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Chapter 4: Problem 37

The correlation between height and weight in a sample of students was found to be \(r=0.865 .\) The correlation between height and weight in a sample of teachers was found to be \(r=0.912 .\) Assuming both associations are linear, which association- the association between height and weight of students, or the association between height and weight of teachers-is stronger? Explain.

Short Answer

Expert verified
The association between height and weight is stronger for teachers (\(r=0.912\)) than for students (\(r=0.865\)) as the absolute value of the correlation coefficient is closer to 1 for the teachers.

Step by step solution

01

Analysis of Correlation Coefficient for Students

Look at the given correlation coefficient for the relationship between height and weight in students. \(r=0.865\) for students shows a very strong positive correlation, as it is close to +1.
02

Analysis of Correlation Coefficient for Teachers

Review the given correlation coefficient for the relationship between height and weight in teachers. \(r=0.912\) for teachers shows an even stronger positive correlation than students, because it is closer to +1.
03

Compare the Correlation Coefficients

Now we compare the correlation coefficients for students and teachers. The coefficient for teachers (\(r=0.912\)) is larger in absolute value than the coefficient for students (\(r=0.865\)). Therefore, the association between height and weight is stronger for teachers than for students.

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Key Concepts

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

Linear Association
When we talk about a linear association, we are referring to a relationship between two variables that can be represented approximately by a straight line in a scatter plot. In simpler terms, if you were to plot the two variables on a graph, the points would align in a straight line pattern.

A linear association can be identified by a positive or negative trend.
  • Positive Linear Association: An increase in one variable is associated with an increase in the other variable.
  • Negative Linear Association: An increase in one variable is linked to a decrease in the other variable.
In the context of our exercise, both the data about students and teachers exhibit a linear association between height and weight.
This linear association suggests that as height increases, weight tends to increase as well.

The strength of this linear association is measured using the correlation coefficient, which we'll explore next.
Positive Correlation
A positive correlation indicates that two variables move in the same direction. When one variable increases, the other variable tends to increase as well.

The correlation coefficient, represented as "r", quantitatively captures the strength and direction of a linear relationship between variables. Its value ranges from -1 to +1:
  • A positive r value (closer to +1) signifies a strong positive correlation.
  • An r of +1 represents a perfect positive correlation.
  • An r of 0 means there is no correlation.
In our exercise, both samples (students and teachers) demonstrate positive correlations. For the student sample, r = 0.865, and for the teacher sample, r = 0.912.

Both values are close to +1, indicating strong positive correlations. However, since 0.912 is closer to 1 than 0.865, it shows that the positive correlation between height and weight is stronger in teachers than in students.
Statistical Analysis
Statistical analysis involves examining and interpreting data to discover patterns and relationships. In this exercise, the focus is on understanding the correlation coefficients to determine the strength of linear associations between height and weight in two different groups.

This type of analysis is essential in multiple fields, including:
  • Education: To study various factors affecting student and teacher metrics.
  • Healthcare: For evaluating health-related variables.
  • Economics: To analyze trends and economic indicators.
The correlation coefficient is a key component in statistical analysis. By examining these coefficients (r = 0.865 for students and r = 0.912 for teachers), we gain insights into how closely connected two variables are in a linear context.

The outcome of the statistical analysis in the exercise demonstrates a stronger linear relationship between teacher height and weight compared to students, suggesting that factors influencing this correlation might be more pronounced or consistent in the teacher group.

Understanding these concepts and their applications can help students develop a solid foundation in analyzing relationships between variables in a statistical framework.

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Most popular questions from this chapter

Answer the questions, using complete sentences. a. What is extrapolation and why is it a bad idea in regression analysis? b. How is the coefficient of determination related to the correlation, and what does the coefficient of determination show? \star c. When testing the \(\mathrm{IQ}\) of a group of adults (aged 25 to 50 ), an investigator noticed that the correlation between IQ and age was negative. Does this show that IQ goes down as we get older? Why or why not? Explain.

