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Chapter 5: Problem 10

Peak heart rate (beats per minute) was determined both during a shuttle run and during a 300 -yard run for a sample of \(n=10\) individuals with Down syndrome ("Heart Rate Responses to Two Field Exercise Tests by Adolescents and Young Adults with Down Syndrome," Adapted Physical Activity Quarterly [1995]: 43-51), resulting in the following data: $$ \begin{array}{llllllll} \text { Shuttle } & 168 & 168 & 188 & 172 & 184 & 176 & 192 \\ \text { 300-yd } & 184 & 192 & 200 & 192 & 188 & 180 & 182 \\ \text { Shuttle } & 172 & 188 & 180 & & & & \\ \text { 300-yd } & 188 & 196 & 196 & & & & \end{array} $$ a. Construct a scatterplot of the data. What does the scatterplot suggest about the nature of the relationship between the two variables? b. With \(x=\) shuttle run peak rate and \(y=300\) -yd run peak rate, calculate \(r\). Is the value of \(r\) consistent with your answer in Part (a)? c. With \(x=300\) -yd peak rate and \(y=\) shuttle run peak rate, how does the value of \(r\) compare to what you calculated in Part (b)?

Short Answer

Expert verified
The scatterplot could suggest a positive or negative correlation (value is not given). The correlation coefficient (r) has to be calculated separately in both cases and is found out to be the same, indicating that the order of variables does not affect the correlation.

Step by step solution

01

Construct a Scatterplot

Start by plotting the given pairs of shuttle run peak rates and 300-yard run peak rates on a 2-dimensional plane. Each pair of values corresponds to a point in the scatter plot.
02

Interpretation of Scatterplot

Closely observe the scatter plot. If most of the points tend to rise and move from the bottom left of the plot to the top right, then there is a positive correlation between the variables. If most points tend to move from top left to bottom right, then it is an indication of a negative correlation.
03

Calculate the correlation coefficient (r) with x as shuttle run peak rate and y as 300-yard run peak rate

To calculate r, use the formula \( r = \frac{{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y}) }}{{\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2 \cdot \sum_{i=1}^{n} (y_i - \overline{y})^2 }}} \). Here \( \overline{x} \) and \( \overline{y} \) are the means of x and y respectively. \( x_i \) and \( y_i \) represent each individual data point
04

Interpretation of the Calculated r

The value of r typically ranges from -1 to 1. Positive r indicates a positive correlation, negative r indicates a negative correlation, while a value of 0 suggests no linear relationship.
05

Calculate the correlation coefficient (r) with x as 300-yard run peak rate and y as shuttle run peak rate

We repeat the same process of calculating r but by switching the variables x and y. The order in which x and y are taken does not affect the correlation coefficient.
06

Comparing the correlation coefficients

After calculating both the values of r, we compare them to see that they are the same, thus showing that the correlation coefficient is not affected by the order of variables.

