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Chapter 18: Problem 7

Confidence intervals 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, higher confidence means a smaller margin of error. b) For a specified confidence level, larger samples provide smaller margins of error. c) For a fixed margin of error, larger samples provide greater confidence. d) For a given confidence level, halving the margin of error requires a sample twice as large.

Short Answer

Expert verified
Statements b and c are true.

Step by step solution

01

Analyze Statement (a)

First, let's consider statement (a). As the confidence level increases (e.g., from 95% to 99%), the interval also expands to ensure it captures the true parameter more reliably. This means the margin of error actually increases with higher confidence levels due to a larger z* value from the normal distribution table.
02

Analyze Statement (b)

For statement (b), a larger sample size decreases the standard error, which is the denominator in the margin of error formula \( \text{Margin of Error} = z* \left( \frac{\sigma}{\sqrt{n}} \right) \). With a smaller standard error, the margin of error decreases if the confidence level is kept constant.
03

Analyze Statement (c)

Statement (c) explores a fixed margin of error at larger sample sizes. With a larger sample, to maintain the same margin of error, the standard deviation of the sample mean decreases, allowing for a higher confidence interval without expanding the margin of error.
04

Analyze Statement (d)

For a given confidence level, halving the margin of error means we need 4 times the original sample size. This is because the sample size \( n \) is inversely proportional to the square of the margin of error in the formula \( \text{Margin of Error} = z* \left( \frac{\sigma}{\sqrt{n}} \right) \).

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Key Concepts

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

Sample Size
Sample size is one of the key factors affecting the accuracy and reliability of confidence intervals. Think of it as the number of observations you include in your data analysis. A larger sample size generally results in more precise estimates of the population parameters.

This is because a larger sample size decreases the variability, or the spread, of the sampling distribution of a statistic like the mean. This in turn affects the standard error, making the confidence interval narrower and the estimates more reliable.

  • A low sample size can lead to high variability and less reliable estimates.
  • A high sample size leads to low variability, giving a clearer picture of the population.

When considering the role of sample size in context, remember that increasing your sample size not only bolsters accuracy but can also reduce your margin of error.
Level of Confidence
The level of confidence reflects how certain we are that our population parameter falls within the calculated confidence interval. It's usually represented as a percentage, like 95% or 99%, meaning there's a 95% or 99% chance the interval contains the true parameter.

The relationship between the level of confidence and the confidence interval is that as you increase your level of confidence, the interval becomes wider. This is because a higher level of confidence requires capturing more of the distribution, thus spreading the interval.
  • A higher confidence level, such as 99%, offers more assurance but results in a wider interval.
  • A lower confidence level, like 90%, produces a narrower interval but with less assurance that it captures the true parameter.

In summary, choosing your confidence level involves a trade-off between certainty and precision.
Margin of Error
The margin of error measures the amount of error you can expect in the estimates of your parameter. It literally forms the boundaries of your confidence interval, indicating the range of uncertainly around your sample statistic.

The margin of error shrinks or grows based on several factors:
  • Sample size: Larger samples mean a smaller margin of error.
  • Confidence Level: A higher level of confidence results in a larger margin of error because the interval widens.
  • Standard deviation: Greater variability in the data leads to a larger margin of error.

This makes the margin of error an important concept as it reflects the confidence and precision of statistical estimates. Lowering the margin of error typically requires either increasing your sample size or accepting a lower confidence level.
Standard Error
Standard error plays a crucial role in determining the precision of sample estimates within confidence intervals. It is the estimated standard deviation of a sample mean distribution, often symbolized as \(SE\). Standard error diminishes as sample size increases, making it essential for calculating both the margin of error and the confidence interval itself.

Here鈥檚 how it works:
  • Formula: The standard error is calculated using \(SE = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
  • Larger samples lead to smaller standard errors, creating a more "centered" confidence interval.
  • Smaller standard errors result in narrower confidence intervals, meaning more precise estimates.

Understanding standard error helps explain why larger sample sizes are typically sought for tighter confidence intervals and more reliable results.

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Most popular questions from this chapter

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?

Gambling A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome. The local newspaper finds that \(53 \%\) of 1200 randomly selected voters plan to vote "yes," while a college Statistics class finds \(54 \%\) of 450 randomly selected voters in support. Both groups will create \(95 \%\) confidence intervals. a) Without finding the confidence intervals, explain which one will have the larger margin of error. b) Find both confidence intervals. c) Which group concludes that the outcome is too close to call? Why?

More conclusions In January \(2002,\) two students made worldwide headlines by spinning a Belgian euro 250 times and getting 140 heads - that's \(56 \% .\) That makes the \(90 \%\) confidence interval \((51 \%, 61 \%) .\) What does this mean? Are these conclusions correct? Explain. a) Between \(51 \%\) and \(61 \%\) of all euros are unfair. b) We are \(90 \%\) sure that in this experiment this euro landed heads on between \(51 \%\) and \(61 \%\) of the spins. c) We are \(90 \%\) sure that spun euros will land heads between \(51 \%\) and \(61 \%\) of the time. d) If you spin a curo many times, you can be \(90 \%\) sure of getting between \(51 \%\) and \(61 \%\) heads. e) \(90 \%\) of all spun euros will land heads between \(51 \%\) and \(61 \%\) of the time.

Another pilot study During routine screening, a doctor notices that \(22 \%\) of her adult patients show higher than normal levels of glucose in their blood-a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within \(4 \%\) with \(98 \%\) confidence. How many randomly selected adults must they test?

Baseball fans In a national poll taken in February 2008 Gallup asked 1006 adults whether they were baseball fans; \(43 \%\) said they were. Two months previously, in December \(2007,40 \%\) of a similar-size sample had reported being baseball fans. a) Find the margin of error for the 2008 poll if we want \(90 \%\) confidence in our estimate of the percent of national adults who are baseball fans. b) Explain what that margin of error means. c) If we wanted to be \(99 \%\) confident, would the margin of error be larger or smaller? Explain. d) Find that margin of error. e) In general, if all other aspects of the situation remain the same, will smaller margins of error produce greater or less confidence in the interval?
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