/*! 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 4 You want to conduct a survey to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 4

You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error?

Short Answer

Expert verified
Increasing the sample size decreases the margin of error.

Step by step solution

01

Understanding Margin of Error

The margin of error in a survey is a measure of how much the survey results are expected to vary from the actual population parameter. It is typically calculated using a statistical formula that incorporates the sample size and a critical value from a statistical distribution, usually the Normal or t-distribution.
02

Link Between Sample Size and Margin of Error

The margin of error (MOE) is inversely related to the square root of the sample size, according to the formula: \[\text{MOE} = \frac{z^* \cdot \sigma}{\sqrt{n}}\]where \(z^*\) is the critical value, \(\sigma\) is the standard deviation, and \(n\) is the sample size. As the denominator increases (i.e., larger sample size), the margin of error decreases.
03

Effect of Increasing Sample Size

Since the margin of error is inversely proportional to the square root of the sample size, increasing the sample size will decrease the margin of error. This means that with a larger sample size, the survey results are expected to be closer to the actual population parameter, increasing confidence in the results.
04

Practical Implication for Surveys

For someone conducting a survey, understanding this relationship means that if they want more precise results with a smaller margin of error, they should increase their sample size. However, it's important to balance the need for precision with the available resources, as larger sample sizes require more time and money to collect data.

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.

Sample Size
The concept of "sample size" is foundational in understanding surveys and statistical studies. Sample size refers to the number of observations or participants included in a survey or study.

It's a critical factor because it directly impacts the reliability of the survey results. A larger sample size tends to give more precise and reliable data, especially when attempting to generalize results to a broader population.

Here's why sample size is so important:
  • Accuracy: Larger samples provide more data points, resulting in a clearer picture of the population being studied.
  • Reduced Variability: With more data, random errors and outliers have less influence on the results.
  • Increased Statistical Power: A larger sample size increases the power of statistical tests, making it easier to detect true effects when they exist.
When planning a survey, researchers must decide on the appropriate sample size by considering the expected variability in the population, the desired level of confidence, and resource constraints.
Statistical Distribution
Understanding "statistical distribution" is crucial when discussing concepts like the margin of error. Statistical distribution describes how data points are spread across a set of values or range.

When conducting a survey, researchers often assume that data follows a certain distribution, such as the Normal distribution, which plays a significant role in determining the margin of error.
  • Normal Distribution: Also known as the bell curve, it is symmetric and describes many natural phenomena. It's commonly used in statistics due to its properties and because it often approximates real-world data.
  • t-Distribution: Similar to the normal distribution but with thicker tails. It's often used when dealing with smaller sample sizes.
  • Role in Margin of Error: The critical value in margin of error calculations is derived from these distributions, affecting the confidence level of the survey.
Recognizing which distribution applies can affect how data is analyzed and interpreted, making it a key consideration in survey design.
Survey Precision
"Survey precision" refers to how closely the results of a survey reflect the true opinions or behaviors of the overall population. This precision is largely influenced by the margin of error and the sample size.

A precise survey has:
  • Low Margin of Error: Indicates that results are likely close to the true population values.
  • High Confidence Level: Reflects that the findings are statistically significant and reproducible.
To enhance survey precision:
  • Increase Sample Size: More data tends to yield more accurate estimates.
  • Optimize Survey Design: Carefully crafted questions and randomized sampling can improve results.
However, increasing survey precision isn't just a mathematical exercise. It's about making sure that the survey truly reflects the population's characteristics and preferences, ensuring the results are both meaningful and useful.

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

As the degrees of freedom increase, what distribution does the Student's \(t\) distribution become more like?

If a \(90 \%\) confidence interval for the difference of means \(\mu_{1}-\mu_{2}\) contains all positive values, what can we conclude about the relationship between \(\mu_{1}\) and \(\mu_{2}\) at the \(90 \%\) confidence level?

A random sample of 5222 permanent dwellings on the entire Navajo Indian Reservation showed that 1619 were traditional Navajo hogans (Navajo Architecture: Forms, History, Distributions, by Jett and Spencer, University of Arizona Press). (a) Let \(p\) be the proportion of all permanent dwellings on the entire Navajo Reservation that are traditional hogans. Find a point estimate for \(p\). (b) Find a \(99 \%\) confidence interval for \(p\). Give a brief interpretation of the confidence interval. (c) Do you think that \(n p>5\) and \(n q>5\) are satisfied for this problem? Explain why this would be an important consideration.

For a binomial experiment with \(r\) successes out of \(n\) trials, what value do we use as a point estimate for the probability of success \(p\) on a single trial?

In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 26 out of 855 fish died when caught and released using barbless hooks on flies or lures. All hooks were removed from the fish. (Source: A National Symposium on Catch and Release Fishing, Humboldt State University Press.) (a) Let \(p\) represent the proportion of all pike and trout that die (i.e., \(p\) is the mortality rate) when caught and released using barbless hooks. Find a point estimate for \(p\). (b) Find a \(99 \%\) confidence interval for \(p\), and give a brief explanation of the meaning of the interval. (c) Is the normal approximation to the binomial justified in this problem? Explain.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.