### Sample Size Required for Comparing Success Rates of Surgical Procedures#### Problem Statement:Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of these procedures, random samples of patients who underwent each type of surgery were collected. The success metric is defined by the number of patients showing no recurrence of the disease after a one-year period. The recorded data is presented in the table below.##### Table: Success Rates for Surgical Procedures| Procedure | $ n $ | Number of Successes ||------------|---------|---------------------|| Procedure A| 100 | 73 || Procedure B| 100 | 92 |The objective is to determine the necessary sample size required to estimate the difference in the true success rates to within 0.10 with 95% reliability.#### Options:- $ n_1 = n_2 = 73 $- $ n_1 = n_2 = 192 $- $ n_1 = n_2 = 53 $- $ n_1 = n_2 = 104 $To solve this, one needs to apply the formula for sample size estimation for two proportions comparing the difference in success rates, along with considerations for confidence levels and margin of error.

We have, E = 1.96*sqrt[(p1(1-p1)/n1)+(p2(1-p2)/n2)]

Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of the two procedures, random samples of the two types of surgical patients were obtained and the numbers of patients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown in the table. How large a sample would be necessary in order to estimate the difference in the true success rates to within 0.10 with 95% reliability? Number of Successes Procedure A 100 73 Procedure B 100 92

Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of the two procedures, random samples of the two types of surgical patients were obtained and the numbers of patients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown in the table. How large a sample would be necessary in order to estimate the difference in the true success rates to within 0.10 with 95% reliability? Number of Successes Procedure A 100 73 Procedure B 100 92

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Concept explainers

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.

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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.

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### Sample Size Required for Comparing Success Rates of Surgical Procedures

#### Problem Statement:
Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of these procedures, random samples of patients who underwent each type of surgery were collected. The success metric is defined by the number of patients showing no recurrence of the disease after a one-year period. The recorded data is presented in the table below.

##### Table: Success Rates for Surgical Procedures
| Procedure | $ n $ | Number of Successes |
|------------|---------|---------------------|
| Procedure A| 100 | 73 |
| Procedure B| 100 | 92 |

The objective is to determine the necessary sample size required to estimate the difference in the true success rates to within 0.10 with 95% reliability.

#### Options:
- $ n_1 = n_2 = 73 $
- $ n_1 = n_2 = 192 $
- $ n_1 = n_2 = 53 $
- $ n_1 = n_2 = 104 $

To solve this, one needs to apply the formula for sample size estimation for two proportions comparing the difference in success rates, along with considerations for confidence levels and margin of error.

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