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Chapter 16: Problem 28

Confidence intervals, again Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? a. For a given sample size, reducing the margin of error will mean lower confidence. b. For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample. c. For a fixed margin of error, smaller samples will mean lower confidence. d. For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

Short Answer

Expert verified
a. True, b. True, c. True, d. False

Step by step solution

01

Analysis of statement a

The statement is true. For a fixed sample size, reducing the margin of error necessitates a decrease in the confidence level because a narrow margin of error suggests greater precision which is only possible at a lower confidence level.
02

Analysis of statement b

The statement is true. For a fixed confidence level, a larger sample size can provide a smaller margin of error. This is because larger samples tend to be more representative of the population, thus reducing the variability and margin of error.
03

Analysis of statement c

Statement c is true. For a given margin of error, smaller samples will result in lower confidence levels. This is because smaller samples are less representative of the population, leading to greater variability and a lower confidence level at the given margin of error.
04

Analysis of statement d

The statement is false. For a given confidence level, increasing the sample size doesn't necessarily decrease the margin of error in the way described. The relationship between sample size and margin of error is more complex and doesn't follow a direct proportionality or inverse square root law. The decrease in the margin of error is less dramatic as the sample size gets larger.

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Key Concepts

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

Sample Size
The concept of sample size is integral to understanding confidence intervals. Essentially, the sample size refers to the number of observations or data points collected from a population to create a sample.
A larger sample size typically provides a more reliable representation of the population. This is because it tends to capture more variations in the data. Here are some key points about sample size:
  • A larger sample size reduces variability, which can lead to more accurate results.
  • Increasing the sample size often means a smaller margin of error and a higher confidence level.
  • While more data is generally better, it can also aid in identifying outliers and trends within a population data set.
Understanding how to manipulate the sample size can significantly impact the precision and reliability of a confidence interval.
Margin of Error
The margin of error is the range within which you expect the true population parameter to fall. It reflects the precision of your estimate and provides a way to quantify uncertainty.
Here are some important insights into the margin of error:
  • A smaller margin of error indicates more precise confidence intervals but usually requires a larger sample size or lower confidence level.
  • A large margin of error might suggest a need for an increased sample size or adjustments to the confidence level to improve precision.
  • The margin of error is typically determined by the standard deviation of the population and the size of the sample.
A tight margin of error is usually desirable as it provides a clearer, more precise measure of the true population parameter.
Confidence Level
Confidence level is the percentage that expresses how certain we are that the true population parameter falls within the confidence interval. It is a critical component of inferential statistics.
Here's what you should know about confidence level:
  • Common confidence levels are 90%, 95%, and 99%, with 95% being the most frequently used in research.
  • A higher confidence level means a wider confidence interval, which suggests more certainty but less precision.
  • The confidence level is directly related to the margin of error: a higher confidence level results in a larger margin of error if the sample size remains constant.
Finding the right balance is key, as the confidence level impacts both the confidence interval's precision and the statistical significance of the results.

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Most popular questions from this chapter

Rickets Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950 s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2700 children randomly selected from all parts of England found \(20 \%\) of them deficient in vitamin D. a. Find a \(98 \%\) confidence interval. b. Explain carefully what your interval means. c. Explain what "98\% confidence" means.

Deer ticks Wildlife biologists inspect 153 deer taken by hunters and find 32 of them carrying ticks that test positive for Lyme disease. a. Create a \(90 \%\) confidence interval for the percentage of deer that may carry such ticks. b. If the scientists want to cut the margin of error in half, how many deer must they inspect? c. What concerns do you have about this sample?

Spanking In a 2015 Pew Research study on trends in marriage and family (www.pewsocialtrends.org/2015/12/17/1the-american-family-today/), \(53 \%\) of randomly selected parents said that they never spank their children. The \(95 \%\) confidence interval is from \(50.6 \%\) to \(55.4 \%(n=1807)\). a. Interpret the interval in this context. b. Explain the meaning of "95\% confident" in this context.

Conclusions A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up 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. Ninety-five percent of all random samples of customers will show that \(88 \%\) of orders arrive on time. c. Ninety-five percent 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.

Pilot study A state's environmental agency worries that many cars may be violating clean air emissions standards. The agency hopes to check a sample of vehicles in order to estimate that percentage with a margin of error of \(3 \%\) and \(90 \%\) confidence. To gauge the size of the problem, the agency first picks 60 cars and finds 9 with faulty emissions systems. How many should be sampled for a full investigation?
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