The table shows the number of people living in a house and the weight of trash (in pounds) at the curb just before trash pickup. $$ \begin{array}{|c|c|} \hline \text { People } & \text { Trash (pounds) } \\ \hline 2 & 18 \\ \hline 3 & 33 \\ \hline 6 & 93 \\ \hline 1 & 23 \\ \hline 7 & 83 \\ \hline \end{array} $$ a. Find the correlation between these numbers by using a computer or a statistical calculator. b. Suppose some of the weight was from the container (each container weighs 3 pounds). Subtract 3 pounds from each weight, and find the new correlation with the number of people. What happens to the correlation when a constant is added (we added negative 3) to cach number? c. Suppose each house contained exactly twice the number of people, but the weight of the trash was the same. What happens to the correlation when numbers are multiplied by a constant?

A car is being driven at an average speed range of \(50-70 \mathrm{kmph}\). The table shows distances between selected cities and the time taken by the car to cover these kilometers. a. Calculate the correlation of the numbers shown in the part a table by using a computer or statistical calculator. $$ \begin{array}{|c|c|} \hline \text { Distance (km) } & \text { Time (hrs) } \\ \hline 120 & 2 \\ \hline 294 & 4 \\ \hline 160 & 3 \\ \hline 340 & 6 \\ \hline 310 & 5 \\ \hline \end{array} $$ b. The table for part b shows the same information, except that the distance was converted to meters by multiplying the number of kilometers by 1000 . What happens to the correlation when numbers are multiplied by a constant? $$ \begin{array}{|c|c|} \hline \text { Distance (m) } & \text { Time (hrs) } \\ \hline 120000 & 2 \\ \hline 294000 & 4 \\ \hline 160000 & 3 \\ \hline 340000 & 6 \\ \hline 310000 & 5 \\ \hline \end{array} $$ c. Suppose the \(0.5\) hour that is lost at toll booths is added to the hours during each travel, no matter how long the distance is. The table for part \(c\) shows the new data. What happens to the correlation when a constant is added to cach number? $$ \begin{array}{|c|c|} \hline \text { Distance (km) } & \text { Time (hrs) } \\ \hline 120 & 2.5 \\ \hline 294 & 4.5 \\ \hline 160 & 3.5 \\ \hline 340 & 6.5 \\ \hline 310 & 5.5 \\ \hline \end{array} $$

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?

The table shows the self-reported number of semesters completed and the number of units completed for 15 students at a community college. All units were counted, but attending summer school was not included as a semester. a. Make a scatterplot with the number of semesters on the \(x\) -axis and the number of units on the \(y\) -axis. Does one point stand out as unusual? Explain why it is unusual. (At most colleges, full-time students take between 12 and 18 units per semester.) Finish cach part two ways, with and without the unusual point, and comment on the differences. b. Find the numerical values for the correlation between semesters and units. c. Find the two equations for the two regression lines. d. Insert the lines. Use technology if possible. e. Report the slopes and intercepts of the regression lines and explain what they show. If the intercepts are not appropriate to report, explain why. $$ \begin{aligned} &\begin{array}{|c|c|} \hline \text { Sems } & \text { Units } \\ \hline 2 & 21.0 \\ \hline 4 & 130.0 \\ \hline 5 & 50.0 \\ \hline 7 & 112.0 \\ \hline 3 & 45.5 \\ \hline 3 & 32.0 \\ \hline 8 & 140.0 \\ \hline 0 & 0.0 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline \text { Sems } & \text { Units } \\ \hline 3 & 30.0 \\ \hline 4 & 60.0 \\ \hline 3 & 45.0 \\ \hline 5 & 70.0 \\ \hline 3 & 32.0 \\ \hline 8 & 70.0 \\ \hline 6 & 60.0 \\ \hline \end{array} \end{aligned} $$
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