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Chapter 10: Problem 9

Over the past 50 years, there has been a strong negative correlation between average annual income and the record time to run 1 mile. In other words, average annual incomes have been rising while the record time to run 1 mile has been decreasing. (a) Do you think increasing incomes cause decreasing times to run the mile? Explain. (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Short Answer

Expert verified
(a) No, correlation does not imply causation. (b) Lurking variables include advances in training, nutrition, and technology.

Step by step solution

01

Interpreting Correlation vs. Causation

Correlation does not imply causation. A strong negative correlation between two variables, such as average annual income and record times to run 1 mile, only indicates that as one increases, the other tends to decrease. It does not mean that the increase in income directly causes the decrease in running times.
02

Analyzing the Relationship for Causation

Evaluate whether increased incomes could logically or directly cause decreased mile-running times. Improvement in running times is more likely due to advances in training, nutrition, and technology rather than simply having a higher income. Thus, it is unlikely that increasing incomes cause decreased run times.
03

Identifying Lurking Variables

Consider variables that might affect both annual income levels and running times. Possible lurking variables include enhanced scientific training methods, improved health and fitness awareness, technological advancements in athletic gear, and increased focus on professional sports.
04

Conclusion on Lurking Variables

The enhancement in athletic performance over the years is likely driven by improvements in sport science, coaching, and nutrition, which might also relate to economic growth that supports these areas. Hence, these are plausible lurking variables that could influence both variables rather than a direct cause-and-effect relationship between income and running times.

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Key Concepts

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

Understanding Lurking Variables
Lurking variables are essentially hidden factors that can influence both variables in a study but are not immediately obvious. They can often explain the relationship between two correlated variables, making it crucial to identify them.
In the context of the exercise, while it's observed that incomes have risen as mile times have decreased, the lurking variables are likely the true drivers of this trend.
  • Enhanced scientific methods in training can lead to faster running times, irrespective of income changes.
  • Advances in nutrition offer improved diets for athletes, which may not directly relate to income levels.
  • Technology in athletic gear has improved performance potential, making it easier to achieve record times.
These lurking variables can cause changes in both athletic performance and income levels as part of broader societal advancements, rather than one directly causing the other.
Exploring Negative Correlation
A negative correlation exists when one variable tends to decrease as the other increases. However, this does not mean one variable causes the other to change. In this exercise, the increase in average annual income and the decrease in one-mile run times are negatively correlated.
However, this does not suggest causation. Simply because two variables move in opposite directions does not mean one influences the other. This is a common misconception in statistical analysis.
Rather than focusing on income causing faster run times, it's more plausible to look at how both variables are independently influenced by lurking factors. Understanding negative correlation helps in properly interpreting data to avoid misconceptions about cause and effect.
The Role of Statistical Analysis
Statistical analysis is a powerful tool used to identify trends, correlations, and relationships between data sets. A crucial part of statistical analysis is determining whether a true cause-and-effect relationship exists between two correlated variables.
  • Through statistical tests, researchers can verify if correlations are strong or weak and if they are merely coincidental.
  • Identifying correlation without assuming causation is a critical component to using statistical analysis effectively.
  • In this exercise, analysis should focus on recognizing the role of lurking variables and not erroneously attributing changes to income alone.
Statistical analysis encourages us to look beyond surface trends and delve deeper into possible external influences and hidden factors that can affect outcomes. This thorough approach ensures a clearer understanding of data and its implications.

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Most popular questions from this chapter

Data for this problem are based on information from STATS Basketball Scoreboard. It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let \(x\) be the number of fouls more than (i.e., over and above) the opposing team. Let \(y\) be the percentage of times the team with the larger number of fouls wins the game. $$ \begin{array}{l|rrrr} \hline x & 0 & 2 & 5 & 6 \\ \hline y & 50 & 45 & 33 & 26 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=13, \quad \Sigma y=154, \quad \Sigma x^{2}=65\), \(\Sigma y^{2}=6290, \Sigma x y=411\), and \(r \approx-0.988 .\) (f) If a team had \(x=4\) fouls over and above the opposing team, what does the least-squares equation forecast for \(y\) ?

