91Ó°ÊÓ

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 for Exercise \(6.82\) given below. The chance experiment would stop after the sixth pair, because Treatment 1 has 2 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 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 represent 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 select pairs, keeping track of the total 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. Estimate 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. Estimate the probability that the researchers will incorrectly conclude that Treatment 2 is the better treatment.

Step by step solution

01

Understanding the Simulation

Each digit 1-7 represents a success in Treatment 1 with a probability of 0.7 and each digit 1-4 represents a success in Treatment 2 with a probability of 0.4. Thus, establish a table to record the results with three columns: Pair, Treatment 1, and Treatment 2. This is where results for each simulation trial will be recorded.

02

Run the Simulation

Randomly select two digits to simulate one pair of subjects in each round, where the first digit represents Treatment 1 and the second represents Treatment 2. Record 'S' for success and 'F' for failure corresponding to each Treatment in the table accordingly. Repeat this process until the number of successes for one treatment exceeds that of the other by 2. This completes one trial. Repeat this whole process for at least 20 trials. The more trials conducted, the more accurate the result, as per the Law of Large Numbers.

03

Estimate Probability for Part (a)

After all the trials, count in how many trials more than five pairs had to be treated before a conclusion could be reached. The probability is estimated as \( P(\) more than 5 pairs\() = \frac{\text{No. of trials with more than 5 pairs}}{\text{Total no. of trials}} \). Alternatively, this could be calculated as 1 minus the probability of 5 or fewer pairs.

04

Estimate Probability for Part (b)

Similarly, count how many times the researchers incorrectly conclude that Treatment 2 is the better treatment. The probability estimate would be \( P(\) Treatment 2 better\() = \frac{\text{No. of trials where Treatment 2 is better}}{\text{Total no. of trials}} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Probability Estimation

Probability estimation is a fundamental concept in statistics where we determine the likelihood of a particular outcome occurring. In the context of our medical research scenario, it involves estimating the chances that one of two treatments will be deemed superior based on their success rates. This process requires a rigorous approach wherein we set up a simulation to mimic the likely outcomes of a real-world experiment.

To estimate the probabilities in the medical study, we use a random digit method. Each digit from 1-7 is mapped to success for Treatment 1, reflecting its success rate of 0.7. Similarly, digits 1-4 represent success for Treatment 2, given its success rate of 0.4. This mapping is key because it allows the simulation to closely replicate the true probability of each treatment's success in an actual trial. By running the simulation multiple times—ideally more than 20 trials—we gather enough data to calculate a reliable estimation of the probability for each treatment's outcome.

Experimental Design

Experimental design refers to planning how to conduct an experiment and gather data in a way that leads to reliable and interpretable results. In the medical study example, the design involves selecting subject pairs and administering the two treatments, recording successes and failures until one treatment's success exceeds the other's by 2.

To translate this into a simulation, we use a structured approach, random digit selection to represent treatment success or failure. The use of this method allows an accurate reflection of the real-world randomness and variability that would be present in a true medical trial. Keeping track of outcomes in a systematically designed table is crucial as it ensures clarity and ease of analysis once the simulation is complete. An effectively designed experiment, simulated or otherwise, is the backbone of quality data collection and consequential analysis, leading to sound conclusions.

Success Rate Analysis

Success rate analysis involves examining the outcomes of treatments or trials to determine which one is more effective. In the context of our simulation, success rate analysis is conducted by comparing the accumulated successes of the two treatments after each simulated trial. By tracking these results, researchers can evaluate which treatment consistently performs better.

The analysis comes down to numerically summarizing the trials, which involves counting the number of times each treatment outcome occurs under simulated conditions. Assessing the success rate gives us an objective measure to compare treatments, which in this case, could lead to identifying the preferred treatment for the disease. Moreover, the analysis includes identifying scenarios where the researchers may incorrectly conclude the effectiveness of a treatment, which is an essential aspect of error evaluation in clinical research.