/*! 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 59 An Associated Press article on p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 59

An Associated Press article on potential violent behavior reported the results of a survey of 750 workers who were employed full time (San Luis Obispo Tribune, September 7 , 1999). Of those surveyed, 125 indicated that they were so angered by a coworker during the past year that they felt like hitting the coworker (but didn't). Assuming that it is reasonable to regard this sample as representative of the population of full-time workers, use this information to construct and interpret a \(90 \%\) confidence interval estimate of \(p,\) the proportion of all full-time workers so angered in the last year that they wanted to hit a coworker.

Short Answer

Expert verified
The \(90 \%\) confidence interval estimate of the proportion of full-time workers who wanted to hit a coworker out of anger in the last year is \([0.1407, 0.1927]\), or equivalently, between 14.07% and 19.27%.

Step by step solution

01

Calculate the Sample Proportion (\(p\))

The sample proportion (\(p\)) is calculated as the number of successful outcomes (workers who wanted to hit a coworker) divided by the total number of outcomes (total workers surveyed). So, \(p = 125/750 = 0.1667\)
02

Calculate the Standard Error (SE)

The Standard Error (SE) for a proportion can be calculated using the formula \(\sqrt{p(1-p)/n}\), where \(n\) is the sample size. Substituting \(p = 0.1667\) and \(n = 750\), we find \(SE = \sqrt{0.1667 * (1 - 0.1667) / 750} = 0.0137\).
03

Determine the Z-Score for a 90% Confidence Level

A 90% confidence level corresponds to a z-score of approximately 1.645. This value can be found in standard z-tables which provides the z-score associated with the desired level of confidence.
04

Calculate the Confidence Interval

The confidence interval can be calculated using the formula \(p 卤 (z*SE)\), where \(z\) is the z-score and SE is the standard error. So, the confidence interval becomes \(0.1667 卤 (1.645*0.0137) = [0.1407, 0.1927]\). This means that we can be 90% confident that the true proportion of all full-time workers who got so angered in the past year that they wanted to hit a coworker lies between 14.07% and 19.27%.

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.

Understanding Sample Proportion
Sample proportion is a statistical estimate of a particular characteristic within a population based on a subset, or sample, of that population. In survey analysis, it represents the fraction of the sample with the desired trait.
For the Associated Press survey, the sample proportion is the number of workers who felt angry enough to hit a coworker divided by the total number of workers surveyed.
This calculation, which resulted in a sample proportion of approximately 0.1667 or 16.67%, provides an estimate of the proportion of all full-time workers feeling similarly.
Remember:
  • The closer the sample proportion is to what you might expect from the entire population, the more reliable the results.
  • Sample size is crucial: larger samples generally provide more accurate estimates.
  • It should always reflect the particular aspect being measured, such as emotion in this case study.
Role of Standard Error
The standard error (SE) measures how much we expect the sample proportion to fluctuate from the true population proportion due to random sampling.
It provides a way to gauge the precision of the sample proportion estimate, calculated using the formula: \[SE = \sqrt{\frac{p(1-p)}{n}}\]where \(p\) is the sample proportion, and \(n\) is the sample size.
In our survey example:
  • \(p = 0.1667\), representing 16.67% of workers.
  • \(n = 750\), the total number sampled.
  • The standard error turns out to be 0.0137.
This low standard error indicates that our sample proportion is a reliable estimate of the population proportion.
The Importance of the Z-Score
A z-score helps in identifying how far away a particular measure in a sample is from the mean of the population.
In constructing confidence intervals, it represents the number of standard errors you need to go from the sample proportion to capture a specific percentage of the population probability.
For a 90% confidence level, our z-score is 1.645.
Key points about z-score include:
  • Z-score increases as the confidence level increases, which means a wider interval.
  • It is essential to determine based on the confidence level. Always reference z-tables for exact values.
  • Higher z-scores ensure that intervals capture more of the population probability.
By understanding your z-score, you can accurately frame how confident you are in your interval estimates.
Significance of Survey Analysis
Survey analysis involves collecting, reviewing, and interpreting responses from a sample group to represent a population.
It's crucial to note that surveys provide estimates that help us infer population trends.
This involves:
  • Identifying biases: How the survey is administered can influence responses. Representation should remain unbiased.
  • Sample selection: Choose a diverse enough sample to mirror the larger population.
  • Data interpretation: Analyze results to garner insights, noting confidence intervals that reflect uncertainty.
Through survey analysis, we conclude that, in this instance, between 14.07% and 19.27% of all full-time workers might feel anger towards a coworker significant enough to lead to the desire to physically react, but additional studies should reaffirm these findings.

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

Data from a representative sample were used to estimate that \(32 \%\) of all computer users in 2011 had tried to get on a Wi-Fi network that was not their own in order to save money (USA Today, May 16,2011 ). You decide to conduct a survey to estimate this proportion for the current year. What is the required sample size if you want to estimate this proportion with a margin of error of 0.05 ? Calculate the required sample size first using 0.32 as a preliminary estimate of \(p\) and then using the conservative value of \(0.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

Describe how each of the following factors affects the width of the large- sample confidence interval for \(p\) : a. The confidence level b. The sample size c. The value of \(\hat{p}\)

USA Today (October 14,2002 ) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a representative sample of 1,004 adult drivers. A margin of error of \(3.1 \%\) was also reported. Is this margin of error correct? Explain.

Will \(\hat{p}\) from a random sample from a population with \(60 \%\) successes tend to be closer to 0.6 for a sample size of \(n=400\) or a sample size of \(n=800 ?\) Provide an explanation for your choice.

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who are registered to vote. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) I. \(\quad n=1,000, p=0.5\) II. \(\quad n=200, p=0.6\) III. \(\quad n=100, p=0.7\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.