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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 6.1.3
Chapter 6, Problem 6.1.3

For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.
a. 90%
b. 95%
c. 98%
d. 99%

Verified step by step guidance
1
Understand the concept of a confidence interval: A confidence interval provides a range of values within which we expect the true population parameter (e.g., mean or proportion) to lie, based on a sample statistic. The level of confidence indicates the probability that the interval contains the true parameter.
Recognize the relationship between the confidence level and the width of the confidence interval: Higher confidence levels require a wider interval to ensure that the true parameter is captured within the range. This is because higher confidence levels demand greater certainty, which necessitates accounting for more variability.
Recall the critical value associated with each confidence level: The critical value (z* or t*) increases as the confidence level increases. For example, the critical value for a 99% confidence level is larger than that for a 90% confidence level. This directly impacts the width of the interval.
Examine the formula for a confidence interval: For a population mean, the formula is typically \( \text{CI} = \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \), where \( z^* \) is the critical value. A larger \( z^* \) results in a wider margin of error, and thus a wider confidence interval.
Compare the given confidence levels (90%, 95%, 98%, 99%): Since 99% is the highest confidence level among the options, it will produce the widest confidence interval because it requires the greatest certainty and accounts for the most variability.

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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 with a certain level of confidence, such as 90%, 95%, or 99%, indicating the probability that the interval includes the parameter. Wider intervals suggest more uncertainty about the parameter's exact value.
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Level of Confidence

The level of confidence represents the degree of certainty that the confidence interval contains the true population parameter. Common levels include 90%, 95%, and 99%. A higher level of confidence means that the interval is constructed to be wider, as it accounts for more variability and uncertainty in the data.
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Critical Value

The critical value is a factor used to calculate the margin of error in a confidence interval. It is derived from the standard normal distribution (Z-distribution) or t-distribution, depending on the sample size and whether the population standard deviation is known. As the level of confidence increases, the critical value also increases, leading to a wider confidence interval.
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Related Practice
Textbook Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (c) construct a 98% confidence interval for the population mean.

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Textbook Question

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.

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Textbook Question

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Textbook Question

Determining a Minimum Sample Size Determine the minimum sample size required when you want to be 99% confident that the sample mean is within two units of the population mean and σ = 1.4. Assume the population is normally distributed.

Textbook Question

Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.

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Textbook Question

Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.