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Chapter 9: Problem 80

Based on data from a survey of 1,200 randomly selected Facebook users (USA Today, March 24, 2010), a \(90 \%\) confidence interval for the proportion of all Facebook users who say it is not OK to "friend" someone who reports to you at work is (0.60,0.64) . What is the meaning of the \(90 \%\) confidence level associated with this interval?

Short Answer

Expert verified
The confidence level of 90% means that if we were to draw many samples from the population in the same way, theoretically 90% of those intervals would include the true population proportion. Thus, we are 90% confident that the real proportion of Facebook users who think it's not okay to 'friend' someone who reports to you at work is between 60% and 64%.

Step by step solution

01

Understand the given Confidence Interval

A confidence interval is an estimated range of values that is likely to include an unknown population parameter. The given confidence interval is (0.60,0.64) with a confidence level of 90%.
02

Understanding Confidence Level

The confidence level is an expression of how confident we are that the procedure would capture the true population parameter if you were to draw many samples from the population in the same way. The confidence level of 90% says that if the same population were sampled an infinite number of times, theoretically 90% of those intervals would include the true population proportion.
03

Interpreting Confidence Level and Interval

In the context of this survey question, we're 90% confident that the true proportion of all Facebook users who think it's not okay to 'friend' a work subordinate is somewhere between 60% and 64%.

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Key Concepts

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

Confidence Level
The confidence level is a crucial part of statistical analysis, especially when interpreting survey data and estimating population parameters.
Simply put, it tells us how certain we should be about the results from a sample that estimates a population parameter, such as a proportion or mean.
  • A confidence level of 90%, as in the exercise, indicates that we believe there is a 90% chance that the calculated confidence interval from our sample contains the true population proportion.
  • It is important to note that this does not guarantee that the true value is within the interval for the single sample. Rather, it means that if we repeated the survey many times, 90% of the confidence intervals calculated from each of those samples would include the true population value.
  • Common levels of confidence typically chosen are 90%, 95%, and 99%, with higher percentages indicating a greater level of certainty about the population parameter.
We can think of confidence levels as a measure of reliability in the sampling process to provide accurate estimations about a population.
Population Proportion
Population proportion is a statistic that estimates the ratio of individuals with a particular characteristic in a population. For instance, in our exercise, it is the proportion of all Facebook users who think it is not okay to "friend" someone who reports to them at work.
  • This population proportion helps us understand the makeup of a population by providing an estimate based on a sample.
  • For example, the specified interval (0.60,0.64) suggests that between 60% and 64% of all Facebook users hold this belief.
  • The key is to use a random sample, which should ideally be representative of the larger population, to reduce biases that might skew this estimation.
By using confidence intervals to estimate population proportions, we can make informed decisions or inferences about the entire population based on limited sample data.
Survey Data Analysis
When conducting a survey to analyze data, there are several steps involved to ensure accurate and reliable results. Survey data analysis is used to interpret and make decisions based on survey data.
  • First, defining the research question is crucial, as it guides the entire survey process from design to analysis.
  • Next, the selection of a sample size matters because it affects the confidence interval. Larger sample sizes typically yield more precise confidence intervals.
  • In this exercise, 1,200 randomly selected Facebook users formed the sample, providing a good-sized data set for estimation.
Surveys should aim for randomness in sampling to represent the population accurately.
This minimizes sampling bias and enhances the validity of the results. Following these principles allows researchers to identify trends, validate hypotheses, and make predictions about larger population behaviors based on survey data.

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Most popular questions from this chapter

For each of the following choices, explain which one would result in a wider large-sample confidence interval for \(p\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

A car manufacturer is interested in learning about the proportion of people purchasing one of its cars who plan to purchase another car of this brand in the future. A random sample of 400 of these people included 267 who said they would purchase this brand again. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1 : The estimate \(\hat{p}=0.668\) will never differ from the value of the actual population proportion by more than \(0.0462 .\) Statement 2 : It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0235 . Statement 3: It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0462 .

The study "Digital Footprints" (Pew Internet \& American Life Project, www.pewinternet.org, 2007) reported that \(47 \%\) of Internet users have searched for information about themselves online. The \(47 \%\) figure was based on a random sample of Internet users. Suppose that the sample size was \(n=300\) (the actual sample size was much larger). Answer the four key questions (QSTN) to confirm that the suggested method in this situation is a large-sample confidence interval for a population proportion.

For estimating a population characteristic, why is an unbiased statistic generally preferred over a biased statistic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain

The article "Career Expert Provides DOs and DON'Ts for Job Seekers on Social Networking" (CareerBuilder.com, August 19,2009 ) included data from a survey of 2,667 hiring managers and human resource professionals. The article noted that more employers are now using social networks to screen job applicants. Of the 2,667 people who participated in the survey, 1,200 indicated that they use social networking sites such as Facebook, MySpace, and LinkedIn to research job applicants. Assume that the sample is representative of hiring managers and human resource professionals. Answer the four key questions (QSTN) to confirm that the suggested method in this situation is a confidence interval for a population proportion.
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