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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 6.1.17
Chapter 6, Problem 6.1.17

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.
c = 0.88
Four horizontal number lines displaying confidence intervals for a population mean, with specific values marked on each line.

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Step 1: Understand the concept of confidence intervals. A confidence interval provides a range of values that is likely to contain the population parameter (e.g., mean) with a certain level of confidence. Higher confidence levels result in wider intervals, while lower confidence levels result in narrower intervals.
Step 2: Analyze the given confidence intervals in the images. Each interval is centered around the same sample statistic (57.2), but the width of the intervals varies. Wider intervals correspond to higher confidence levels, and narrower intervals correspond to lower confidence levels.
Step 3: Recall that the confidence level c = 0.88 is relatively low compared to common levels like 0.95 or 0.99. Therefore, the interval associated with c = 0.88 should be narrower than those for higher confidence levels.
Step 4: Compare the intervals in the images. The interval in option 'c' (55.6 to 58.8) is narrower than the intervals in options 'a' and 'b', but slightly wider than the interval in option 'd'. This suggests that option 'c' is likely to correspond to c = 0.88.
Step 5: Match the confidence level c = 0.88 with the appropriate interval. Based on the analysis, the interval in option 'c' (55.6 to 58.8) is the best match for c = 0.88.

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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 true population parameter. It is expressed as an interval estimate, typically calculated using the sample mean and a margin of error, which is influenced by the desired confidence level. For example, a 95% confidence interval suggests that if we were to take many samples, approximately 95% of the calculated intervals would contain the true mean.
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Introduction to Confidence Intervals

Confidence Level (c)

The confidence level, denoted as 'c', represents the probability that the confidence interval will contain the true population parameter. Common confidence levels include 90%, 95%, and 99%. A higher confidence level results in a wider interval, reflecting greater uncertainty about the exact value of the parameter. In this case, a confidence level of 0.88 indicates that we expect the true mean to fall within the interval 88% of the time.
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Introduction to Confidence Intervals

Margin of Error

The margin of error is the amount of error that is allowed in the estimation of a population parameter. It is calculated based on the standard error of the sample and the critical value associated with the desired confidence level. A larger margin of error results in a wider confidence interval, which can provide more assurance that the interval captures the true parameter, but may also reduce the precision of the estimate.
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Finding the Minimum Sample Size Needed for a Confidence Interval
Related Practice
Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.95, xbar = 31.39, σ = 0.80, n = 82.

Textbook Question

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

c = 0.95, σ = 2.5, E = 1.

Textbook Question

In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.

c = 0.98, xbar = 4.3, s = 0.34, n = 14

Textbook Question

In Exercises 13 and 14, use the confidence interval to find the margin of error and the sample mean.

(14.7, 22.1)

Textbook Question

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

c = 0.90, σ = 6.8, E = 1.

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 = 0.80, n = 51