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Chapter 9: Problem 51

The USA Snapshot titled "Social Media Jeopardizing Your Job?" (USA TODAY, November 12,2014\()\) summarized data from a survey of 1855 recruiters and human resource professionals. The Snapshot indicted that \(53 \%\) of the people surveyed had reconsidered a job candidate based on his or her social media profile. Assume that the sample is representative of the population of recruiters and human resource professionals in the United States. a. Use the given information to estimate the proportion of recruiters and human resource professionals who have reconsidered a job candidate based on his or her social media profile using a \(95 \%\) confidence interval. Give an interpretation of the interval in context and an interpretation of the confidence level of \(95 \%\). b. Would a \(90 \%\) confidence interval be wider or narrower than the \(95 \%\) confidence interval from Part (a)?

Short Answer

Expert verified
The 95% confidence interval for the proportion of recruiters and human resource professionals who have reconsidered a job candidate based on their social media profile is (0.507, 0.553), meaning we are 95% confident that this proportion falls between 50.7% and 55.3% in the United States. A 90% confidence interval would be narrower than the 95% confidence interval because it has less uncertainty and a smaller margin of error.

Step by step solution

01

a. Estimating the proportion with a 95% confidence interval

Given that 53% of the 1855 people surveyed have reconsidered a job candidate based on their social media profile, we can find the sample proportion \(p = 0.53\). The sample size \(n = 1855\). To calculate the 95% confidence interval, we can use the formula: Confidence Interval = \(p \pm z \times \sqrt{\frac{p(1-p)}{n}}\) where \(z\) is the critical value for a 95% confidence interval, which is 1.96. Plugging in the values, we get: Confidence Interval = \(0.53 \pm 1.96 \times \sqrt{\frac{0.53(1-0.53)}{1855}}\) Now, calculate the margin of error: Margin of Error = \(1.96 \times \sqrt{\frac{0.53(0.47)}{1855}} = 0.023\) So, the confidence interval is: Confidence Interval = \(0.53 \pm 0.023\) Therefore, the 95% confidence interval is \((0.507, 0.553)\). Interpretation of the confidence interval in context: We are 95% confident that the proportion of recruiters and human resource professionals who have reconsidered a job candidate based on his or her social media profile falls between 50.7% and 55.3% in the United States. Interpretation of the 95% confidence level: If we were to repeat this survey many times with different samples of recruiters and human resource professionals, 95% of the confidence intervals calculated would contain the true proportion of those who have reconsidered job candidates based on their social media profiles.
02

b. Comparing a 90% confidence interval with the 95% confidence interval

To determine if a 90% confidence interval is wider or narrower than the 95% confidence interval, we need to compare their respective critical values. A 90% confidence interval has a smaller critical value (z = 1.645) than a 95% confidence interval (z = 1.96), which means less uncertainty. Therefore, since the margin of error is calculated using the critical value, a 90% confidence interval will have a smaller margin of error compared to a 95% confidence interval. Thus, a \(90\%\) confidence interval will be narrower than the \(95\%\) confidence interval.

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Key Concepts

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

Sample Proportion
Understanding sample proportion is crucial when dealing with statistics related to a population. To put it simply, the sample proportion is the fraction of individuals in a sample that exhibit a particular attribute or trait. It's commonly represented by the symbol \( p \).
In the context of our example, where recruiters and human resource professionals were surveyed about reconsidering job candidates based on their social media profiles, the sample proportion was found to be \( p = 0.53 \) or 53%. This means that 53% of the surveyed group had indeed reconsidered a candidate based on social media.
It's important to recognize that the sample proportion is an estimate of the true proportion within the entire population. This estimate comes from the selected sample and is used as the basis for confidence interval calculations. Especially when the sample is large enough and deemed representative, the sample proportion can provide valuable insights about the whole population's behavior or characteristics.
Margin of Error
The margin of error is a measure that expresses the amount of random sampling error in a survey's results. It's the radius of the confidence interval and represents the uncertainty in the sample proportion. The margin of error increases with more variability within a sample and decreases as the sample size grows.
In calculating the margin of error, we encountered the formula \(1.96 \times \sqrt{\frac{p(1-p)}{n}}\), where \(1.96\) is the critical value associated with a 95% confidence level and \(n\) is the number of people surveyed. This calculation yielded a margin of error of 0.023, or 2.3%, for our particular example. This figure tells us how far we can expect the true population proportion to be from our sample proportion of 53%, with a 95% confidence interval being \(0.53 \pm 0.023\). This is a pivotal concept for students to grasp because it affects how confidently one can make statements about the entire population based on the sample.
Confidence Level Interpretation
Interpreting the confidence level properly is imperative in understanding what confidence intervals represent. A confidence level of \(95\%\) means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to include the true population proportion.
In simpler terms, it indicates a degree of certainty in the likelihood that the calculated interval contains the true parameter. Therefore, a higher confidence level shows a greater extent of certainty; though, this also causes the margin of error to increase, leading to a wider interval. Ultimately, confidence levels give us a way to communicate how much faith we can put in our interval estimates.
Applying this to our social media profile example, the interpretation is that we are 95% confident the interval from 50.7% to 55.3% captures the true proportion of professionals who would reconsider a candidate based on social media. It is essential to note that confidence levels do not express the probability that a particular interval captures the population parameter. Instead, they indicate the reliability of the estimation process over many repetitions.

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Most popular questions from this chapter

Suppose that county planners are interested in learning about the proportion of county residents who would pay a fee for a curbside recycling service if the county were to offer this service. Two different people independently selected random samples of county residents and used their sample data to construct the following confidence intervals for the proportion who would pay for curbside recycling: Interval 1:(0.68,0.74) Interval 2:(0.68,0.72) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals are associated with a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

A large online retailer is interested in learning about the proportion of customers making a purchase during a particular month who were satisfied with the online ordering process. A random sample of 600 of these customers included 492 who indicated they were satisfied. For each of the three following statements, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1: It is unlikely that the estimate \(\hat{p}=0.82\) differs from the value of the actual population proportion by more than 0.016 . Statement 2: It is unlikely that the estimate \(\hat{p}=0.82\) differs from the value of the actual population proportion by more than 0.031 . Statement 3: The estimate \(\hat{p}=0.82\) will never differ from the value of the actual population proportion by more than 0.031 .

In mid-2016 the United Kingdom (UK) withdrew from the European Union (an event known as "Brexit"), causing economic concerns throughout the world. One indicator that economists use to monitor the health of the economy is the proportion of residential properties offered for sale at auction that are successfully sold. An article titled "Going, going, gone through the roof-sky's the limit at auction" (October \(22,2016,\) www.estateagenttoday .co.uk/features/2016/10/going-going-gone-through-theroof-the-skys-the-limit- at-auction, retrieved May 4,2017 ) reported the success rate of a sample of 26 residential properties offered for sale at auctions in the UK in the summer of \(2016 .\) For this sample of properties, 14 of the 26 residential properties were successfully sold. Suppose it is reasonable to consider these 26 properties as representative of residential properties offered at auction in the post-Brexit UK. a. Would it be appropriate to use the large-sample confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. b. Would it be appropriate to use a bootstrap confidence interval for a population proportion to estimate the proportion of residential properties successfully sold at auction in the post-Brexit UK? Explain. c. Use the accompanying output from the "Bootstrap Confidence Interval for One Proportion" Shiny app to report a \(95 \%\) bootstrap confidence interval for the population proportion of residential properties successfully

Describe how each of the following factors affects the width of the large- sample confidence interval for \(p\) : a. The confidence level b. The sample size c. The value of \(\hat{p}\)

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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