91Ó°ÊÓ

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.

Step by step solution

01

Compute the statistics

Calculate the following statistics: \ sum of \(x\), \ sum of \(y\), \ sum of \(xy\), \ sum of \(x^2\), \ sum of \(y^2\), \ and the total number of observations \(n\), using the given data.

02

Compute the slope of the least-squares line

To calculate the slope of the least-squares line \(b\) use the following formula: \[ b = \frac{n \sum(xy) - \sum(x) \sum(y)}{n \sum(x^2) - (\sum(x))^2} \]

03

Compute the y-intercept of the least-squares line

To calculate the y-intercept \(a\) of the least-squares line use the formula: \[ a = \frac{\sum(y) - b \sum(x)}{n} \]

04

Write the equation of the least-squares line

The equation of the line is given by \(y = a + bx\), with \(a\) as the y intercept computed in step 3 and \(b\) as the slope computed in step 2.

05

Compute the value of \(r^{2}\)

The formula for \(r^{2}\) is given by \[ r^{2} = \frac{(n \sum(xy) - \sum(x) \sum(y))^2}{(n \sum(x^2) - (\sum(x))^2)(n \sum(y^2) - (\sum(y))^2)} \] where \( \sum(x)\) is the sum of \(x\), \( \sum(y)\) is the sum of \(y\) and \( \sum(xy)\) is the sum of the product of \(x\) and \(y\) respectively.

06

Interpretation of \(r^{2}\)

To interpret \(r^{2}\) value, remember it signifies the proportion of the variance in the dependent variable that is predictable from the independent variable(s). A low \(r^{2}\) indicates that the linear relationship does not account for much of the variability in the dependent variable, while a high \(r^{2}\) indicates the opposite.

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Key Concepts

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

Perceived Stress

Perceived stress is a subjective measure of how stressful you find your life. This concept is important in understanding behavioral responses and impacts on physical health. Researchers often utilize questionnaires to assess perceived stress levels among various subjects. These questionnaires typically include questions focused on mindfulness of stress within the last month, which allows individuals to share their personal experiences. The perceived stress index then quantifies this data, producing numerical values that can be used for statistical analyses, such as in our study concerning telomere length. Understanding perceived stress is fundamental to psychological studies because it helps researchers identify correlations between stress and various health outcomes. For instance, high levels of perceived stress have frequently been linked to negative health effects, demonstrating important implications for both mental and physical well-being.

Telomere Length

Telomere length refers to the length of the protective caps found at the ends of DNA strands within our cells. Often considered a marker of cellular aging, telomeres naturally shorten as cells divide over time. Shorter telomeres have been associated with various age-related diseases and early mortality. In the context of stress and cellular aging, telomere length becomes an essential factor. Studies suggest that chronic stress may accelerate the shortening of telomeres, thus accelerating the aging process at the cellular level. Long telomeres typically indicate a healthier and potentially longer lifespan, as they enable cells to continue dividing without losing vital genetic information. Therefore, maintaining telomere length can be a significant focus in preventative healthcare strategies. Researchers might pursue interventions that potentially improve or maintain telomere length, such as lifestyle changes, stress reduction techniques, and nutritional adjustments. This highlights why studying telomere length in response to perceived stress is so significant.

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), quantifies the degree to which two variables are related. This statistic ranges from -1 to 1, where:- 1 indicates a perfect positive linear relationship,- 0 indicates no linear relationship,- -1 indicates a perfect negative linear relationship.In the context of perceived stress and telomere length, the correlation coefficient helps determine whether higher stress correlates with shorter telomeres. This information can be incredibly valuable, as it provides insights into whether stress management could potentially influence aging at the cellular level.Also, by calculating the correlation coefficient, researchers can assess the strength and direction of their data's relationships. In real-world applications, this means determining if a high level of perceived stress is likely to accompany a significant decrease in telomere length.Thus, the correlation coefficient offers a concise metric to evaluate the association between complex variables like stress and physical health outcomes.