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Suppose that 10 patients with meningitis received treatment with large doses of penicillin. Three days later, temperatures were recorded, and the treatment was considered successful if there had been a reduction in a patient's temperature. Denoting success by \(S\) and failure by \(\mathrm{F}\), the 10 observations are \(\begin{array}{llllllllll}\text { S } & \text { S } & \text { F } & \text { S } & \text { S } & \text { S } & \text { F } & \text { F } & \text { S } & \text { S }\end{array}\) a. What is the value of the sample proportion of successes? b. Replace each \(S\) with a 1 and each \(F\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(\hat{p}\) ? c. Suppose that it is decided to include 15 more patients in the study. How many of these would have to be S's to give \(\hat{p}=.80\) for the entire sample of 25 patients?

Step by step solution

01

Computing Sample Proportion

Count the number of successes (S's) out of the total sample. It equals to 7 out of 10. The sample proportion of successes \(\hat{p}\) is then \(\hat{p}=\frac{7}{10}=0.70\).

02

Computing Sample Mean and Comparison

Replace each S by 1 and F by 0. Compute the sample mean \(\bar{x}\). For example, the mean of {\[1, 1, 0, 1, 1, 1, 0, 0, 1, 1 \]will also equal to 0.70. It can be observed \(\bar{x} = \hat{p}\)

03

Compute Additional Successes Needed

We want to find out how many more successes (S's) are necessary out of 15 additional patients to make the proportion of successful outcomes 0.80, based on all 25 patients. Let's assume the necessary number of successful cases is x. Thus, we have:\[\frac{7 + x}{25} = 0.80\]Solving for x gives: \(x = 20 - 7 = 13\). So 13 out of 15 new patients need to have successful outcomes

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Key Concepts

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

Binary Outcomes

Binary outcomes refer to two possibilities in an observation or experiment. In our example, the treatment's outcome for a patient is either successful (denoted by S) or unsuccessful (denoted by F).
This simplifies assessments by allowing a clear distinction between two states.
Binary data is often recorded as 1s and 0s, which facilitates easier processing and analysis in statistical computations.

• 1 represents success or the occurrence of an event.
• 0 represents failure or the non-occurrence.

Binary outcomes are foundational in statistics and probability because they allow researchers to quantify and analyze the chance of different outcomes easily.

Sample Mean

In statistics, the sample mean is a way to summarize data by finding the average value of a set of numbers.
When you convert binary outcomes into numerical values (1s and 0s), the sample mean can be calculated by summing all these values and dividing by the total number of observations.
For instance, in our exercise, the sample mean \(\bar{x}\) is based on the binary conversion of the outcomes: \[\bar{x} = \frac{1 + 1 + 0 + 1 + 1 + 1 + 0 + 0 + 1 + 1}{10} = 0.70\].
The sample mean in this context directly correlates to the sample proportion, giving insight into the average level of success across the sample group.

Statistical Computation

Statistical computation is the process of analyzing and interpreting numerical data to gain insights or make predictions.
In our exercise, computing the sample proportion and sample mean are examples of basic statistical computations.
These calculations help summarize the data so we can make informed assumptions or predictions about a larger population.
In statistical computation, finding out how many additional successes are needed to reach a new sample proportion involves equations: \[\frac{7 + x}{25} = 0.80\].
Solving such equations helps determine required conditions or outcomes in future experiments.

Probability

Probability helps us understand the likelihood of different outcomes. In terms of binary outcomes, it gauges the chance of one outcome over the other.
Probability is expressed as a number between 0 and 1, with 0 meaning an event is impossible and 1 meaning it is certain.
The probability of success in our sample (which is also the sample proportion) is \(0.70\), meaning there's a 70% chance of treatment success for a given patient.
In planning experiments or treatments, understanding probability allows for the prediction of outcomes and assessment of risks and benefits.
It is a cornerstone of data-driven decision-making in scientific research and numerous other fields.