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Chapter 9: Problem 12

Pepcid A study of 74 patients with ulcers was conducted in which they were prescribed \(40 \mathrm{mg}\) of Pepcid \(^{\mathrm{TM}}\). After 8 weeks, 58 reported ulcer healing. (a) Obtain a point estimate for the proportion of patients with ulcers receiving Pepcid who will report ulcer healing. (b) Verify that the requirements for constructing a confidence interval about \(\hat{p}\) are satisfied. (c) Construct a \(99 \%\) confidence interval for the proportion of patients with ulcers receiving Pepcid who will report ulcer healing. (d) Interpret the confidence interval.

Short Answer

Expert verified
The point estimate for the proportion is 0.7838. The 99% confidence interval is (0.6688, 0.8988). This means we are 99% confident that the true proportion of patients reporting ulcer healing is between 66.88% and 89.88%.

Step by step solution

01

Calculate Point Estimate

The point estimate for the proportion of patients with ulcers who report healing is given by \ \ \ \ \( \ \ \hat{p} = \ \ \frac{x}{n} \ \ \) \ \ where: \ \ \( \ \ x \ \ \) = number of patients who reported ulcer healing = 58 \ \ \( \ \ n \ \ \) = total number of patients = 74 \ \ \( \ \ \hat{p} = \ \ \frac{58}{74} \ \ = \ \ 0.7838 \ \ \)
02

Verify Requirements for Confidence Interval

To construct a confidence interval for \( \ \ \hat{p} \ \ \), the following conditions must be met: \ \ 1. The sample must be a simple random sample. \ \ 2. The conditions for the sampling distribution of \( \ \ \hat{p} \ \ \) should be approximately normal, which requires \( \ \ np \ \ \geq 5 \ \ \) and \( \ \ n(1 - p) \ \ \geq 5 \ \ \). \ \ Check: \ \ \( \ \ np \ \ = 74 \ \ \times 0.7838 \ \ = 57.9852 \ \ \approx \ \ 58 \ \ \geq 5 \ \ \) \ \ \( \ \ n(1 - p) \ \ = 74 \ \ \times (1 - 0.7838) \ \ = 74 \ \ \times 0.2162 \ \ = 16.0048 \ \ \approx \ \ 16 \ \ \geq 5 \ \ \) \ \ Both conditions are satisfied.
03

Construct the Confidence Interval

The formula for the confidence interval for a proportion is given by \ \ \( \ \ \hat{p} \ \ \pm z_{\alpha/2} \ \ \sqrt{\ \ \frac{\ \ \hat{p}(1 - \ \ \hat{p})}{n}} \ \ \) \ \ Here, \( \ \ z_{\alpha/2} \ \ \) is the z-score that corresponds to the desired confidence level. For a 99% confidence interval, \( \ \ z_{\alpha/2}\approx2.576 \ \ \). \ \ Calculate the margin of error (E): \ \ \( \ \ E \ \ = z_{\alpha/2} \ \ \sqrt{\ \ \frac{\ \ \hat{p}(1 - \ \ \hat{p})}{n}} \ \ \) \ \ \( \ \ = 2.576 \ \ \sqrt{\ \ \frac{0.7838 \times 0.2162}{74}} \ \ \approx 0.115 \ \ \) \ \ Finally, the confidence interval is: \ \ \( \ \ \hat{p} \ \ \pm E = 0.7838 \pm 0.115 \ \ \) \ \ or \( \ \ (0.6688, 0.8988) \ \ \)
04

Interpret the Confidence Interval

With 99% confidence, the proportion of patients with ulcers receiving Pepcid who will report ulcer healing is between 66.88% and 89.88%.

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Key Concepts

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

headline of the respective core concept
When we talk about a point estimate in statistics, it refers to a single value that serves as an estimate of a population parameter. In this exercise, we are dealing with the point estimate for the proportion of patients who report ulcer healing after being prescribed Pepcid.
headline of the respective core concept
The sampling distribution of a statistic is the probability distribution of that statistic when obtained from a large number of samples of the same size from the same population. To construct a sampling distribution for the proportion \(\hat{p}\), we need the sample to be selected randomly, and the sample size should be large enough for the distribution to be approximately normal. These requirements ensure that the sample proportion is unbiased and accurate in representing the population proportion.
headline of the respective core concept
The term margin of error quantifies the amount by which we expect our estimate (e.g., a sample proportion) to differ from the true population proportion. It provides a range within which we expect the true population parameter to fall. The margin of error is calculated using the critical value (z-score), the sample proportion, and the size of the sample. In this case, the margin of error for a 99% confidence level was found to be approximately 0.115.
headline of the respective core concept
In statistics, a proportion is a type of ratio that compares a part to the whole. It is used to describe the fraction of the population that meets a certain criterion. The proportion of patients reporting ulcer healing is calculated by dividing the number of patients who reported healing by the total number of patients. The proportion in this study was found to be \(\frac{58}{74} = 0.7838\). This is an important measure because it provides a direct estimate of how effective the treatment might be.

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Most popular questions from this chapter

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