement does not affect the strength and direction of the relationship.
This allows us to compare datasets even when expressed in different units, as the correlation remains consistent.

Unit Conversion in Statistics

In statistics, converting units can sometimes be necessary to better interpret data, especially when your dataset comes from different systems of measurement, like metric and imperial.
Converting units involves applying consistent factors like "multiply by 2.54" to go from inches to centimeters, and "divide by 2.205" to convert pounds to kilograms.

• This process does not alter the underlying correlation of the data.
• It ensures universal comprehension and ease of calculations.

Despite the change in scales, essential relationships (expressed through correlation and regression) remain the same.
For instance, the calculated slope in the regression line might change due to scaling, but the interpretation in terms of relationship logic remains consistent.
In the exercise, unit conversion allows one to test the robustness of linear associations and statistical interpretations across different measurement systems, reinforcing our understanding regardless of the units.