Parts \((\mathrm{a})\) and \((\mathrm{b})\) relate to testing \(\rho .\) Part \((\mathrm{c})\) requests the value of \(S_{e} .\) Parts (d) and (e) relate to confidence intervals for prediction. Parts (f) and (g) relate to testing \(\beta\) and finding confidence intervals for \(\beta\). Answers may vary due to rounding. Let \(x\) be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let \(y\) be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of \(n=6\) professional basketball players gave the following information. (Reference: The Official NBA Basketball Encyclopedia, Villard Books.) $$ \begin{array}{c|cccccc} \hline x & 67 & 65 & 75 & 86 & 73 & 73 \\ \hline y & 44 & 42 & 48 & 51 & 44 & 51 \\ \hline \end{array} $$ (a) Verify that \(\Sigma x=439, \Sigma y=280, \Sigma x^{2}=32,393, \Sigma y^{2}=13,142, \Sigma x y=\) 20,599 , and \(r \approx 0.784\). (b) Use a \(5 \%\) level of significance to test the claim that \(\rho>0\). (c) Verify that \(S_{e} \approx 2.6964, a \approx 16.542, b \approx 0.4117\), and \(\bar{x} \approx 73.167\). (d) Find the predicted percentage \(\hat{y}\) of successful field goals for a player with \(x=70 \%\) successful free throws. (e) Find a \(90 \%\) confidence interval for \(y\) when \(x=70\). (f) Use a \(5 \%\) level of significance to test the claim that \(\beta>0\). (g) Find a \(90 \%\) confidence interval for \(\beta\) and interpret its meaning.

(a) Suppose you are given the following \(x, y\) data pairs: $$ \begin{array}{l|lll} \hline x & 1 & 3 & 4 \\ \hline y & 2 & 1 & 6 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=1.071 x+0.143\) (rounded to three digits after the decimal). (b) Now suppose you are given these \(x, y\) data pairs: $$ \begin{array}{l|lll} \hline x & 2 & 1 & 6 \\ \hline y & 1 & 3 & 4 \\ \hline \end{array} $$ Show that the least-squares equation for these data is \(y=0.357 x+1.595\) (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the \(x\) and \(y\) values of each data pair? (d) Solve \(y=0.143+1.071 x\) for \(x .\) Do you get the least-squares equation of part (b) with the symbols \(x\) and \(y\) exchanged? (e) In general, suppose we have the least-squares equation \(y=a+b x\) for a set of data pairs \(x, y .\) If we solve this equation for \(x\), will we necessarily get the least-squares equation for the set of data pairs \(y, x\) (with \(x\) and \(y\) exchanged)? Explain using parts (a) through (d).

Over the past few years, there has been a strong positive correlation between the annual consumption of diet soda drinks and the number of traffic accidents. (a) Do you think increasing consumption of diet soda drinks causes traffic accidents? Explain. (b) What lurking variables might be causing the increase in one or both of the variables? Explain.

Do larger universities tend to have more property crime? University crime statistics are affected by a variety of factors. The surrounding community, accessibility given to outside visitors, and many other factors influence crime rate. Let \(x\) be a variable that represents student enrollment (in thousands) on a university campus, and let \(y\) be a variable that represents the number of burglaries in a year on the university campus. A random sample of \(n=8\) universities in California gave the following information about enrollments and annual burglary incidents. (Reference: Crime in the United States, Federal Bureau of Investigation.) $$ \begin{array}{c|clllllll} \hline x & 12.5 & 30.0 & 24.5 & 14.3 & 7.5 & 27.7 & 16.2 & 20.1 \\ \hline y & 26 & 73 & 39 & 23 & 15 & 30 & 15 & 25 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or high? positive or negative? (c) Using a calculator, verify that \(\Sigma x=152.8, \Sigma x^{2}=3350.98, \Sigma y=246\), \(\Sigma y^{2}=10,030\), and \(\Sigma x y=5488.4\). Compute \(r\). As \(x\) increases, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.
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