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Chapter 6: Problem 23

\(6.23\) A medical research team wishes to evaluate two different treatments for a disease. Subjects are selected two at a time, and then one of the pair is assigned to each of the two treatments. The treatments are applied, and each is either a success (S) or a failure (F). The researchers keep track of the total number of successes for each treatment. They plan to continue the chance experiment until the number of successes for one treatment exceeds the number of successes for the other treatment by \(2 .\) For example, they might observe the results in the table on the next page. The chance experiment would stop after the sixth pair, because Treatment 1 has two more successes than Treatment \(2 .\) Treatment 1 and success for Treatment 2 . Continue to select pairs, keeping track of the cumulative number of successes for each treatment. Stop the trial as soon as the number of successes for one treatment exceeds that for the other by 2 . This would complete one trial. Now repeat this whole process until you have results for at least 20 trials [more is better]. Finally, use the simulation results to estimate the desired probabilities.) a. What is the probability that more than 5 pairs must be treated before a conclusion can be reached? (Hint: \(P\) (more than 5\()=1-P(5\) or fewer \() .)\) b. What is the probability that the researchers will incorrectly conclude that Treatment 2 is the better treatment?

Short Answer

Expert verified
The probabilities for parts a and b will depend on the outcome of the simulation and would differ for different runs. However, in general, we can say that the probability for a is the portion of the runs in which more than 5 pairs were treated before a conclusion, and for b, the portion of trials where Treatment 2 was incorrectly deemed better. These results are obtained from the analysis of the results of your simulation.

Step by step solution

01

Identify variables

We have two treatments, named as 'Treatment 1' and 'Treatment 2'. The success or failure of each treatment on the pairs forms our observations. Our experiment ends when any one treatment has two more successes than the other.
02

Setup Simulation

We can set up a simulation (for instance in a program or online simulation tool) to randomly assign success or failure to both treatments for each pair. Carry on this experiment until one treatment has two more successes than the other. Repeat this for at least 20 trials.
03

Count Trials Exceeding 5 pairs

For each simulation run, count the number of pairs needed to come to a conclusion. Then, find the portion of the trials in which more than 5 pairs are treated before a conclusion is reached. Note that since this is a probability, it will be a value between 0 and 1.
04

Calculate P(more than 5)

This can be calculated by subtracting from 1 the proportion of trials that concluded within 5 pairs (or fewer). This value is the answer for part a.
05

Evaluate Treatment 2's success

Count the total number of trials in which Treatment 2 had more successes than Treatment 1 and divide it by the total number of trials. This gives us the probability of incorrectly concluding (given the same chance of success for both treatments) that Treatment 2 is the better treatment. This is the answer for part b.

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Key Concepts

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

Simulation in Statistics
Simulation in statistics is a powerful tool used to mimic real-world processes through computational models. In medical research, simulations help in understanding complex scenarios without directly impacting actual subjects. For instance, when evaluating two different treatments, simulations can replicate countless scenarios to deduce probable outcomes.

By generating random samples and outcomes for each treatment, a researcher can determine the likelihood of various events. In the exercise provided, simulations are pivotal to create a virtual environment where treatments are assigned and the success or failure is logged. This helps to draw conclusions over multiple imaginary experiments efficiently.
  • Simulations allow handling scenarios that might be impractical in actual settings.
  • They provide insight into potential trends over numerous hypothetical trials.
With their ability to model unobservable quantities, simulations have become an invaluable resource in statistical research, particularly in fields requiring ethical considerations, such as medical studies.
Statistical Trials and Simulation
Statistical trials, especially when coupled with simulation, are essential in evaluating treatments or interventions. Each trial mimics the experiment, providing a snapshot of potential outcomes based on random probability.

In our exercise, a trial continues until the calculated difference in treatment successes is significant. Each pair of subjects represents a single data point, and their results influence the continuation of the trial.
  • A trial ends when one treatment shows a clear advantage with two additional successes over the other.
  • At least 20 trials are needed to ensure that the results truly represent different probability outcomes.
As these trials are repeated through simulations, researchers can identify patterns and derive conclusions on treatment efficacy, ensuring they make evidence-based decisions. This layered approach enhances the robustness and credibility of the statistical analysis.
Probability Estimation in Experiments
Estimating probabilities in experiments, like in the given medical research scenario, involves determining the likelihood of certain outcomes over others. It is crucial in deciding the effectiveness of a treatment.

