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Chapter 8: Problem 1

Health care A study dealing with health care issues plans to take a sample survey of 1500 Americans to estimate the proportion who have health insurance and the mean dollar amount that Americans spent on health care this past year. a. Identify two population parameters that this study will estimate. b. Identify two statistics that can be used to estimate these parameters.

Short Answer

Expert verified
The parameters are the proportion with insurance (\( P \)) and mean expenditure (\( \)). The statistics are the sample proportion (\( p \)) and sample mean (\( ar{x} \)).

Step by step solution

01

Understanding the Problem

The study aims to gather data from a sample of 1500 Americans. From this sample, it will estimate certain characteristics鈥攕pecifically, the proportion of Americans with health insurance and the average amount spent on health care by Americans in the last year.
02

Identify the Population Parameters

Population parameters are the actual values in the full population which we are aiming to estimate with our study. In this case: 1. The proportion of all Americans who have health insurance (let's call it \( P \)).2. The mean amount of dollars the entire American population spent on health care last year (let's call it \( \)).
03

Identify the Statistics for Estimation

Statistics are the calculated values from the sample used to estimate population parameters. In this study:1. The sample proportion of the 1500 surveyed who have health insurance, which estimates \( P \), will be denoted as \( p \).2. The sample mean expenditure on health care from the 1500 surveyed, which estimates \( ar{X} \), will be denoted as \( ar{x} \).

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Key Concepts

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

Population Parameters
In health research, population parameters are crucial because they represent the characteristics of an entire group鈥攆rom which conclusions will be drawn. For instance, when calculating the proportion of Americans with health insurance or determining the mean health care expenditure across the nation, these values are deemed population parameters.

Population parameters have two main properties:
  • They are theoretical constructs representing an entire population.
  • They are often unknown and need estimation through studies and surveys.
In the context of the health care study mentioned, two specific population parameters have been identified:
  • The proportion of all Americans who have health insurance ( \( P \)).
  • The mean amount spent by all Americans on health care in the past year ( \( \mu \)).
Understanding these population parameters helps researchers form strategies to improve health services and policy making.
Sample Statistics
To estimate unknown population parameters, researchers rely on sample statistics. These are calculations based on data collected from a subset of the population, known as a sample.

In the study described, a sample of 1500 Americans is used to gather necessary data. From this sample, researchers identify specific sample statistics:
  • The sample proportion of those with health insurance ( \( \hat{p} \)), which serves as an estimate for the true population proportion ( \( P \)).
  • The sample mean of health care expenditure ( \( \bar{x} \)), used to estimate the overall population mean expenditure ( \( \mu \)).
Sample statistics act as stand-ins to understand and infer population parameters. Their accuracy depends on factors like the sample size and how well the sample represents the entire population.
Health Care Expenditure
Health care expenditure refers to the amount of money spent by individuals or the nation as a whole on health services and products. The exercise focuses on estimating the mean expenditure of all Americans.

Calculating the mean expenditure helps to:
  • Understand spending patterns across different demographics.
  • Identify trends and predict future spending needs.
  • Fight financial inefficiencies within the health care system.
  • Inform policy decisions and resource allocation in healthcare.
Estimating national averages through a well-designed sample helps bridge the gap between assumptions and evidence-based policy making.
Proportion Estimation
Proportion estimation is a statistical technique that predicts the likelihood or fraction of a population having a particular characteristic. In this study, it gauges the fraction of Americans with health insurance.

Proportion estimates are vital for:
  • Recognizing health coverage gaps among populations.
  • Understanding sociodemographic disparities in insurance coverage.
  • Helping agencies craft targeted interventions to increase coverage.
To estimate a proportion, the formula used is \[\hat{p} = \frac{x}{n}\]where \( x \) is the number of successes (e.g., having insurance) in the sample, and \( n \) is the total sample size. Accurate proportion estimation leads to actionable insights for policymakers and healthcare providers.

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Most popular questions from this chapter

Abstainers The Harvard study mentioned in the previous exercise estimated that \(19 \%\) of college students abstain from drinking alcohol. To estimate this proportion in your school, how large a random sample would you need to estimate it to within 0.05 with probability \(0.95,\) if before conducting the study a. You are unwilling to predict the proportion value at your school. b. You use the Harvard study as a guideline. c. Use the results from parts a and \(\mathrm{b}\) to explain why strategy (a) is inefficient if you are quite sure you'll get a sample proportion that is far from \(0.50 .\)

How often do women feel sad? A recent GSS asked, "How many days in the past seven days have you felt sad?" The 816 women who responded had a median of \(1,\) mean of 1.81 , and standard deviation of \(1.98 .\) The 633 men who responded had a median of \(1,\) mean of \(1.42,\) and standard deviation of \(1.83 .\) a. Find a \(95 \%\) confidence interval for the population mean for women. Interpret. b. Do you think that this variable has a normal distribution? Does this cause a problem with the confidence interval method in part a? Explain.

Women's satisfaction with appearance A special issue of Newsweek in March 1999 on women and their health reported results of a poll of 757 American women aged 18 or older. When asked, "How satisfied are you with your overall physical appearance?" \(30 \%\) said very satisfied, \(54 \%\) said somewhat satisfied, \(13 \%\) said not too satisfied, and \(3 \%\) said not at all satisfied. True or false: Since all these percentages are based on the same sample size, they all have the same margin of error.

Males watching TV Refer to the previous exercise. The 626 males had a mean of 2.87 and a standard deviation of \(2.61 .\) The \(95 \%\) confidence interval for the population mean is \((2.67,3.08) .\) Interpret in context.

Vegetarianism Time magazine (July 15,2002 ) quoted a poll of 10,000 Americans in which only \(4 \%\) said they were vegetarians. a. What has to be assumed about this sample to construct a confidence interval for the population proportion of vegetarians? b. Construct a \(99 \%\) confidence interval for the population proportion. Explain why the interval is so narrow, even though the confidence level is high. c. In interpreting this confidence interval, can you conclude that fewer than \(10 \%\) of Americans are vegetarians? Explain your reasoning.
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