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Chapter 5: Problem 21

The data given in the previous exercise on \(x=\) call-to-shock time (in minutes) and \(y=\) survival rate (percent) were used to compute the equation of the leastsquares line, which was $$ \hat{y}=101.33-9.30 x $$ The newspaper article "FDA OKs Use of Home Defibrillators" (San Luis Obispo Tribune. November \(13 .\) 2002) reported that "every minute spent waiting for paramedics to arrive with a defibrillator lowers the chance of survival by 10 percent." Is this statement consistent with the given least-squares line? Explain.

Short Answer

Expert verified
Yes, the statement made in the newspaper article is reasonably consistent with the given least-squares line because both suggest a decrease in survival rate with an increase in call-to-shock time. The least-squares line suggests a decrease of around 9.3% per minute, while the article suggests a decrease of 10% per minute.

Step by step solution

01

Understand the equation and its components

The given least-squares line is an equation of a line that minimizes the sum of the squared residual between the observed survival rate and the survival rate predicted by the linear equation, which in this case is \(\hat{y}=101.33-9.30x\). Here, \(x\) is the 'call-to-shock' time in minutes, and \(y\) is the survival rate in percent.
02

Interpret the Slope and the Y-intercept

In this least squares line, 101.33 is the y-intercept, which is the expected survival rate when the call-to-shock time (x) is 0. The slope of the line is -9.30, meaning that for every additional minute (increase in x), the survival rate (y) decreases by 9.3%.
03

Compare the Data to the Statement in the Newspaper Article

The newspaper article suggests that every minute spent waiting for the paramedics to arrive with a defibrillator lowers the chance of survival by 10%. The slope of the line derived from the data is -9.3, suggesting a 9.3% decrease for each additional minute. While the two percentages are not exactly the same, they are very close.
04

Make conclusion

Because the slope of the least-squares line (a decrease of 9.3% per minute) is close to the 10% decrease per minute stated in the newspaper article, the statement in the article is reasonably consistent with the given least-squares line.

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Key Concepts

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

Understanding Call-to-Shock Time
Call-to-shock time is a critical component when considering the effectiveness of cardiac emergency response. It refers to the duration from when emergency medical services (EMS) are called to when a defibrillator delivers an electric shock to a cardiac arrest patient. The importance of this metric stems from its direct influence on survival rates.

In the context of the given exercise, we've interpreted data showing how survival rates of patients are affected as call-to-shock time increases. Studies infer that the quicker a defibrillator shock is administered after cardiac arrest, the greater the chance of survival. This factor is so significant that it's often measured and reported in medical studies and health news articles, such as the one referenced in the exercise.

To evaluate the impact of increased call-to-shock time, a least-squares regression analysis is typically employed. This statistical tool allows us to examine the relationship between the time it takes to deliver a shock (independent variable) and patient survival rates (dependent variable). The exercise reveals that every additional minute seems to decrease survival odds, which is consistent with medical expectations and emphasizes the necessity for prompt emergency responses.
Decoding Survival Rate Statistics
Survival rate statistics provide valuable insights into the likelihood of patients surviving a medical condition over a specified period. In the case of cardiac arrest, these statistics are utilized to determine the effectiveness of treatment protocols, like the timely use of defibrillators.

For healthcare professionals and emergency response teams, understanding survival rate statistics is essential for improving interventions and saving lives. Statistics, like those provided in the least-squares regression of the exercise, convey the urgency and importance of rapid medical response.

How Survival Rates are Communicated

Survival rates are often presented as percentages, indicating how many patients out of a hundred survive a condition or make a recovery. In our exercise, the equation suggests that if the call-to-shock time were zero, the model predicts a survival rate of 101.33%, an evidently theoretical value exceeding 100%. However, as time increases, we see a decrease of roughly 9.3% in survival likelihood per minute. While survival rate statistics can sometimes seem abstract, their impact on patients' lives is very real and shapes emergency care practices.
Slope and Y-intercept Interpretation
When interpreting the components of a linear regression equation like the least-squares line, the slope and y-intercept play essential roles. The slope of the line indicates how much the dependent variable (in our case, survival rate) is expected to change with a one-unit change in the independent variable (call-to-shock time). A negative slope, such as the -9.30 in our exercise, signals a decrease in survival rate with each passing minute.

Understanding the Y-Intercept

The y-intercept is the point where the regression line crosses the y-axis, representing the value of the dependent variable when the independent variable is zero. Similarly, in our exercise, the y-intercept of 101.33 represents an idealized survival rate when there is no delay in defibrillation鈥攕uggesting immediate action theoretically results in the highest chance of survival.

The slope and y-intercept not only help us visualize the relationship between variables but also aid in making predictions. For example, if emergency response times improve, we may predict an increase in survival rates based on the established relationship, highlighting the impact of reducing call-to-shock times for cardiac arrest patients.

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Most popular questions from this chapter

The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003 ): \begin{tabular}{cc} Year & Number of Transplants (in thousands) \\ \hline 1 (1990) & \(15.0\) \\ 2 & \(15.7\) \\ 3 & \(16.1\) \\ 4 & \(17.6\) \\ 5 & \(18.3\) \\ 6 & \(19.4\) \\ 7 & \(20.0\) \\ 8 & \(20.3\) \\ 9 & \(21.4\) \\ 10 (1999) & \(21.8\) \\ \hline \end{tabular} a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between \(y=\) number of transplants performed and \(x=\) year. Describe how the number of transplants performed has changed over time from 1990 to \(1999 .\) b. Compute the 10 residuals, and construct a residual plot. Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.

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Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\).
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