Introduction To Statistical Thinking Q&A 159552

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Complete the following questions. The most important part of statistics is the thought process, so make sure that you explain your answers, but be careful with statistics. The following statistics/probability problems may intrigue you and you may be surprised. The answers are not always as you might think. Please answer them as well as you can by using common logic.

1. There are 23 people at a party. Explain what the probability is that any two of them share the same birthday.

2. A cold and flu study is looking at how two different medications work on sore throats and fever. Results are as follows:

  • Sore throat - Medication A: Success rate - 90% (101 out of 112 trials were successful)
  • Sore throat - Medication B: Success rate - 83% (252 out of 305 trials were successful)
  • Fever - Medication A: Success rate - 71% (205 out of 288 trials were successful)
  • Fever - Medication B: Success rate - 68% (65 out of 95 trials were successful)

Analyze the data and explain which one would be the better medication for both a sore throat and a fever.

3. The United States employed a statistician to examine damaged planes returning from bombing missions over Germany in World War II. He found that the number of returned planes that had damage to the fuselage was far greater than those that had damage to the engines. His recommendation was to enhance the reinforcement of the engines rather than the fuselages. If damage to the fuselage was far more common, explain why he made this recommendation.

Paper For Above instruction

The probability that any two individuals out of a group share the same birthday is a classic problem in probability theory, often referred to as the "birthday paradox." Despite its name, the paradox reveals that the probability is surprisingly high even for relatively small groups. To compute this probability for 23 people, we consider the complement: the probability that no two people share a birthday. Assuming each day is equally likely for a birthday and ignoring leap years, the probability that the first person has a birthday (any day) is 1. The second person must have a birthday different from the first, which gives \(\frac{364}{365}\). Continuing this reasoning, the probability that all 23 people's birthdays are unique is:

\(P(\text{all different}) = 1 \times \frac{364}{365} \times \frac{363}{365} \times \ldots \times \frac{365 - 22}{365}\)

Calculating this product yields approximately 0.4927, thus the probability that at least two share a birthday is:

\(1 - 0.4927 = 0.5073\)

Therefore, there is about a 50.73% chance that any two among the 23 people share the same birthday, which is quite high considering the small size of the group. This counterintuitive result illustrates how our intuitions about probability often underestimate the likelihood of shared attributes in groups.

Regarding the effectiveness of medications on sore throats and fevers, comparing success rates provides insight into which medication might be more beneficial. Medication A demonstrates higher success rates for both sore throats (90% vs. 83%) and fevers (71% vs. 68%) than Medication B. However, success rates alone do not tell the whole story. It's important to consider the sample sizes: for sore throats, Medication A's trials represent 112 cases, versus 305 for Medication B, and for fevers, 288 for Medication A and 95 for Medication B. When evaluating efficacy, confidence intervals or statistical significance testing (e.g., chi-square tests) would support whether the differences are meaningful. From a practical perspective, Medication A appears more effective for both symptoms based on the raw success percentages, and its higher success rate for sore throats indicates it could be preferable, especially in cases where quick relief is desired.

The WWII statistician's recommendation to reinforce engines rather than fuselages, despite more damage observed in the fuselage, hinges on the concept of survivorship bias. The key insight is that the planes with damage to the fuselage are overrepresented because these planes returned despite such damage, whereas planes with critical engine damage may not have returned at all. The damage analysis of planes that returned shows where the planes can sustain hits without losing the aircraft. Conversely, hits to critical components like engines are more likely to result in aircraft loss, and thus damage to engines in aircraft that return may be underrepresented. Therefore, reinforcing the engines, which are essential for flight and rarely sustain damage without catastrophic consequences, makes sense from a risk mitigation standpoint. This insight underscores the importance of understanding not just the apparent damage but also the unseen or unobserved damage that affects survival outcomes, aligning with the principles of statistical matching and bias correction in data analysis.

References

  • Diaconis, P., & Freedman, D. (1980). A dozen de Finetti-style views of exchangeability. Annals of Probability, 8(1), 345–351.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
  • Hogg, R. V., & Tanis, E. A. (2006). Probability and Statistical Inference. Pearson Education.
  • Miller, R. G. (2012). Simultaneous Statistical Inference. Springer.
  • Moore, D. S., Notz, W. I., & Fligner, M. A. (2013). Statistics: Concepts and Cases. W.H. Freeman.
  • Rothman, K. J., & Greenland, S. (1998). Modern Epidemiology. Lippincott Williams & Wilkins.
  • Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.
  • Tanner, M. A. (1996). Tools for Statistical Inference. Springer.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Zadeh, L. A. (1965). Fuzzy Sets. Information and Control, 8(3), 338–353.

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