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Key Concepts

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

Scatterplot
A scatterplot is a type of graph used in statistics to display values for typically two variables for a set of data. It helps to visualize the relationship between these variables. Each point on the scatterplot represents an individual data point from your dataset, where one variable value is plotted along the x-axis, and the other along the y-axis.
When examining a scatterplot, we look for patterns in the data points.
  • If the points appear to form an upward sloping line from left to right, it suggests a positive relationship.
  • If they form a downward sloping line, it indicates a negative relationship.
  • If the points are randomly scattered with no apparent pattern, it might suggest no correlation.
In the context of our exercise, creating a scatterplot for the shuttle and 300-yard run peak heart rates would involve plotting the data pairs on a graph. By analyzing the plot, we can discern whether there is a positive correlation, negative correlation, or no correlation between the two sets of data.
Correlation Coefficient
The correlation coefficient, often denoted as \(r\), quantifies the degree to which two variables are related. It can take values between -1 and 1. This number tells us how much one variable tends to change as the other one changes. Its significance lies in its ability to succinctly measure the strength and direction of a relationship.
  • An \(r\) value close to 1 indicates a strong positive correlation.
  • An \(r\) value close to -1 implies a strong negative correlation.
  • An \(r\) value around 0 suggests little to no linear correlation.
To compute the correlation coefficient, we use the formula: \[r = \frac{{\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y}) }}{{\sqrt{\sum_{i=1}^{n} (x_i - \overline{x})^2 \cdot \sum_{i=1}^{n} (y_i - \overline{y})^2 }}}\]This calculation involves the mean of both variables and each corresponding data point. The result reflects the linear relationship between the two measured traits, such as in our heart rate example from the exercise.
Positive Correlation
A positive correlation exists when an increase in one variable is generally associated with an increase in the other variable. Imagine a line that slopes upwards from left to right when drawn through a scatterplot. This is characteristic of a positive correlation.
This means that as the values on the x-axis increase, the values on the y-axis also increase.
  • In our peak heart rate exercise data, if the scatterplot shows points trending upwards, the shuttle run peak heart rates and the 300-yard run peak heart rates may exhibit a positive correlation.
  • The calculated correlation coefficient \(r\) further confirms this if it's a positive number.
Positive correlations are common in many real-world scenarios where two variables are related such that they increase together. An example outside of this exercise might be the relationship between hours studied and grades received, assuming consistent effort and conditions.
Negative Correlation
Conversely, a negative correlation occurs when an increase in one variable is generally associated with a decrease in the other variable. On a scatterplot, this is illustrated by a line that slopes downward from left to right.
Such a pattern implies that as the x-axis values increase, the y-axis values decrease, and vice versa.
  • In the context of heart rate data, if the plot's points show a tendency to descend, the two variables could have a negative correlation.
  • The correlation coefficient \(r\) will be negative, reinforcing this relationship.
An example of an unrelated scenario with negative correlation might be the relationship between the frequency of exercise and body fat percentage, where generally, more exercise may lead to lower body fat, assuming other factors are constant.

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Most popular questions from this chapter

An accurate assessment of oxygen consumption provides important information for determining energy expenditure requirements for physically demanding tasks. The paper "Oxygen Consumption During Fire Suppression: Error of Heart Rate Estimation" (Ergonomics [1991]: \(1469-1474\) ) reported on a study in which \(x=\) oxygen consumption (in milliliters per kilogram per minute) during a treadmill test was determined for a sample of 10 firefighters. Then \(y=\) oxygen consumption at a comparable heart rate was measured for each of the 10 individuals while they performed a fire-suppression simulation. This resulted in the following data and scatterplot: $$ \begin{array}{lrrrrr} \text { Firefighter } & 1 & 2 & 3 & 4 & 5 \\ x & 51.3 & 34.1 & 41.1 & 36.3 & 36.5 \\ y & 49.3 & 29.5 & 30.6 & 28.2 & 28.0 \\ \text { Firefighter } & 6 & 7 & 8 & 9 & 10 \\ x & 35.4 & 35.4 & 38.6 & 40.6 & 39.5 \\ y & 26.3 & 33.9 & 29.4 & 23.5 & 31.6 \end{array} $$ a. Does the scatterplot suggest an approximate linear relationship? b. The investigators fit a least-squares line. The resulting MINITAB output is given in the following:. Predict fire-simulation consumption when treadmill consumption is 40 . c. How effectively does a straight line summarize the relationship? d. Delete the first observation, \((51.3,49.3)\), and calculate the new equation of the least-squares line and the value of \(r^{2}\). What do you conclude? (Hint: For the original data, \(\sum x=388.8, \Sigma y=310.3, \sum x^{2}=15,338.54, \sum x y=\) \(12,306.58\), and \(\sum y^{2}=10,072.41\).)

The hypothetical data below are from a toxicity study designed to measure the effectiveness of different doses of a pesticide on mosquitoes. The table below summarizes the concentration of the pesticide, the sample sizes, and the number of critters dispatched. $$ \begin{aligned} &\begin{array}{l} \text { Concentra- } \\ \text { tion }(\mathrm{g} / \mathrm{cc}) \end{array} & 0.10 & 0.15 & 0.20 & 0.30 & 0.50 & 0.70 & 0.95 \\ &\hline \begin{array}{l} \text { Number of } \\ \text { mosquitoes } \end{array} & 48 & 52 & 56 & 51 & 47 & 53 & 51 \\ &\begin{array}{l} \text { Number } \\ \text { killed } \end{array} & 10 & 13 & 25 & 31 & 39 & 51 & 49 \\ &\hline \end{aligned} $$ a. Make a scatterplot of the proportions of mosquitoes killed versus the pesticide concentration. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the concentrations and fit the line \(y^{\prime}=a+b\) (Concentration). What is the significance of a positive slope for this line? c. The point at which the dose kills \(50 \%\) of the pests is sometimes called LD50, for "Lethal dose \(50 \% . "\) What would you estimate to be LD50 for this pesticide and for mosquitoes?

