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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 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 at the bottom of the page. The experiment would stop after the sixth pair, because Treatment 1 has two more successes than Treatment \(2 .\) The researchers would conclude that Treatment 1 is preferable to Treatment 2 . Suppose that Treatment 1 has a success rate of .7 (that is, \(P(\) success \()=.7\) for Treatment 1\()\) and that Treatment 2 has a success rate of .4. Use simulation to estimate the probabilities requested in Parts (a) and (b). (Hint: Use a pair of random digits to simulate one pair of subjects. Let the first digit represent Treatment 1 and use \(1-7\) as an indication of a success and \(8,9,\) and 0 to indicate a failure. Let the second digit represcnt Treatment \(2,\) with \(1-4\) representing a success. For example, if the two digits selected to represent a pair were 8 and 3 , you would record failure for Treatment 1 and success for Treatment 2. Continue to sclect pairs, kecping 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 five 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?

Step by step solution

01

Assign random digits

Assign random digits to simulate one pair of subjects. For Treatment 1, use 1-7 to indicate a success and 8-0 to indicate a failure. For Treatment 2, use 1-4 to represent a success.

02

Simulate Trials

Initiate a trial. Start picking pairs of numbers and keep track of the total number of successes for each treatment. Once the number of successes for one treatment exceed the other treatment by 2, stop the trial. Repeat this process for at least 20 trials.

03

Calculate Probabilities

Use the simulation results from Step 2 to calculate the two probabilities. To calculate the probability that more than five pairs must be treated before a conclusion can be reached (P(more than 5)), use the formula \(P(\(more than 5)\) = 1 - P(\(5 or fewer)\) .\) For the second probability, count the number of times Treatment 2 was incorrectly concluded as the better treatment from the simulation results and divide it by the total number of trials.

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Key Concepts

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

Probability Estimation

Understanding probability estimation is essential in determining how likely it is for an event to occur based on experimental or theoretical data. Probability estimation involves running experiments or simulations to gather data.
In the context of the medical research example, the probability estimation begins with defining the success rates for Treatment 1 and Treatment 2. Specifically, Treatment 1 has a success probability of 0.7, while Treatment 2 has a probability of 0.4.
To estimate the probability that more than five treatment pairs might be needed to establish a conclusion, researchers use repeated simulation trials. Repeating the experiment multiple times aids in refining the probability estimation by offering a clearer picture of the distribution of outcomes.
Thus, probability estimation is both a powerful tool and a fundamental concept in statistics, helping researchers reach more accurate conclusions.

Random Digit Simulation

Random digit simulation is a technique used to mimic the outcome of random events, which in this case are the results of treatment successes or failures.
By utilizing random digits, researchers can replicate real-world scenarios through a controlled and repeatable method. For Treatment 1, digits 1 to 7 are designated for a success, while 8 to 0 (where 0 represents digits 10) account for a failure. Similarly, Treatment 2 uses 1 to 4 as success indicators.
In executing the simulation, researchers draw pairs of random digits to simulate a pair of treatment results. The sequence of success and failure is recorded throughout the trials, with each pair adding to the cumulative count of successes for each treatment.
Random digit simulation effectively removes bias and allows for focus on probability through the lens of random chance.

Treatment Comparison

Treatment comparison is a critical component of clinical research aiming to determine which treatment option is more effective or preferable.
For this problem, treatments are compared based on their success rates evaluated through simulated trials. The key metric is the difference in numbers of successes between the two treatments. The trials cease when one treatment exceeds the other's success count by two.
This simulation approach allows researchers to compare treatments not through a single test, but through variability across multiple replications of the trial. This reduces the chance of any single anomalous result skewing the overall outcome.

• Objective: Establish superiority of one treatment over another based on empirical success rates.
• Method: Simulate numerous trials for reliable comparison.
• Outcome: The treatment with consistently higher success in the trials is deemed preferable.

Thus, treatment comparison in this exercise culminates in confidently concluding which treatment is statistically better.

Success Rates Analysis

Success rates analysis involves critically evaluating the outcomes of various trials to determine the effective success rate of each treatment.
At its core, it measures the proportion of favorable outcomes, which in this scenario are the successful treatments for the disease.
The procedure begins by cataloging the success for each treatment across multiple simulation trials. Researchers also pinpoint cases where Treatment 2 is incorrectly deemed better, despite having a lower inherent success rate.
Through success rates analysis,

• Results are shown in terms of the frequency of one treatment's success compared to the other.
• Errors (incorrect results) in the decision of treatment preference are meticulously recorded and analyzed.

This analytical process helps strengthen the conclusions drawn from the trials, offering actionable insights into the effectiveness of treatments in the study.