/*! 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 37 A researcher for the U.S. Depart... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 37

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with \(98 \%\) confidence if (a) he uses a 2006 estimate of \(15 \%\) obtained from a Coinstar National Currency Poll? (b) he does not use any prior estimate?

Short Answer

Expert verified
a) 1727, b) 3393.

Step by step solution

01

Identify the Given Information and Required Formula

For both parts (a) and (b), the margin of error (E) is 0.02 (2 percentage points), the confidence level is 98%, the critical value for 98% confidence is 2.33 (from the Z-table), and the formula for the sample size (n) is \[ n = \frac{{Z^2 \times p \times (1 - p)}}{{E^2}} \]
02

Step 2a: Calculate Sample Size Using the Prior Estimate

For part (a), use the prior estimate from 2006, where \( p = 0.15 \): \[ n = \frac{{2.33^2 \times 0.15 \times (1 - 0.15)}}{{0.02^2}} \] Calculate as follows: \[ n = \frac{{5.4289 \times 0.15 \times 0.85}}{{0.0004}} \] \[ n = \frac{{0.6905}}{{0.0004}} \] \[ n 鈮� 1726.25 \] Since the sample size needs to be a whole number, round up to 1727.
03

Step 2b: Calculate Sample Size Without a Prior Estimate

For part (b), use \( p = 0.5 \) (the most conservative estimate): \[ n = \frac{{2.33^2 \times 0.5 \times (1 - 0.5)}}{{0.02^2}} \] Calculate as follows: \[ n = \frac{{5.4289 \times 0.25}}{{0.0004}} \] \[ n = \frac{{1.3572}}{{0.0004}} \] \[ n 鈮� 3393 \]

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
Imagine you are trying to estimate the percentage of people who support abolishing the penny. A margin of error helps gauge how close your survey results are likely to come to the true population value. For example, if the margin of error is 2%, and your survey says 15% support abolishing the penny, the true value is likely between 13% and 17%.
  • It represents the range of the estimation error.
  • It is denoted as E in formulas.
  • The formula for margin of error in a proportion estimate is \(E = Z \times \sqrt{\frac{p(1-p)}{n}}\)
The margin of error depends not just on sample size but also on the level of confidence and variability in the data. A smaller margin of error means more precision but often requires a larger sample size.
confidence level
The confidence level is another crucial concept in statistics. It indicates how certain you are that the true value lies within your margin of error.
  • A 98% confidence level means you are 98% sure the true percentage of people who support abolishing the penny is within your 2% margin of error.
  • Higher confidence levels lead to larger margins of error and require larger sample sizes.
  • Common confidence levels are 90%, 95%, and 99%.
The formula to find the required sample size combines the confidence level, margin of error, and the proportion of the population. For example, if your confidence level is 98%, then your critical value (Z-score) is 2.33, taken from statistical Z-tables.
critical value
The critical value is a key number used in estimating sample size. It is derived from the confidence level and shows how many standard deviations away from the mean you need to look.
  • For a 98% confidence level, the critical value is 2.33.
  • This value comes from a Z-table in statistics.
  • The critical value affects the margin of error: higher confidence levels mean larger critical values and larger required sample sizes.
The critical value is found using Z-tables and is part of the formula for calculating sample size: \(n = \frac{Z^2 \times p \times (1 - p)}{E^2}\). By understanding and using the correct critical value, margin of error, and confidence level, one can accurately estimate the required sample size for surveys.

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

A researcher wishes to estimate the proportion of households that have broadband Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with \(99 \%\) confidence if (a) she uses a 2009 estimate of 0.635 obtained from the National Telecommunications and Information Administration? (b) she does not use any prior estimates?

In a survey conducted by the marketing agency 11 mark, 241 of 1000 adults 19 years of age or older confessed to bringing and using their cell phone every trip to the bathroom (confessions included texting and answering phone calls). (a) What is the sample in this study? What is the population of interest? (b) What is the variable of interest in this study? Is it qualitative or quantitative? (c) Based on the results of this survey, obtain a point estimate for the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (d) Explain why the point estimate found in part (c) is a statistic. Explain why it is a random variable. What is the source of variability in the random variable? (e) Construct and interpret a \(95 \%\) confidence interval for the population proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (f) What ensures that the results of this study are representative of all adults 19 years of age or older?

A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates? (c) Why are the results from parts (a) and (b) so close?

An urban economist wishes to estimate the proportion of Americans who own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with \(90 \%\) confidence if (a) he uses a 2010 estimate of 0.669 obtained from the U.S. Census Bureau? (b) he does not use any prior estimates?

A \(90 \%\) confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) \(90 \%\) of full-time college students sleep between 7.8 hours and 8.8 hours. (b) We are \(90 \%\) confident that the mean number of hours of sleep that full- time college students get any day of the week is between 7.8 hours and 8.8 hours. (c) There is a \(90 \%\) probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours. (d) We are \(90 \%\) confident that the mean hours of sleep that fulltime college students get during a weekday is between 7.8 hours and 8.8 hours.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.