/*! 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 51 Pilot study A state's environmen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 51

Pilot study A state's environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of \(3 \%\) and \(90 \%\) confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emissions systems. How many should be sampled for a full investigation?

Short Answer

Expert verified
Based on the given confidence level, margin of error, and pilot study data, you need to sample approximately 816 cars for a full investigation.

Step by step solution

01

Calculate the Pilot Study Proportion

First, calculate the proportion of cars with faulty emission systems from the pilot study. This is done by dividing the number of faulty cars by the total number of cars sampled. In this case, 9 of the 60 cars sampled had faulty emission systems, so the proportion is calculated as \( p = \frac{9}{60} = 0.15 \).
02

Determine the Z-Score for the Confidence Level

Next, find the Z-score that corresponds to the desired confidence level. A \(90\%\) confidence level corresponds to a Z-score of \(1.645\). This is because \(90\%\) of the area under the standard normal curve lies within \(1.645\) standard deviations of the mean.
03

Calculate the Margin of Error

The margin of error is given as \(3\%\), or \(0.03\), in decimal format. This is the maximum acceptable difference between the true population proportion and the sample proportion. It is given as a percentage, but we will use it in decimal form in calculations.
04

Calculate the Sample Size

Finally, use the following formula to calculate the necessary sample size \( n \) for the full investigation: \[ n = \frac{(Z^2) \cdot p \cdot (1-p)}{(E^2)} \] Where: \( Z \) is the Z-score, \( p \) is the pilot study proportion, and \( E \) is the margin of error. Substitute \( Z = 1.64 \), \( p = 0.15 \), and \( E = 0.03 \) into the equation to find: \[ n = \frac{(1.645^2) \cdot 0.15 \cdot (1 - 0.15)}{(0.03^2)} \] Rounded up to the nearest whole number (because we can't sample a fraction of a car), we find we need to sample approximately 816 cars for the full investigation.

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.

Pilot Study
A pilot study acts as a small-scale preliminary survey or experiment conducted to evaluate feasibility, duration, cost, adverse events, and improve upon the study design prior to performance of a full-scale research project or examination. In the context of the state's environmental agency, this pilot study of 60 cars gave an initial proportion of faulty emissions, used as a basis for estimating the size of a much larger sample needed to assess compliance with clean air emissions standards. By identifying a subset, in this case, 9 out of 60 cars with issues, the agency can estimate the expected proportion of emissions failures within the overall population of vehicles.
Confidence Level
In statistics, the confidence level represents how sure a researcher can be about the accuracy of their estimates. A higher confidence level indicates greater certainty that the population parameter lies within the confidence interval. For instance, a 90% confidence level means there's a 90% chance that the true population proportion falls within the calculated range, given the margin of error. Establishing a confidence level is crucial for determining the Z-score, which in turn is used to calculate an appropriate sample size for the study.
Z-score
A Z-score, also referred to as a standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of sample size calculation, the Z-score corresponds to the selected confidence level. It is used to look up the critical value which defines the distance from the mean needed to encapsulate the desired percentage of data under the bell curve of a standard normal distribution. The Z-score for a 90% confidence level is 1.645, meaning that we are looking for a range that captures the central 90% of the possible percentages that could occur in the larger population.
Margin of Error
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents the radius of the confidence interval for a given statistic. For the agency's requirements, a margin of error of 3% indicates that the true population proportion is expected to be within 3 percentage points of the observed sample proportion with a 90% confidence level. This is a crucial figure in determining the sample size needed for a population estimate, as smaller margins of error require larger sample sizes for the same level of confidence.
Population Proportion
In the exercise, the population proportion refers to the fraction of all cars in the entire population that would be found to have faulty emissions if all were tested. It's what the agency is aiming to estimate by sampling vehicles. The initial estimate from the pilot study suggested that 15% (0.15 in decimal form) of the cars have faulty emissions. This proportion, while only an estimate from a small sample, serves as an essential input in the formula to calculate a fuller and more accurate sample size for the agency's full investigation.
Statistics
Statistics is the field of science that deals with the collection, analysis, interpretation, presentation, and organization of data. It allows the environmental agency in question to make informed decisions based on the pilot study. The principles and methodologies of statistics, such as measuring central tendencies like the mean, as well as variability through standard deviations and Z-scores, are instrumental in calculating the required sample size to meet the specified margin of error and confidence level.
Clean Air Emissions Standards
Clean air emissions standards are regulations that set limits on the amount of pollution a vehicle is legally allowed to emit. These standards are crucial for maintaining public health and environmental quality. The state's environmental agency is tasked with ensuring that vehicles meet these standards. Through their investigation, they utilize statistical methods to determine how widespread non-compliance might be, hence protecting both the environment and public wellbeing by ensuring adherence to these standards.

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

Marketing The proportion of adult women in the United States is approximately \(51 \%\). A marketing survey telephones 400 people at random. a. What proportion of the sample of 400 would you expect to be women? b. What would the standard deviation of the sampling distribution be? c. How many women, on average, would you expect to find in a sample of that size?

Another margin of error A medical researcher estimates the percentage of children exposed to lead-based paint, adding that he believes his estimate has a margin of error of about \(3 \%\). Explain what the margin of error means.

Conditions For each situation described below, identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a. Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe. b. A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public. c. A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change. d. A college admits 1632 freshmen one year, and four years later, 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Conclusions A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a \(95 \%\) confidence interval for the proportion of all orders that arrive on time is \(88 \% \pm 6 \%\). What does this mean? Are these conclusions correct? Explain. a. Between \(82 \%\) and \(94 \%\) of all orders arrive on time. b. Ninety-five percent of all random samples of customers will show that \(88 \%\) of orders arrive on time. c. Ninety-five percent of all random samples of customers will show that \(82 \%\) to \(94 \%\) of orders arrive on time. d. We are \(95 \%\) sure that between \(82 \%\) and \(94 \%\) of the orders placed by the sampled customers arrived on time. e. On \(95 \%\) of the days, between \(82 \%\) and \(94 \%\) of the orders will arrive on time.

More conditions Consider each situation described. Identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a.A consumer group hoping to assess customer experiences with auto dealers surveys 167 people who recently bought new cars; \(3 \%\) of them expressed dissatisfaction with the salesperson. b. What percent of college students have cell phones? 2883 students were asked as they entered a football stadium, and 2430 said they had phones with them. c. Two hundred forty potato plants in a field in Maine are randomly checked, and only 7 show signs of blight. How severe is the blight problem for the U.S. potato industry? d. Twelve of the 309 employees of a small company suffered an injury on the job last year. What can the company expect in future years?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied 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.