Question

Studies have shown that people who suffer sudden cardiac arrest have a better chance of survival if a defibrillator shock is administered very soon after cardiac arrest. How is survival rate related to the length of time between cardiac arrest and the defibrillator shock being delivered? This question is addressed in the paper "Improving Survival from Sudden Cardiac Arrest: The Role of Home Defibrillators" (www.heartstarthome.com). The accompanying data give \(y=\) survival rate (percent) and \(x=\) mean call-to-shock time (minutes) for a cardiac rehabilitation center (in which cardiac arrests occurred while victims were hospitalized and so the call-to-shock time tended to be short) and for four other communities of different sizes. Mean call-to-shock time, \(x\) 1 12 30 \begin{tabular}{l} \hline \end{tabular} Survival rate, \(y\) 90 45 5 a. Find the equation of the least squares line. b. Interpret the slope of the least squares line in the context of this study. c. Does it make sense to interpret the intercept of the least squares regression line? If so, give an interpretation. If not, explain why it is not appropriate for this data set. d. Use the least squares line to predict survival rate for a community with a mean call-to-shock time of 10 minutes.

Short answer

a. The equation of the least squares line is obtained by using the calculated slope and y-intercept in the equation \( y = mx + b \). b. The slope of the line represents the decrease in survival rate with each minute increase in mean call-to-shock time. c. The y-intercept represents the survival rate when the mean call-to-shock time is zero. However, because a mean call-to-shock time of zero is not practical in real world situations, the y-intercept might not be a meaningful number in this context. d. The predicted survival rate for a mean call-to-shock time of 10 minutes can be found by substituting \( x = 10 \) in the equation of the line.

Step-by-step solution

Plot the data points and calculate the slope of the least squares line

First, plot the given data points on a graph. Then calculate the slope of the least squares line using the formula: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] where \( n \) is the number of data points, \( x \) represents the mean call-to-shock time, and \( y \) is the survival rate. The sums of \( xy \), \( x \), \( y \), \( x^2 \), and \( x \) squared are calculated using the given data.

Calculate the y-intercept of the least squares line

Next, calculate the y-intercept of the least squares line using the formula: \[ b = \frac{\sum y - m(\sum x)}{n} \] where \( m \) is the slope of the line calculated in the previous step, \( x \) and \( y \) are as defined previously, and \( n \) is the number of data points.

Interpret the slope and y-intercept

The slope of the line represents the change in survival rate for each minute increase in mean call-to-shock time. The y-intercept represents the predicted survival rate when the mean call-to-shock time is zero.

Predict the survival rate for a mean call-to-shock time of 10 minutes

Using the least squares line equation \( y = mx + b \), substitute \( x = 10 \) to predict the survival rate for a mean call-to-shock time of 10 minutes.

Key concepts

Survival Rate Analysis

Survival rate analysis is a crucial tool in understanding and improving outcomes in healthcare, particularly in emergency situations like sudden cardiac arrest. In our exercise, the survival rate refers to the percentage of people who survive a cardiac arrest. The analysis involves examining the relationship between survival after cardiac arrest and various intervening factors, one of which is the call-to-shock time.

The data from the study provided in the exercise enables us to conduct a statistical analysis to see how the time it takes to deliver a defibrillation shock affects the chances of survival. By plotting these data points and finding a pattern, we can draw conclusions that could be pivotal for emergency response practices. For instance, a negative correlation between the call-to-shock time and survival rate would suggest that faster responses could increase survival chances, highlighting the importance of quick action in such cases.

Call-to-Shock Time

Call-to-shock time is a critical metric in the assessment of cardiac arrest cases. It is defined as the duration between the call for help (typically made immediately after a cardiac arrest is identified) and the administration of an electric shock using a defibrillator.

In the context of the exercise, a shorter call-to-shock time is hypothesized to correlate with higher survival rates. With each minute that passes, the survival rate is expected to diminish, as indicated by the negative slope in the least squares regression analysis. This timeframe is a pivotal factor in emergency medical services and underscores the need for rapid response and readily accessible defibrillation equipment in public spaces and homes.

Statistical Data Interpretation

Data interpretation is fundamental in making sense of numerical findings and drawing practical conclusions. In the analysis of the call-to-shock time study, statistical data interpretation involves processing the provided call-to-shock times and the corresponding survival rates.

Through methods like the least squares regression, we quantify the relationship between the two variables. The slope derived from the regression tells us the rate at which survival chances decrease as the call-to-shock time increases. Meanwhile, the y-intercept gives us an extrapolated survival rate for when the call-to-shock time is hypothetically zero. Interpreting this information requires understanding that, while the intercept offers a theoretical starting point, it may not always be meaningful in a real-world context if it doesn't align with feasible scenarios. Further interpretation might involve assessing the strength of the relationship between the variables and evaluating how well the regression line fits the data.

Linear Regression Prediction

Linear regression prediction utilizes an established relationship between two variables to estimate or predict the outcome for a given input. In our exercise, after calculating the equation of the least squares regression line, we use it to predict the survival rate for a community with a mean call-to-shock time of 10 minutes.

The equation, represented as 'y = mx + b', where 'm' is the slope and 'b' is the y-intercept, provides a means to estimate the survival rate (y) for any given call-to-shock time (x). This prediction can inform community health officials and emergency planners about the potential impact of improving response times and can ultimately contribute to saving lives.