In the exercise, calculating the probability of needing more than five pairs to decide between treatments (Part a) and misidentifying the superior treatment (Part b) forms the core analysis.
  • Probability gives researchers a quantitative basis to decide how likely an event is to occur.
  • In the simulation, results are collected and analyzed to deduce these probabilities empirically.
The main question in probability estimation is: given the repeated trials, how often does a specific outcome arise? This helps predict real-world outcomes and informs medical decisions. Ultimately, these estimated probabilities play a key role in shaping the conclusions about treatment efficacy and their potential impact on patient care.

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Most popular questions from this chapter

The Los Angeles Times (June 14,1995 ) reported that the U.S. Postal Service is getting speedier, with higher overnight on-time delivery rates than in the past. Postal Service standards call for overnight delivery within a zone of about \(60 \mathrm{mi}\) for any first-class letter deposited by the last collection time posted on a mailbox. Two-day delivery is promised within a 600 -mi zone, and three-day delivery is promised for distances over \(600 \mathrm{mi}\). The Price Waterhouse accounting firm conducted an independent audit by "seeding" the mail with letters and recording on-time delivery rates for these letters. Suppose that the results of the Price Waterhouse study were as follows (these numbers are fictitious, but they are compatible with summary values given in the article): $$\begin{array}{lcc} & \begin{array}{c} \text { Number } \\ \text { of Letters } \\ \text { Mailed } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Letters Arriving } \\ \text { on Time } \end{array} \\ \hline \text { Los Angeles } & 500 & 425 \\ \text { New York } & 500 & 415 \\ \text { Washington, D.C. } & 500 & 405 \\ \text { Nationwide } & 6000 & 5220 \end{array}$$ Use the given information to estimate the following probabilities: a. The probability of an on-time delivery in Los Angeles b. The probability of late delivery in Washington, D.C. c. The probability that both of two letters mailed in New York are delivered on time d. The probability of on-time delivery nationwide

In a small city, approximately \(15 \%\) of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible, and the same individual cannot be called more than once in the same year. What is the probability that an eligible person in this city is selected 2 years in a row? 3 years in a row?

Students at a particular university use a telephone registration system to select their courses for the next term. There are four different priority groups, with students in Group 1 registering first, followed by those in Group 2, and so on. Suppose that the university provided the accompanying information on registration for the fall semester. The entries in the table represent the proportion of students falling into each of the 20 priority-unit combinations. $$\begin{array}{c|ccccc} \begin{array}{c} \text { Priority } \\ \text { Group } \end{array} & \mathbf{0 - 3} & \mathbf{4 - 6} & \mathbf{7 - 9} & \mathbf{1 0 - 1 2} & \begin{array}{c} \text { More } \\ \text { Than 12 } \end{array} \\ \hline \mathbf{1} & .01 & .01 & .06 & .10 & .07 \\ \mathbf{2} & .02 & .03 & .06 & .09 & .05 \\ \mathbf{3} & .04 & .06 & .06 & .06 & .03 \\ \mathbf{4} & .04 & .08 & .07 & .05 & .01 \\ & & & & & \end{array}$$ a. What proportion of students at this university got 10 or more units during the first call? b. Suppose that a student reports receiving 11 units during the first call. Is it more likely that he or she is in the first or the fourth priority group? c. If you are in the third priority group next term, is it likely that you will get more than 9 units during the first call? Explain.

Three friends \((\mathrm{A}, \mathrm{B}\), and \(\mathrm{C})\) will participate in a round-robin tournament in which each one plays both of the others. Suppose that \(P(\mathrm{~A}\) beats \(\mathrm{B})=.7, P(\mathrm{~A}\) beats \(\mathrm{C})\) \(=.8\), and \(P(\) B beats \(C\) ) \(=.6\) and that the outcomes of the three matches are independent of one another. a. What is the probability that \(\mathrm{A}\) wins both her matches and that B beats \(C\) ? b. What is the probability that A wins both her matches? c. What is the probability that A loses both her matches? d. What is the probability that each person wins one match? (Hint: There are two different ways for this to happen. Calculate the probability of each separately, and then add. )

Jeanie is a bit forgetful, and if she doesn't make a "to do" list, the probability that she forgets something she is supposed to do is 1 . Tomorrow she intends to run three errands, and she fails to write them on her list. a. What is the probability that Jeanie forgets all three errands? What assumptions did you make to calculate this probability? b. What is the probability that Jeanie remembers at least one of the three errands? c. What is the probability that Jeanie remembers the first errand but not the second or third?3
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