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Chapter 22: Problem 1

Staph Infections. A study investigated ways to prevent staph infections in surgery patients. In a first step, the researchers examined the nasal secretions of a random sample of 6771 patients admitted to various hospitals for surgery. They found that 1251 of these patients tested positive for Staphylococcus aureus, a bacterium responsible for most staph infections. \(.^{4}\) a. Describe the population and explain in words what the parameter \(p\) is. b. Give the numerical value of the statistic \(\widehat{p}\) that estimates \(p\).

Short Answer

Expert verified
(a) Population: All surgery patients; parameter \(p\) is the proportion testing positive for Staphylococcus aureus. (b) \(\widehat{p} \approx 0.1847\).

Step by step solution

01

Define the Population

The population in this study is all surgery patients admitted to various hospitals. This includes any patient who might undergo surgery and have the potential to test either positive or negative for Staphylococcus aureus.
02

Identify the Parameter

The parameter \(p\) is the proportion of all surgery patients who test positive for Staphylococcus aureus. It represents the entire population of surgery patients and is what the study aims to estimate.
03

Find the Sample Statistic

The sample statistic \(\widehat{p}\) is the proportion of patients in the sample who tested positive for Staphylococcus aureus. This is calculated by dividing the number of positive tests by the total number of patients sampled. Thus, \(\widehat{p} = \frac{1251}{6771} \approx 0.1847\). This statistic is an estimate of the parameter \(p\).

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Key Concepts

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

Population
In the context of statistics, the term 'Population' refers to the complete set of subjects that share a characteristic of interest. It encompasses all potential observations in the study.
In the given exercise about staph infections in surgery patients, the population is all the surgery patients admitted to hospitals. This means every patient entering the hospital for surgical procedures forms part of this group.
  • When conducting a study, researchers aim to learn something about the entire population.
  • However, as it's often impossible to study every individual within a population, samples are used.
  • By understanding the population, researchers aim to provide results that can be generalized beyond the sample studied.
Recognizing the population in a study helps determine the scope of the research and provides clarity on the context in which results should be applied.
Parameter
A 'Parameter' in statistics is a value that helps describe a particular feature of a population. Parameters are fixed and pertain to populations, not samples. For example, a parameter could be the average age, median income, or, as in the original exercise, the proportion of individuals with a specific trait.
In the staph infection study, the parameter of interest, denoted as \( p \), is the true proportion of all surgery patients who test positive for Staphylococcus aureus.
  • This proportion represents a characteristic about the entire population.
  • It is what the study ultimately aims to measure.
  • Parameters are often unknown and are estimated using sample statistics.
Understanding parameters is critical since they constitute the target of inferential statistics鈥攖he process by which we make predictions or inferences about populations based on sample data.
Sample Statistic
A 'Sample Statistic' is used to make an inference or guess about a population parameter. In statistical studies, the statistic is calculated from sample data and reflects a certain property of the data set.
The sample statistic, denoted by \( \widehat{p} \), is a specific measure calculated from the data collected from a sample. In the staph infection example, this was the proportion of patients within the sampled group who tested positive for Staphylococcus aureus. The sample statistic in this case is calculated as \( \widehat{p} = \frac{1251}{6771} \approx 0.1847 \).
  • Sample statistics are crucial as they offer a means to estimate population parameters.
  • They provide insights without needing to survey the entire population.
  • These statistics are subject to variation, as they change with different samples.
Grasping the concept of sample statistics is fundamental because they offer a snapshot of the population through the lens of the sample data.

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Most popular questions from this chapter

Based on the sample, the large-sample \(90 \%\) confidence interval for the proportion of all American adults who actively try to avoid drinking regular soda or pop is a. \(0.61 \pm 0.015\). b. \(0.61 \pm 0.025\). c. \(0.61 \pm 0.029\).

No Confidence Interval. The 2017 Youth Risk Behavioral Survey, in a random sample of 497 high school seniors in Connecticut, found that \(0.8 \%\) (that's \(0.008\) as a decimal fraction) smoked cigarettes daily. \({ }^{2}\) Explain why we can't use the large-sample confidence interval to estimate the proportion \(p\) in the population of all Connecticut high school seniors in 2017 who smoked cigarettes daily.

Stopping Traffic with a Smile! Throughout Europe, more than 8000 pedestrians are killed each year in road accidents, with approximately \(25 \%\) of these dying when using a pedestrian crossing. Although failure to stop for pedestrians at a pedestrian crossing is a serious traffic offense in France, more than half of drivers do not stop when a pedestrian is waiting at a crosswalk. In this experiment, a female research assistant was instructed to stand at a pedestrian crosswalk and stare at the driver's face as a car approached the crosswalk. In 400 trials, the research assist ant maintained a neutral expression, and in a second set of 400 trials, the research assistant was instructed to smile. The order of smiling or not smiling was randomized, and several pedestrian crossings were used in a town on the coast in the west of France. The research assistant was dressed in normal attire for her age (jeans, t-shirt, and sneakers). \({ }^{24}\) a. In the 400 trials in which the assistant maintained a neutral expression, the driver stopped in 229 out of the 400 trials. Find a 95\% confidence interval for the proportion of drivers who would stop when a neutral expression is maintained. b. In the 400 trials in which the assistant smiled at the driver, the driver stopped in 277 out of the 400 trials. Find a \(95 \%\) confidence interval for the proportion of drivers who would stop when the assistant is smiling. c. What do your results in parts (a) and (b) suggest about the effect of a smile on a driver stopping at a pedestrian crosswalk? Explain briefly. (In Chapter 23 , we will consider formal methods for comparing two proportions.)

Do You Listen to Podcasts? In January and February 2019, Edison Research conducted a national telephone survey of 1500 Americans aged 12 and older, using random digit dialing techniques to both cell phones and landlines. The survey included questions about the use of mobile devices, Internet audio, podcasting, social media, smart speakers, and more. Of the 1500 people surveyed, 480 said they had listened to a podcast in the past month. 12 a. What is the margin of error of the large-sample \(95 \%\) confidence interval for the proportion of Americans aged 12 and older who had listened to a podcast in the past month? b. How large a sample is needed to get the common \(\pm 3\) percentage point margin of error? Use the January/February survey of 1500 as a pilot study to get \(p^{*}\)

Usage of the Olympic National Park. U.S. National Parks that contain designated wilderness areas are required by law to develop and maintain a wilderness stewardship plan. The Olympic National Park, containing some of the most biologically diverse wilderness in the United States, had a survey conducted in 2012 to collect information relevant to the development of such a plan. National Park Service staff visited 30 wilderness trailheads in moderate-to high-use areas over a 60-day period and asked visitors as they completed their hike to complete a questionnaire. The 1019 completed questionaires, giving a response rate of \(50.4 \%\), provided each subject's opinions on the use and management of wilderness. In particular, there were 694 day users and 325 overnight users in the sample. a. Why do you think the National Park staff only visited trailheads in moderate-to high-use areas to obtain the sample? b. Assuming that the 1019 subjects represent a random sample of users of the wilderness areas in Olympic National Park, give a \(90 \%\) confidence interval for the proportion of day users. c. The response rate was \(49 \%\) for day users and \(52 \%\) for overnight users. Does this lessen any concerns you might have regarding the effect of nonresponse on the interval you obtained in part (b)? Explain briefly. d. Do you think it would be better to refer to the interval in part (b) as a confidence interval for the proportion of day users or the proportion of day users on the most popular trails in the park? Explain briefly.
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