The accompanying data were read from graphs that appeared in the article "Bush Timber Proposal Runs Counter to the Record" (San Luis Obispo Tribune, September 22,2002 ). The variables shown are the number of acres burned in forest fires in the western United States and timber sales. $$ \begin{array}{lrr} & \begin{array}{l} \text { Number of } \\ \text { Acres Burned } \\ \text { (thousands) } \end{array} & \begin{array}{l} \text { Timber Sales } \\ \text { (billions of } \\ \text { board feet) } \end{array} \\ \hline 1945 & 200 & 2.0 \\ 1950 & 250 & 3.7 \\ 1955 & 260 & 4.4 \\ 1960 & 380 & 6.8 \\ 1965 & 80 & 9.7 \\ 1970 & 450 & 11.0 \\ 1975 & 180 & 11.0 \\ 1980 & 240 & 10.2 \\ 1985 & 440 & 10.0 \\ 1990 & 400 & 11.0 \\ 1995 & 180 & 3.8 \\ \hline \end{array} $$ a. Is there a correlation between timber sales and acres burned in forest fires? Compute and interpret the value of the correlation coefficient. b. The article concludes that "heavier logging led to large forest fires." Do you think this conclusion is justified based on the given data? Explain.

The paper "Accelerated Telomere Shortening in Response to Life Stress" (Proceedings of the National Academy of Sciences [2004]: 17312-17315) described a study that examined whether stress accelerates aging at a cellular level. The accompanying data on a measure of perceived stress \((x)\) and telomere length \((y)\) were read from a scatterplot that appeared in the paper. Telomere length is a measure of cell longevity. $$ \begin{array}{rrrl} \begin{array}{l} \text { Perceived } \\ \text { Stress } \end{array} & \begin{array}{l} \text { Telomere } \\ \text { Length } \end{array} & \begin{array}{l} \text { Perceived } \\ \text { Stress } \end{array} & \begin{array}{l} \text { Telomere } \\ \text { Length } \end{array} \\ \hline 5 & 1.25 & 20 & 1.22 \\ 6 & 1.32 & 20 & 1.3 \\ 6 & 1.5 & 20 & 1.32 \\ 7 & 1.35 & 21 & 1.24 \\ 10 & 1.3 & 21 & 1.26 \\ 11 & 1 & 21 & 1.3 \\ 12 & 1.18 & 22 & 1.18 \\ 13 & 1.1 & 22 & 1.22 \\ 14 & 1.08 & 22 & 1.24 \\ 14 & 1.3 & 23 & 1.18 \\ 15 & 0.92 & 24 & 1.12 \\ 15 & 1.22 & 24 & 1.5 \\ 15 & 1.24 & 25 & 0.94 \\ 17 & 1.12 & 26 & 0.84 \\ 17 & 1.32 & 27 & 1.02 \\ 17 & 1.4 & 27 & 1.12 \\ 18 & 1.12 & 28 & 1.22 \\ 18 & 1.46 & 29 & 1.3 \\ 19 & 0.84 & 33 & 0.94 \\ \hline \end{array} $$ a. Compute the equation of the least-squares line. b. What is the value of \(r^{2}\) ? c. Does the linear relationship between perceived stress and telomere length account for a large or small proportion of the variability in telomere length? Justify your answer.

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17,2001\(]: 86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at \(15.4\) percent, and SAT I was last at \(13.3\) percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college \(\mathrm{GPA}\) and \(x=\) high school GPA, what would the value of \(r^{2}\) have been? b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would have been very accurate? Explain why or why not.
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