Question

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 by colon cancer in virtually 100% of patients before the age of 40. A group of 14 people suffering from FAP and being treated at the Cleveland Clinic drank black raspberry powder in a slurry of water every day for nine months. The number of polyps was reduced in 11 out of 14 of these patients.

Why can’t we use the large-sample confidence interval for the proportion p of patients suffering from FAP who will have the number of polyps reduced after 9 months of treatment?

a. Among the 20 observations, we have 18 successes and 2 failures. The number of successes and failures should both be at least 21 for the Normal approximation to be valid.
b. Among the 14 observations, we have 12 successes and 2 failures. The number of successes and failures should both be at least 18 for the Normal approximation to be valid.
c. Among the 18 observations, we have 15 successes and 3 failures. The number of successes and failures should both be at least 20 for the Normal approximation to be valid.
d. Among the 14 observations, we have 11 successes and 3 failures. The number of successes and failures should both be at least 15 for the Normal approximation to be valid.

Answer 1

The correct answer is d. We cannot use the large-sample confidence interval for the proportion of FAP patients responding to treatment because the sample size is too small with only 11 successes and 3 failures out of 14 observations.

Step-by-step explanation:

The large-sample confidence interval is not suitable for the study of FAP patients because the number of observations is too small therefore, we cannot use the large-sample confidence interval for the proportion of patients suffering from FAP who will have the number of polyps reduced after 9 months of treatment. The plus-four method adjusts the sample size and number of successes to improve the accuracy of confidence intervals when the sample size is small. This technique is especially helpful when conducting scientific studies or polls where estimating a population proportion with less variability is necessary.

This method adds four 'imaginary' observations – two successes and two failures – to help mitigate the error introduced when using point estimates to calculate the standard deviation of the sampling distribution.