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Chapter 18: Problem 2

Margin of error A medical researcher estimates the percentage of children exposed to lead-based paint, adding that he believes his estimate has a margin of error of about \(3 \% .\) Explain what the margin of error means.

Short Answer

Expert verified
The margin of error indicates the range within which the true percentage is likely to fall based on the estimate, in this case 卤3%.

Step by step solution

01

Understanding the Margin of Error

The margin of error indicates how much we expect the estimated results from the sample to vary from the actual population proportions. In this context, the researcher claims a margin of error of 3%, suggesting that the estimated percentage is likely to fall within 3% above or below the actual percentage.
02

Interpreting the Range

If the researcher estimates that a certain percentage of children are exposed to lead-based paint, the margin of error tells us the range in which the true percentage likely falls. For example, if the researcher estimates the exposure at 25%, the actual percentage is likely between 22% and 28% (25% 卤 3%).

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Key Concepts

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

Confidence Interval
A confidence interval provides a range of values that are believed to encompass the true value of an unknown parameter in a population. This interval is calculated from the sample data and includes the margin of error to signify a level of certainty about where the true parameter lies. For example, when a medical researcher reports a percentage of children exposed to lead-based paint, a confidence interval might be given as 25% with a 3% margin of error. This implies that the researcher is confident that the true percentage of children exposed falls between 22% and 28%, representing a 95% confidence level in most cases.
  • The confidence level indicates how certain you are that the interval contains the true population parameter.
  • Common confidence levels are 90%, 95%, and 99%.
  • Higher confidence levels require wider intervals.
Understanding confidence intervals helps gauge the precision and reliability of statistical estimates.
Statistical Estimation
Statistical estimation is the process of inferring the value of a population parameter based on a sample statistic. There are two main types of statistical estimations: point estimation and interval estimation. Point estimation involves using a single value, like the sample mean, to estimate the population mean. Interval estimation, on the other hand, provides a range of values using a confidence interval to estimate the population parameter.
  • The process relies on random sampling, assuming that the sample data represents the wider population.
  • Statistical estimation helps researchers make educated predictions and decisions based on data.
  • Estimations are more accurate with larger sample sizes, as they better represent the population.
By understanding statistical estimation methods, researchers can derive meaningful conclusions that aid in decision-making.
Population Proportion
A population proportion refers to the fraction of individuals in a population that display a certain characteristic. It is a key component in statistical studies, often used in surveys and experiments to measure how prevalent certain traits or behaviors are within a larger group.
  • To estimate the population proportion, researchers collect sample data and calculate the sample proportion.
  • Tools like the confidence interval and margin of error are used to understand the precision of this estimation.
  • The sample proportion can fluctuate depending on the sample size and variation within the population.
For instance, if a researcher is trying to estimate how many children are exposed to lead paint, the population proportion would denote the percentage of all children with that exposure. Estimating population proportions is vital for understanding and addressing large-scale issues in various fields.

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Most popular questions from this chapter

Conclusions A catalog sales company promises to deliver orders placed on the Internet within 3 days. Followup calls to a few randomly selected customers show that a \(95 \%\) confidence interval for the proportion of all orders that arrive on time is \(88 \% \pm 6 \% .\) What does this mean? Are these conclusions correct? Explain. a) Between \(82 \%\) and \(94 \%\) of all orders arrive on time. b) \(95 \%\) of all random samples of customers will show that \(88 \%\) of orders arrive on time. c) \(95 \%\) of all random samples of customers will show that \(82 \%\) to \(94 \%\) of orders arrive on time. d) We are \(95 \%\) sure that between \(82 \%\) and \(94 \%\) of the orders placed by the sampled customers arrived on time. e) On \(95 \%\) of the days, between \(82 \%\) and \(94 \%\) of the orders will arrive on time.

Mislabeled seafood In December \(2011,\) Consumer Reports published their study of labeling of seafood sold in New York, New Jersey, and Connecticut. They purchased 190 pieces of seafood from various kinds of food stores and restaurants in the three states and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(22 \%\) of these packages of seafood were mislabeled, incompletely labeled, or misidentified by store or restaurant employees. a) Construct a \(95 \%\) confidence interval for the proportion of all seafood packages in those three states that are mislabeled or misidentified. b) Explain what your confidence interval says about seafood sold in these three states. c) A 2009 report by the Government Accountability Board says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 190 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Teenage drivers An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. a) Create a \(95 \%\) confidence interval for the percentage of all auto accidents that involve teenage drivers. b) Explain what your interval means. c) Explain what "95\% confidence" means. d) A politician urging tighter restrictions on drivers" licenses issued to teens says, "In one of every five auto accidents, a teenager is behind the wheel." Does your confidence interval support or contradict this statement? Explain.

Conditions For each situation described below, identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a) Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe, b) A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public. c) A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change. d) A college admits 1632 freshmen one year, and four years later 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

Teachers A 2011 Gallup poll found that \(76 \%\) of Americans believe that high achieving high school students should be recruited to become teachers. This poll was based on a random sample of 1002 Americans. a) Find a \(90 \%\) confidence interval for the proportion of Americans who would agree with this. b) Interpret your interval in this context. c) Explain what "90\% confidence" means. d) Do these data refute a pundit's claim that \(2 / 3\) of Americans believe this statement? Explain.
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