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Chapter 22: Problem 17

Based on the sample, the large-sample \(90 \%\) confidence interval for the proportion of all American adults who actively try to avoid drinking regular soda or pop is (a) \(0.611 \pm 0.020\). (b) \(0.611 \pm 0.025\). (c) \(0.611 \pm 0.031\).

Short Answer

Expert verified
The answer is (b) \(0.611 \pm 0.025\).

Step by step solution

01

Understanding the Problem

We need to find the correct confidence interval for the given proportion. The confidence interval is given as a point estimate plus/minus the margin of error.
02

Identifying the Components of a Confidence Interval

The confidence interval is represented as \( \hat{p} \pm E \), where \( \hat{p} \) is the sample proportion and \( E \) is the margin of error. Here, \( \hat{p} = 0.611 \).
03

Decoding the Options

Each option is presented in the format \( 0.611 \pm x \), where \( x \) is the margin of error. We need to select the option where the expression matches the confidence interval stated in the problem.
04

Selecting the Correct Option

Compare the given options:\( 0.611 \pm 0.020 \), \( 0.611 \pm 0.025 \), and \( 0.611 \pm 0.031 \). Determine which one fits the correct confidence interval description by considering the typical range of a 90% confidence interval margin for the proportion.
05

Final Decision

The correct option is \( 0.611 \pm 0.025 \), as this is a reasonable margin of error for a 90% confidence interval with a given point estimate close to 0.6.

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Key Concepts

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

Proportion
In statistics, a proportion is a type of ratio that compares a part to the whole. For instance, when we talk about the proportion of American adults who avoid drinking soda, we're comparing the number of adults who avoid soda to the total number of American adults. Understanding proportions is crucial to interpret data in a meaningful way.
A proportion is often expressed as a fraction, decimal, or percentage. Here, it is represented as decimal such as the sample proportion of 0.611 in our example. This means that about 61.1% of the sample group avoids drinking soda.
It's important to note that proportions offer insights into data trends and are fundamental in constructing confidence intervals for predictions about a population. By analyzing sample data, we can make educated inferences about the larger group based on the observed proportion.
Margin of Error
The margin of error is a vital part of interpreting confidence intervals in statistics, providing a range within which the true population proportion is expected to lie. It quantifies the uncertainty or variability inherent in the sample estimate.
For our example, the margin of error was given in options such as 0.020, 0.025, and 0.031. It is generally expressed as a plus or minus value (卤) attached to the sample proportion, which creates an interval demonstrating potential fluctuation in the data.
The smaller the margin of error, the more precise the estimate of the population proportion is. A 90% confidence interval like the one in our example means that we are 90% confident that the true proportion of all American adults avoiding soda is within the margin of error range around the point estimate.
Sample Proportion
The sample proportion is a statistical estimate that provides an approximation of the true proportion of the population. It is symbolized as \( \hat{p} \) and calculated by dividing the number of successful outcomes by the total number of trials in a sample.
In our example, the sample proportion \( \hat{p} \) is 0.611, meaning 61.1% of the sample surveyed tries to avoid drinking soda.
This measure is a cornerstone of inferential statistics, as it serves as the best estimate for the population proportion when constructing a confidence interval. It plays a crucial role by serving as the reference point from where the upper and lower bounds of the confidence interval are calculated, allowing researchers to make predictions about an entire population based on a sample.

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Most popular questions from this chapter

A study investigated ways to prevent staph infections in surgery patients. In a first step, the researchers examined the nasal secretions of a random sample of 6771 patients admitted to various hospitals for surgery. They found that 1251 of these patients tested positive for Staphylococcus aureus, a bacterium responsible for most staph infections. \({ }^{4}\) (a) Describe the population and explain in words what the parameter \(p\) is. (b) Give the numerical value of the statistic \(\mathrm{p}^{\wedge \hat{p}}\) that estimates \(p\).

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. \({ }^{18}\) (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 \(\mathrm{p}^{-\bar{p}}\) 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.

The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each state. One variable of interest is the proportion of returns claiming itemized deductions. The total number of tax returns in a state varies from almost 30 million in California to approximately 500,000 in Wyoming. (a) Will the margin of error for estimating the population proportion change from state to state if an SRS of 2000 tax returns is selected in each state? Explain your answer. (b) Will the margin of error change from state to state if an SRS of \(1 \%\) of all tax returns is selected in each state? Explain your answer.

\(\hat{p}\). Greenville County, South Carolina, has 461,299 adult residents, of which 59,969 are 65 years or older. A survey wants to contact \(n=689\) residents. \({ }^{5}\) (a) Find \(p\), the proportion of Greenville county adult residents who are 65 years or older. (b) If repeated simple random samples of 689 residents are taken, what would be the range of the sample proportion of adults over 65 in the sample according to the 95 part of the 68-95-99.7 rule? (c) The actual survey contacted 689 adults using random digit dialing of residential numbers using a database of exchanges, with no cell phone numbers contacted. The 689 respondents represent a response rate of approximately \(30 \%\). In the sample obtained, 253 of the 689 adults contacted were over 65 . Do you have any concerns treating this as a simple random sample from the population of adult residents of Greenville County? Explain briefly.

In Exercise 22.17, suppose we computed a large-sample 99\% confidence interval for the proportion of all American adults who actively try to avoid drinking regular soda or pop. This \(99 \%\) confidence interval (a) would have a smaller margin of error than the \(90 \%\) confidence interval. (b) would have a larger margin of error than the \(90 \%\) confidence interval. (c) could have either a smaller or a larger margin of error than the \(90 \%\) confidence interval. This varies from sample to sample.
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