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Chapter 9: Problem 46

Based on data from a survey of 1200 randomly selected Facebook users (USA TODAY, March 24, 2010), a \(98 \%\) confidence interval for the proportion of all Facebook users who say it is OK to ignore a coworker's "friend" request is \((0.35,0.41) .\) What is the meaning of the confidence level of \(98 \%\) that is associated with this interval?

Short Answer

Expert verified
The 98% confidence level means that if we repeatedly sampled 1200 Facebook users, 98% of the confidence intervals would contain the true proportion of users who think it's OK to ignore a coworker's "friend" request. With this confidence level, we can say that we are 98% confident that the true proportion is between 35% and 41%.

Step by step solution

01

Define Confidence Interval and Confidence Level

A confidence interval is an interval estimate of a parameter. In this case, the parameter is the proportion of all Facebook users who say it is okay to ignore a coworker's "friend" request. The confidence level is the probability that the interval estimate contains the true parameter. In this exercise, the confidence level is 98%.
02

Explain the Meaning of 98% Confidence Level

The 98% confidence level means that if we were to conduct the same survey many times with new random samples of 1200 Facebook users each time, 98% of these samples would produce confidence intervals that contain the true proportion of all Facebook users who say it's OK to ignore a coworker's "friend" request.
03

Describe the Confidence Interval in Context

Given the 98% confidence level and the confidence interval (0.35, 0.41), we can say that we are 98% confident that the true proportion of all Facebook users who say it is OK to ignore a coworker's "friend" request is between 35% and 41%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Level
When you hear the term "confidence level," think of the likelihood that an estimate falls within a specific range. In statistics, a confidence level quantifies this likelihood and is expressed as a percentage.
For example, in the given exercise, we are dealing with a confidence level of 98%. This means that if we conducted the survey repeatedly, 98% of the time, the computed confidence intervals would include the true proportion of all Facebook users who find ignoring a coworker's friend request acceptable.
  • A higher confidence level means a higher likelihood that the interval contains the true parameter, but it may result in a wider interval.
  • In contrast, a lower confidence level results in a narrower interval but with less certainty that it includes the true parameter.
Thus, choosing a confidence level is a trade-off between certainty and precision.
Survey Sampling
Survey sampling is a method used to gather data from a subset of a larger population, aiming to infer information about the whole.
In the context of the exercise, 1200 randomly selected Facebook users represent the larger population of all Facebook users. This is a classic example of survey sampling.
  • Random selection is crucial because it ensures that every individual has an equal chance of being chosen, reducing bias.
  • The sample size of 1200 is quite substantial, helping to increase the accuracy of the estimates made about the entire population.
Through survey sampling, we can make educated judgments about the whole group based on data collected from a smaller portion.
Parameter Estimation
Parameter estimation involves using data from a sample to make inferences or estimates about a population parameter.
In this exercise, the parameter in question is the proportion of all Facebook users who think it is okay to ignore a coworker's friend request.
  • The estimation process begins by collecting a sample, such as the 1200 Facebook users surveyed.
  • Then, statistical methods, like calculating a confidence interval, are used to estimate the population parameter.
The confidence interval \(0.35, 0.41\) suggests that we are 98% certain the true proportion lies within this range. Parameter estimation helps transform raw data into actionable insights.

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Most popular questions from this chapter

Business Insider reported that a study commissioned by eBay Motors found that nearly \(40 \%\) of millennials who drive a car that is more than 5 years old have named their cars ("Millennials Have an Odd Habit When It Comes to Their Cars," April 14,2016 ). a. Assuming that the sample was selected to be representative of the population of millennials who drive a car that is more than 5 years old, what is an estimate of the population proportion who have named their car? b. Suppose that the sample size for the study described was 800. Calculate and interpret the margin of error associated with your estimate in Part (a).

Use the formula for the standard error of \(\hat{p}\) to explain why increasing the sample size decreases the standard error.

A survey of a representative sample of 478 U.S. employers found that 359 ranked stress as their top health and productivity concern (June \(29,2016,\) www.globenewswire .com/news-release/2016/06/29/852338/0/en/Seventy-five- percent-of-U-S-employers-say-stress-is-their-number-one-workplace-health- concern.html?print=1, retrieved May 4,2017) a. Use the accompanying output from the "Bootstrap Confidence Interval for One Proportion" Shiny app to report a \(95 \%\) bootstrap confidence interval for the proportion of all U.S. employers who would rank stress at their top health and productivity concern. Interpret the confidence interval in context. b. A number of international employers were also surveyed. If the international employers had a similar rate of identifying stress as their top health and productivity concern, and if the results from international employers were included in the sample, would the width of the resulting confidence interval remain the same, decrease, or increase? Explain your reasoning.

In \(2010,\) the National Football League adopted new rules designed to limit head injuries. In a survey conducted in 2015 by the Harris Poll, 1216 of 2096 adults indicated that they were football fans and followed professional football. Of these football fans, 692 said they thought that the new rules were effective in limiting head injuries (December 21 , \(2015,\) www.theharrispoll.com/sports/Football-Injuries.html, retrieved May 6,2017 ). a. Assuming that the sample is representative of adults in the United States, construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults who consider themselves to be football fans. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of football fans who think that the new rules have been effective in limiting head injuries. c. Explain why the confidence intervals in Parts (a) and (b) are not the same width even though they both have a confidence level of \(95 \%\)

It probably wouldn't surprise you to know that Valentine's Day means big business for florists, jewelry stores, and restaurants. But did you know that it is also a big day for pet stores? In January \(2015,\) the National Retail Federation conducted a survey of consumers in a representative sample of adult Americans ("Survey of Online Shopping for Valentine's Day 2015," nrf.com/news/delivering-customer-delight-valentines-day, retrieved November 14,2016)\(.\) One of the questions in the survey asked if the respondent planned to spend money on a Valentine's Day gift for his or her pet. a. The proportion who responded that they did plan to purchase a gift for their pet was 0.212 . Suppose that the sample size for this survey was \(n=200 .\) Construct and interpret a \(95 \%\) confidence interval for the proportion of all adult Americans who planned to purchase a Valentine's Day gift for their pet. b. The actual sample size for the survey was much larger than \(200 .\) Would a \(95 \%\) confidence interval calculated using the actual sample size have been narrower or wider than the confidence interval calculated in Part (a)?
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