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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.1.55
Chapter 6, Problem 6.1.55

When estimating the population mean, why not construct a 99% confidence interval every time?

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Understand the concept of a confidence interval: A confidence interval provides a range of values within which the true population parameter (e.g., the population mean) is likely to fall, based on a given confidence level (e.g., 95%, 99%). The confidence level represents the proportion of times the interval would capture the true parameter if the process were repeated many times.
Recognize the trade-off between confidence level and interval width: A higher confidence level (e.g., 99%) results in a wider confidence interval, meaning the estimate is less precise. Conversely, a lower confidence level (e.g., 90%) results in a narrower interval, providing a more precise estimate but with less certainty.
Consider the purpose of the analysis: If the goal is to make a highly precise estimate of the population mean, a narrower confidence interval (e.g., 95%) might be more appropriate. If the goal is to ensure a higher level of certainty, a wider interval (e.g., 99%) might be preferred.
Account for sample size and variability: A 99% confidence interval requires a larger sample size to maintain precision compared to a 95% confidence interval. If the sample size is small or the data is highly variable, constructing a 99% confidence interval may result in an interval that is too wide to be practically useful.
Balance precision and certainty: Constructing a 99% confidence interval every time may not always be necessary or efficient. The choice of confidence level should depend on the context of the problem, the desired balance between precision and certainty, and the resources available for data collection.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals, approximately 99% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals

Trade-off Between Confidence Level and Precision

When constructing confidence intervals, there is a trade-off between the confidence level and the width of the interval. A higher confidence level, such as 99%, results in a wider interval, which may be less precise. Conversely, a lower confidence level, like 90%, yields a narrower interval but with less certainty that it contains the true mean.
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Sample Size Considerations

The sample size plays a crucial role in determining the width of a confidence interval. Larger sample sizes lead to more precise estimates of the population mean and narrower confidence intervals. Therefore, consistently using a 99% confidence interval may not be practical if the sample size is small, as it could result in overly broad intervals that are less informative.
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Related Practice
Textbook Question

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

Textbook Question

What happens to the shape of the chi-square distribution as the degrees of freedom increase?

Textbook Question

Finding Critical Values for χ2 In Exercises 3–8, find the critical values χR2 and χL2 for the level of confidence c and sample size n.

c = 0.99, n = 30

Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(21.61, 30.15)

Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.80, σ = 4.1, E = 2.

Textbook Question

In Exercise 31, the population mean salary is \$67,319. Does the t-value fall between -t0.98 and t0.98? (Source: Salary.com)