/*! 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 46 Do You Have Confidence? A report... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 24: Problem 46

Do You Have Confidence? A report of a survey distributed to randomly selected email addresses at a large university says: "We have collected 427 responses from our sample of 2,100 as of April 30, 2004. This number of responses is large enough to achieve a \(95 \%\) confidence interval with \(\pm 5 \%\) margin of sampling error in generalizing the results to our study population." 14 Why would you be reluctant to trust a confidence interval based on these data?

Short Answer

Expert verified
The sample may not be representative due to response bias and potential coverage error.

Step by step solution

01

Understanding Confidence Intervals

A confidence interval provides a range of values that is likely to contain the population parameter with a certain level of confidence, typically 95%. A margin of error quantifies the uncertainty level around the sample estimate. A typical formula used is: \[ \text{Margin of Error (MoE)} = Z \cdot \sqrt{\frac{p(1-p)}{n}} \]where \(Z\) is the Z-score corresponding to the confidence level, \(p\) is the sample proportion, and \(n\) is the sample size.
02

Evaluating the Sample Size

The sample size of 427 is used to generalize to a larger population of 2,100. While the survey obtained responses from about 20.33% of the population, whether this is sufficient depends on how representative the sample is of the entire population, not just its size.
03

Potential Response Bias

There may be response bias if the respondents who chose to reply are systematically different from those who did not. For instance, those with strong opinions might be more likely to respond, leading to non-representative results.
04

Coverage Error

Coverage error occurs if the sample is not drawn from the entire population or if some members have a lower chance of being included in the sample. The random selection process should ideally ensure that every email address has an equal chance of being chosen.

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 crucial when conducting surveys and interpreting confidence intervals. In simple terms, the sample size refers to the number of observations or responses collected from the population. A larger sample size generally provides more reliable results because it is more likely to represent the entire population accurately.

In the context of the survey with 427 responses out of a potential 2,100, our sample size is 427. This might seem substantial; however, it's important to ask if these 427 individuals represent the diversity and characteristics of the entire 2,100.
  • A larger sample size can lead to a smaller margin of error, meaning we have more confidence in our results.
  • But size isn't everything; how the sample is collected also matters greatly.
  • If the sample isn鈥檛 representative, even a large sample size won't prevent inaccurate conclusions.
Response Bias
Response bias occurs when there is a systematic difference between those who choose to respond to a survey and those who don鈥檛. This is an important factor to consider in surveys and questionnaires. In many cases, those with stronger opinions or particular characteristics may be more inclined to respond.

For instance, if a survey about campus services reaches out to all students, those who frequently use those services might be more likely to reply, skewing results. This can lead to a biased representation of the true population's opinions.
  • Always consider who is likely to respond and who might be left out.
  • Look for ways to encourage responses from a wide range of individuals to minimize bias.
  • Ensuring that questions are clear and unbiased is also critical to reducing response bias.
Margin of Error
The margin of error is a key element of confidence intervals, highlighting the range within which the true population parameter is expected to lie. Essentially, it quantifies the uncertainty of the sample estimate.

The formula for the margin of error depends on the level of confidence desired and the number of observations in the sample:\[ \text{Margin of Error (MoE)} = Z \cdot \sqrt{\frac{p(1-p)}{n}} \]Where:
  • Z is the Z-score associated with your confidence level (e.g., 1.96 for a 95% confidence interval).
  • p represents the sample proportion.
  • n is the sample size.
A smaller margin of error means that the sample estimate is closer to the actual population parameter. It is worth noting that increasing the sample size or reducing response bias can help decrease the margin of error.
Coverage Error
Coverage error arises when the surveyed sample does not properly reflect the whole population. This might happen if certain segments of the population are not included or have a lesser chance of selection.

Imagine a survey distributed via email to students, but some students never or rarely check their emails. These individuals are less likely to be represented, introducing coverage error into the results.
  • Ensuring equal chances of selection is critical to minimizing coverage error.
  • Consider using multiple channels for reaching respondents to increase representativeness.
  • Random sampling methods, where every individual has an equal likelihood of being chosen, help reduce this error.
Addressing coverage error is essential because such errors can severely limit the accuracy and generalizability of the survey 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

Preventing Drowning. Drowning in bathtubs is a major cause of death in children less than five years old. A random sample of parents was asked many questions related to bathtub safety. Overall, \(85 \%\) of the sample said they used baby bathtubs for infants. Estimate the percentage of all parents of young children who use baby bathtubs.

Starting to Talk. At what age do infants speak their first word of English? Here are data on 20 children (ages in months): 18 | 1STWORD (In fact, the sample contained one more child, who began to speak at 42 months. Child development experts consider this abnormally late, so the investigators dropped the outlier to get a sample of "typical" children. We are willing to treat these data as an SRS.) Is there good evidence that the mean age at first word among all typical children is greater than one year?

The numerical value of the \(z\) test for comparing the proportions of all VLBW and normal-birth-weight (control) babies who would graduate from high school is a. \(z=-1.65\). b. \(z=-2.34 .\) c. \(z=-2.77\). d. \(z=-3.14\).

Healthcare. If we want to estimate \(p\), the population proportion of likely voters who believe healt hcare to be the most urgent national concern, with \(95 \%\) confidence and a margin of error no greater than \(2 \%\), how many likely voters need to be surveyed? Assume that you have no idea of the value of \(p\). a. 188 b. 1691 c. 2401 d. 4802

(Optional Topic) Drug-Detecting Rats? Dogs are big and expensive. Rats are small and cheap. Might rats be trained to replace dogs in sniffing out illegal drugs? A first st udy of this idea trained rats to rear up on their hind legs when they smelled simulated cocaine. To see how well rats performed after training, they were let loose on a surface with many cups sunk in it, one of which contained simulated cocaine. Four out of six trained rats succeeded in 80 out of 80 trials. \(\frac{16}{\text { How }}\) should we estimate the long- term success rate \(p\) of a rat that succeeds in every one of 80 trials? a. What is the rat's sample proportion \(\vec{p}\) ? What is the large-sample \(95 \%\) confidence interval for \(p\) ? It's not plausible that the rat will always be successful, as this interval says. b. Find the plus four estimate \(\tilde{p}\) and the plus four \(95 \%\) confidence interval for \(p\). These results are more reasonable.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.