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Black Raspberries and Cancer. Sample surveys usually contact large samples, so we can use the large-sample confidence interval if the sample design is close to an SRS. Scientific studies often use smaller samples that require the plus four method. For example, Familial Adenomatous Polyposis (FAP) is a rare inherited disease characterized by the development of an extreme number of polyps early in life and colon cancer in virtually \(100 \%\) of patients before the age of 40 . A group of 14 people suffering from FAP being treated at the Cleveland Clinic drank black raspberry powder in a slurry of water every day for nine months. The numbers of polyps were reduced in 11 out of 14 of these patients. 19 a. Why can't we use the large-sample confidence interval for the proportion \(p\) of patients suffering from FAP that will have the number of polyps reduced after nine months of treatment? b. The plus four method adds four observations: two successes and two failures. What are the sample size and the number of successes after you do this? What is the plus four estimate \(\tilde{p}\) of \(p\) ? c. Give the plus four \(90 \%\) confidence interval for the proportion of patients suffering from FAP who will have the number of polyps reduced after nine months of treatment.

Step by step solution

01

Understand Why Large-Sample Confidence Interval Can't Be Used

The large-sample confidence interval requires a sample size that is large enough so that the sampling distribution of the sample proportion is approximately normal. Typically, the rule of thumb is that both the number of successes and the number of failures should be at least 10. In this case, the sample size is only 14, with 11 successes, which does not meet this rule. Hence, we cannot rely on the large-sample confidence interval.

02

Apply Plus Four Method to Sample Size

For the plus four method, we add two successes and two failures to the data. The original sample had 14 people and 11 successes. Adding the four yields a new sample size: \[ n = 14 + 4 = 18 \]The new number of successes becomes: \[ x = 11 + 2 = 13 \]

03

Calculate Plus Four Estimate, \( \tilde{p} \)

The plus four estimate is calculated by dividing the adjusted number of successes by the adjusted sample size:\[ \tilde{p} = \frac{x}{n} = \frac{13}{18} \approx 0.7222 \]

04

Calculate Standard Error for \( \tilde{p} \)

The standard error (SE) of \( \tilde{p} \) is calculated using the formula:\[ SE = \sqrt{ \frac{\tilde{p} (1 - \tilde{p})}{n} } = \sqrt{ \frac{0.7222 \times (1 - 0.7222)}{18} } \approx 0.1082 \]

05

Determine the Critical Value for 90% Confidence Interval

For a 90% confidence interval using a normal distribution, the critical value \( z^* \) is approximately 1.645.

06

Calculate Margin of Error

The margin of error (MOE) is calculated as:\[ MOE = z^* \times SE = 1.645 \times 0.1082 \approx 0.178 \]

07

Calculate the Plus Four Confidence Interval

The confidence interval is given by:\[ \tilde{p} \pm MOE = 0.7222 \pm 0.178 \]Thus, the 90% confidence interval is approximately \([0.5442, 0.9002]\).

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Key Concepts

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

Plus Four Method

The Plus Four Method is a statistical technique used to adjust small sample sizes when estimating the confidence interval for proportions. In situations where the sample size is too small to satisfy the criteria for a large-sample confidence interval, this method becomes invaluable. It is particularly useful because it adds stability to the estimate by introducing two hypothetical successes and two failures to the existing data.
This adjustment inflates the sample size by four, making it more reliable under conditions where the approximation to the normal distribution would otherwise be questionable. For example, if a study on Familial Adenomatous Polyposis (FAP) uses a very small sample, the Plus Four Method helps provide a more accurate estimation of the proportion by recalibrating this small dataset. In practice, this adjustment allows researchers to derive meaningful confidence intervals even from limited data.

Proportion Estimation

Proportion estimation involves calculating a value that represents the fraction of the sample exhibiting a particular trait or characteristic, such as the success of a treatment. In the context of Familial Adenomatous Polyposis (FAP) treatment studies, researchers are interested in estimating the proportion of patients showing a reduction in polyps.
To achieve this, researchers count the number of successful outcomes (e.g., reduction in polyps) and divide by the total sample size. In situations where data is limited, such as having 14 patients in the study, confidence about the estimate can be improved using methods like the Plus Four. The estimated proportion, \(\tilde{p}\), becomes more stable with these adjustments, leading to a more reliable understanding of treatment effectiveness.

Familial Adenomatous Polyposis

Familial Adenomatous Polyposis (FAP) is a genetic disorder that significantly increases the risk of colon cancer due to the development of numerous polyps in the colon and rectum. It's a rare, inherited condition that often manifests early in life, markedly increasing the likelihood of cancer by age 40 if left untreated.
The rarity and severity of FAP necessitate special statistical methods when conducting research, such as proportion estimation, because sample sizes in studies tend to be small due to limited patient availability. In studies aiming to reduce polyps through various treatments, understanding the genetic and health dynamics of FAP is crucial for interpreting the outcomes and estimates correctly.

Sample Size Calculation

Sample size calculation is a critical step in designing any study or experiment. It determines the number of participants or observations needed to ensure that the results are statistically significant and reliable. In smaller sample studies, such as those involving rare diseases like Familial Adenomatous Polyposis (FAP), standard methods may not apply.
By adding four additional hypothetical observations (two successes and two failures), the Plus Four Method effectively increases the sample size, helping to overcome small sample constraints. This adjusted sample size not only makes the derived estimates more accurate but also enhances the confidence intervals' reliability, drawing more robust conclusions from the data. Proper sample size calculation ensures that the study maintains sufficient power to detect meaningful effects, thereby helping in the accurate prognosis and treatment planning for diseases like FAP.