/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 The Internal Revenue Service pla... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 22: Problem 33

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.

Short Answer

Expert verified
(a) No, MOE is constant. (b) Yes, MOE varies by state.

Step by step solution

01

Understanding Margin of Error in SRS

The margin of error (MOE) in a Simple Random Sample (SRS) of size \( n \) when estimating a proportion is given by the formula \( MOE = z \cdot \sqrt{\frac{p(1-p)}{n}} \), where \( z \) is the z-score for the desired confidence level and \( p \) is the sample proportion. Notice that \( n \), the sample size, is a crucial factor here.
02

Analyzing State Invariance with Fixed Sample Size

When an SRS of 2000 tax returns is selected in each state (like a fixed sample size \( n = 2000 \)), the MOE will not change across states despite varying total tax returns, because the formula for MOE depends only on \( n \) and \( p \). Thus, with the same \( n \) across all states, MOE remains the same.
03

Impact of Proportional Sample Sizes

If an SRS of \(1\%\) of all tax returns is selected in each state, the sample size \( n \) will vary based on the state鈥檚 total number of tax returns. Since the sample size directly affects the MOE, different \( n \) for each state implies that MOE will vary from state to state. Larger states like California will have a larger \( n \) and thus a smaller MOE, while smaller states like Wyoming will have a smaller \( n \) and a larger MOE.
04

Conclusion

(a) The margin of error will not change from state to state when a fixed SRS of 2000 is taken, because \( n \) is constant across states. (b) The margin of error will change from state to state when selecting \(1\%\) of all tax returns, as \( n \) varies with the state's total tax return counts.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Margin of Error
The Margin of Error (MOE) is a measure used to express the uncertainty or potential error in a statistical estimate, typically seen in surveys or polls. It is crucial in understanding how sample results can vary from the actual population parameter. The formula for calculating the MOE when estimating a population proportion is given by:
  • \( MOE = z \cdot \sqrt{\frac{p(1-p)}{n}} \)
In this formula, \( z \) represents the z-score corresponding to the desired confidence level, \( p \) is the sample proportion, and \( n \) is the sample size.

A larger MOE implies greater uncertainty about the estimate's accuracy to the true population proportion. Keeping the sample size consistent across different samples, like using an SRS of 2000 tax returns in each state, means the MOE remains stable across these groups despite varying population sizes.
Population Proportion
The population proportion \( p \) is the fraction of the total population that exhibits a particular characteristic. For instance, in a study seeking to determine what proportion of tax filers claims itemized deductions, \( p \) would be the proportion of the entire population of tax returns from which the sample is drawn.

Understanding and estimating the true population proportion is critical, as it is often an unknown that researchers aim to infer from sample data. In the given exercise, the task is to examine what percentage of tax returns in different states claim itemized deductions.
  • By using a sample, researchers estimate \( p \), introducing the need for statistical tools, such as MOE, to account for potential deviation from the true population proportion.
  • Sampling error can influence this proportion, making MOE an essential aspect of reliable estimations.
Sample Size
Sample size, denoted by \( n \), is a vital element in statistical analysis, particularly when estimating proportions using samples. In the formula for the Margin of Error, \( n \) plays a crucial role: a larger sample size generally results in a smaller MOE, offering a more accurate estimate of the population parameter.

In the context of the exercise:
  • When a fixed sample size (e.g., 2000 tax returns) is used for surveys across states, the variability in total population size in each state does not affect the MOE.
  • However, if the sample size becomes proportional to each state鈥檚 population (e.g., \(1\%\) of all tax returns), then \( n \) varies by state.
This variation in sample size across different populations significantly affects the MOE, demonstrating the impact of \( n \) on statistical estimates.
Proportional Sampling
Proportional sampling refers to the selection of a sample such that it mirrors certain characteristics of the overall population. In this context, it means selecting a sample size that represents a fixed percentage of the total population.

For the exercise involving tax returns:
  • Proportional sampling adjusts the sample size according to the population size鈥攕o for California with nearly 30 million tax returns, \(1\%\) results in a larger sample than for Wyoming, with about 500,000 returns.
  • This approach ensures that the sample reflects the population's structure but also results in varying sample sizes, impacting the MOE for each state.
The use of proportional sampling is marked by balancing the need for representation against the fluctuating precision of estimates due to varying sample sizes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

Canada has much stronger gun control laws than the United States, and Canadians support gun control more strongly than do Americans. A sample survey asked a random sample of 1505 adult Canadians, "Do you agree or disagree that all firearms should be registered?" Of the 1505 people in the sample, 1288 answered either "Agree strongly" or "Agree somewhat."9 (a) The survey dialed residential telephone numbers at random in all 10 Canadian provinces (omitting the sparsely populated northern territories). Based on what you know about sample surveys, what is likely to be the biggest weakness in this survey? (b) Nonetheless, act as if we have an SRS from adults in the Canadian provinces. Give a \(95 \%\) confidence interval for the proportion who support registration of all firearms.

You are planning a survey of students at a large university to determine what proportion favor an increase in student fees to support an expansion of the student newspaper. Using records provided by the registrar, you can select a random sample of students. You will ask each student in the sample whether he or she is in favor of the proposed increase. Your budget will allow a sample of 100 students. (a) For a sample of size 100, construct a table of the margins of error for \(95 \%\) confidence intervals when \(\mathrm{p}^{\wedge \hat{p}}\) takes the values \(0.1,0.3,0.5,0.7\), and \(0.9\). (b) A former editor of the student newspaper offers to provide funds for a sample of size 500 . Repeat the margin of error calculations in part (a) for the larger sample size. Then write a short thank-you note to the former editor describing how the larger sample size will improve the results of the survey.

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.

Although more than \(50 \%\) of American adults believe the maxim that breakfast is the most important meal of the day, only about \(30 \%\) eat breakfast daily. \({ }^{6}\) A cereal manufacturer contacts an SRS of 1500 American adults and calculates the proportion \(\mathrm{p}^{\wedge \hat{p}}\) in this sample who eat breakfast daily. (a) What is the approximate distribution of \(\mathrm{p}^{\wedge \hat{p}}\) ? (b) If the sample size were 6000 rather than 1500 , what would be the approximate distribution of \(\mathrm{p}^{\wedge \hat{p}}\) ?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.