Question

An article in the New York Times (March 2, 1994) reported that people who suffer cardiac arrest in New York City have only a 1 in 100 chance of survival. Using probability notation, an equivalent statement would be \(P\) (survival \()=.01\) for people who suffer a cardiac arrest in New York City. (The article attributed this poor survival rate to factors common in large cities: traffic congestion and the difficulty of finding victims in large buildings.) a. Give a relative frequency interpretation of the given probability. b. The research that was the basis for the New York Times article was a study of 2329 consecutive cardiac arrests in New York City. To justify the " 1 in 100 chance of survival" statement, how many of the 2329 cardiac arrest sufferers do you think survived? Explain.

Short answer

a. The probability of \(P=0.01\) means that in the long run, for every 100 people who suffer cardiac arrest in New York City, approximately 1 person is expected to survive. b. Based on this probability, out of 2329 cardiac arrest sufferers in New York City, around 23 are expected to have survived.

Step-by-step solution

Understand the Relative Frequency Interpretation

The relative frequency interpretation of a probability refers to the long-term proportion of a certain event in a series of identical and independent trials. In this context, the survival rate \(P=0.01\) indicates that, in the long run, for every 100 people who suffer cardiac arrest in New York City, 1 person is expected to survive.

Estimate the Number of Survivors

To estimate the number of survivors out of 2329 cardiac arrest cases, multiply the total number of cardiac arrests by the probability of survival. This can be calculated using the formula: \(n \times P\) where \(n\) is the total number of trials (or in this case, the total number of cardiac arrest cases) and \(P\) is the probability of success (or in this case, the survival rate).

Calculation

Perform the calculation: \(2329 \times 0.01= 23.29 \). Since the number of survivors must be a whole number, round down to 23. This indicates that, in this case, approximately 23 people are expected to have survived out of 2329 consecutive cardiac arrest cases in New York City.

Key concepts

Relative Frequency Interpretation

When exploring the concept of probability in statistics, the relative frequency interpretation is fundamental. At its core, this interpretation looks at the frequency with which a particular event occurs over a large number of trials and uses this information to predict the likelihood of that event happening in the future. For instance, if we toss a fair coin 100 times and observe that it lands heads up 55 times, we might interpret the relative frequency of getting a head as 55%.

In the case of the exercise, where the survival probability of cardiac arrest in New York City is given as 1%, the relative frequency interpretation means that historically, out of a similar large number of cases, approximately 1 out of every 100 individuals has survived. This statistical approach is crucial in many fields, including healthcare, where it informs the expected outcomes and is used to guide policy and procedure development to improve survival rates.

However, it is important to note that while this interpretation can help in understanding trends and probabilities, it does not guarantee the exact same outcome in every set of a hundred instances. The nature of probability means there will always be some level of variability, and the actual number could slightly differ from the expected value due to chance.

Probabilistic Modeling

Delving into the realm of probabilistic modeling, it's about creating mathematical models that incorporate randomness and uncertainty to predict outcomes of various scenarios. It serves as a tool to understand and manage the inherent unpredictability in processes or events, like predicting weather patterns, stock market fluctuations, or, as in our textbook exercise, medical survival rates.

Using the probability of survival from the New York Times article, we apply probabilistic modeling by assuming that the chance of survival is consistent across all cardiac arrest cases in New York City. This model simplifies a complex reality—it doesn't consider individual differences or varying circumstances, but rather treats each case as equally likely to adhere to the average probability. The model then becomes useful in estimating the number of people who might survive from a given number of cases. Despite its simplicity, probabilistic modeling requires careful consideration of its assumptions and limitations, as they can greatly influence the accuracy of the model's predictions in real-world applications.

Statistical Analysis

Statistical analysis embodies the process of collecting, exploring, interpreting, and presenting data to discover underlying patterns and trends. It's a cornerstone of decision-making in a wide array of disciplines. In this statistical analysis example involving cardiac arrest cases, the analysis begins with the interpretation of a given probability and proceeds to estimate real-world outcomes based on that probability.

To conduct the statistical analysis in our exercise, we multiply the total number of cases (2329) by the given probability of survival (1%), resulting in an expected value of 23.29 survivors. This is a clear application of statistical analysis—taking probabilistic information and applying it to a real-world quantity to estimate an outcome. We also practice rounding because fractional people don't exist in practical terms, which means applying mathematical principles to produce a result that makes sense in our everyday experience.

Finally, this type of analysis is instrumental in evaluating the effectiveness of interventions, like improving emergency response systems in urban environments to increase cardiac arrest survival rates. By continually applying statistical analysis to new data, we can constantly refine our understanding and enhance the solutions we develop.