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Chapter 8: Problem 14

Body measurements, Part II. The scatterplot below shows the relationship between weight measured in kilograms and hip girth measured in centimeters from the data described in Exercise 8,13 . (a) Describe the relationship between hip girth and weight. (b) How would the relationship change if weight was measured in pounds while the units for hip girth remained in centimeters?

Short Answer

Expert verified
The relationship is positive and linear. Changing to pounds doesn't alter correlation.

Step by step solution

01

Interpreting the scatterplot

Examine the scatterplot provided, which shows the relationship between weight (in kilograms) and hip girth (in centimeters). Look for any patterns, trends, or correlations in the data points. Pay attention to the direction (positive or negative), form (linear or nonlinear), and strength (strong or weak) of the relationship.
02

Describing the Current Relationship

Based on the observation of the scatterplot, determine if there's a positive or negative correlation between hip girth and weight. A positive correlation would mean as hip girth increases, weight also increases. Identify if the relationship is linear, which would suggest a straight diagonal trend, and estimate the strength of the correlation by considering how closely the data points follow this trend.
03

Converting Weight Units

To understand the effect of measuring weight in pounds instead of kilograms, apply the conversion where 1 kilogram is approximately 2.20462 pounds. This conversion will be used to express the weight, and it will merely change the scale of the weight measurement.
04

Analyzing Effect of Conversion

Recognize that although converting kilograms to pounds changes the scale (making the numerical values larger), it does not change the overall relationship, trend, or shape of the scatterplot between hip girth and weight. The correlation remains unchanged, but the slope or numerical value representing weight will be larger.

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Key Concepts

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

Correlation Interpretation
When working with scatterplots, understanding the correlation is key to interpreting the data correctly. Correlation tells us about the direction and strength of a linear relationship between two variables.
In the case of weight (in kilograms) and hip girth (in centimeters), you would look at the scatterplot to determine whether there is a positive or negative correlation.
If the data points tend to rise from left to right, this indicates a positive correlation, meaning as hip girth increases, so does the weight.
Also, consider the strength of the correlation.
  • If the points are closely packed around a line, this suggests a strong correlation.
  • If they are more scattered, the correlation is weaker.
Visualizing this helps in understanding the extent to which the two variables are related.
Relationship Between Variables
Analyzing the relationship between variables involves understanding how one variable moves in relation to another. Here, we're looking at weight and hip girth.
The scatterplot can reveal a linear relationship if the points form a roughly straight path. Linear relationships are quite straightforward: as one variable changes, the other changes in a consistent way.
This consistent pattern can be described by a line of best fit, which represents the average relationship between the two variables. However, relationships can also be nonlinear, indicating more complex interactions, but in this exercise, we are examining a mainly linear relationship.
Understanding these relationships is crucial in predicting how changes in one aspect might influence another.
Unit Conversion Effects
In data interpretation, unit conversion is often necessary, and it's important to know how this affects data analysis. When you change the units of a variable, for example, converting weight from kilograms to pounds, the numerical values will be affected, as pounds are larger than kilograms.
The conversion factor here is 1 kilogram equals approximately 2.20462 pounds. This impacts only the numerical scale, not the data's fundamental relationship.
  • The scatterplot's shape remains the same.
  • The correlation (whether the relationship is strong or weak, positive or negative) does not change.
So while the numbers on the axes of your scatterplot may look different, the underlying data relationship remains unchanged.
Understanding this ensures clarity in interpreting and comparing data across different measurement systems.

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Most popular questions from this chapter

Body measurements, Part IV. The scatterplot and least squares summary below show the relationship between weight measured in kilograms and height measured in centimeters of 507 physically active individuals. (a) Describe the relationship between height and weight. (b) Write the equation of the regression line. Interpret the slope and intercept in context. (c) Do the data provide strong evidence that an increase in height is associated with an increase in weight? State the null and alternative hypotheses, report the p-value, and state your conclusion. (d) The correlation coefficient for height and weight is \(0.72 .\) Calculate \(R^{2}\) and interpret it in context.

Over-under, Part II. Suppose we fit a regression line to predict the number of incidents of skin cancer per 1,000 people from the number of sunny days in a year. For a particular year, we predict the incidence of skin cancer to be 1.5 per 1,000 people, and the residual for this year is \(0.5 .\) Did we over or under estimate the incidence of skin cancer? Explain your reasoning.

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?
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