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Chapter 5: Problem 36

More exercise, more weight loss. In the study described in Example 5.5, the researchers found that, in general, subjects who engaged in more physical activity had higher total energy expenditures. In particular, they found that physical activity explained \(3.3 \%\) of the variation in total energy expenditure. What is the numerical value of the correlation between physical activity and total energy expenditure?

Short Answer

Expert verified
The correlation coefficient is approximately 0.182.

Step by step solution

01

Understand the Relationship Between Variables

In the given exercise, we need to calculate the correlation coefficient between two variables: physical activity and total energy expenditure. We know that the percentage of the variation in the dependent variable (total energy expenditure) explained by the independent variable (physical activity) is given by the coefficient of determination, denoted as \(R^2\). Here, \(R^2 = 3.3\%\), which is equal to 0.033 as a decimal.
02

Find the Correlation Coefficient

The correlation coefficient \(r\) is related to \(R^2\) by the formula:\[ r = \sqrt{R^2}\]This is because \(R^2\) is simply the square of the correlation coefficient. We need to calculate \(r\) by taking the square root of \(0.033\).
03

Calculate the Square Root

Find \(r\) by calculating the square root of \(0.033\):\[ r = \sqrt{0.033} \approx 0.1817\]This gives the absolute value of correlation coefficient. Since correlation could be positive or negative depending on the direction of the relationship, and the context suggests a positive relationship (more activity, more energy expenditure), we assume \(r = 0.1817\).

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Key Concepts

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

Physical Activity
Physical activity is a crucial part of maintaining a healthy lifestyle. It encompasses any movement that consumes energy, ranging from casual walking to intense workouts. This not only affects your muscles and bones but also impacts various bodily systems.
Different types of physical activity include:
  • Aerobic exercises like running and swimming, which improve cardiovascular health.
  • Strength training such as weightlifting, which enhances muscle strength and endurance.
  • Flexibility exercises like yoga, which increase the range of motion and reduce the risk of injury.

Engaging in regular physical activity helps increase the total energy expenditure of an individual, implying that more energy is burned even at rest. Over time, this can lead to numerous health benefits such as improved mental health, better weight management, and a reduced risk of chronic diseases.
Total Energy Expenditure
Total Energy Expenditure (TEE) refers to the total amount of calories burned by the body in a day. It is composed of several components:
  • Basal Metabolic Rate (BMR): the energy expended while at rest, which is necessary for maintaining vital body functions.
  • Thermic Effect of Food (TEF): the energy required for digesting and processing food.
  • Physical Activity Level (PAL): the energy used during physical activities.

In the context of the problem, increased physical activity directly influences TEE by raising the amount of energy burned through physical activity. When subjects engage in more physical activity, they increase their TEE, which is essential for weight management and overall health.
Understanding TEE allows individuals to balance their energy intake and expenditure effectively, helping in creating tailored plans for health goals, such as weight loss or muscle gain.
Coefficient of Determination
The coefficient of determination, symbolized as \(R^2\), indicates how well data fit a statistical model. It represents the proportion of the variance in the dependent variable that can be explained by the independent variable.

Key insights about \(R^2\) include:
  • A higher \(R^2\) value suggests a stronger relationship between variables.
  • A value of \(R^2 = 1\) implies a perfect fit, meaning all observations fall on the regression line.
  • In this exercise, \(R^2\) is \(0.033\) or \(3.3\%\), indicating that physical activity explains \(3.3\%\) of the variation in total energy expenditure.

When calculating the correlation coefficient \(r\), by taking the square root of \(R^2\), we can assume that since the context suggests a positive relationship, \(r = 0.1817\). This measure is useful in understanding the strength and direction of the linear relationship between physical activity and total energy expenditure.

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Most popular questions from this chapter

Death by Intent. Homicide and suicide are both intentional means of ending a life. However, the reason for committing a homicide is different from that for suicide, and we might expect homicide and suicide rates to be uncorrelated. On the other hand, both can involve some degree of violence, so perhaps we might expect some level of correlation in the rates. Here are data from 2008-2011 for 26 counties in Ohio. \({ }^{5}\) Rates are per 100,000 people.(a) Make a scatterplot that shows how suicide rate can be predicted from homicide rate. There is a weak linear relationship, with correlation \(r=0.17\). (b) Find the least-squares regression line for predicting suicide rate from homicide rate. Add this line to your scatterplot. (c) Explain in words what the slope of the regression line tells us. (d) Another Ohio county has a homicide rate of \(8.0\) per 100,000 people. What is the county's predicted suicide rate?

The price of diamond rings. A newspaper advertisement in the Straits Times of Singapore contained pictures of diamond rings and listed their prices, diamond weight (in carats), and gold purity. Based on data for only the 20 -carat gold ladies' rings in the advertisement, the least-squares regression line for predicting price (in Singapore dollars) from the weight of the diamond (in carats) is 17 $$ \text { price }=259.63+3721.02 \text { carats } $$ (a) What does the slope of this line say about the relationship between price and number of carats? (b) What is the predicted price when number of carats = 0? How would you interpret this price?

The points on a scatterplot lie close to the line whose equation is \(y=3-4 x\). The slope of this line is (a) 3 . (b) 4 . (c) \(-4\).

What's my grade? In Professor Krugman's economics course, the correlation between the students' total scores prior to the final examination and their finalexamination scores is \(r=0.5\). The pre-exam totals for all students in the course have mean 280 and standard deviation 40 . The final-exam scores have mean 75 and standard deviation 8. Professor Krugman has lost Julie's final exam but knows that her total before the exam was 300 . He decides to predict her finalexam score from her pre-exam total. (a) What is the slope of the least-squares regression line of final-exam scores on pre-exam total scores in this course? What is the intercept? Interpret the slope in the context of the problem. (b) Use the regression line to predict Julie's final-exam score. (c) Julie doesn't think this method accurately predicts how well she did on the final exam. Use \(r^{2}\) to argue that her actual score could have been much higher (or much lower) than the predicted value.

Penguins diving. A study of king penguins looked for a relationship berween how deep the penguins dive to seek food and how long they stay underwater. 16 For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled, "The relation of dive duration (DD) to depth (D)." Duration DD is measured in minutes, and depth \(D\) is in meters. The report then says, "The regression equation for this bird is \(\mathrm{DD}=2.69+0.0138 \mathrm{D} . "\) (a) What is the slope of the regression line? Explain in specific language what this slope says about this penguin's dives. (b) According to the regression line, how long does a typical dive to a depth of 200 meters last? (c) The dives varied from 40 meters to 300 meters in depth. Use the regression equation to determine \(D D\) for \(D=40\) and \(D=300\) and then plot the regression line from \(D=40\) to \(D=300\).
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