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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 6.1.43a
Chapter 6, Problem 6.1.43a

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.
a. Increase in the level of confidence

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1
Understand the relationship between the level of confidence and the width of a confidence interval: A higher confidence level means we want to be more certain that the interval contains the true population parameter. This requires a wider interval to account for more variability.
Recall the formula for a confidence interval: \( \text{Confidence Interval} = \bar{x} \pm z^* \frac{s}{\sqrt{n}} \), where \( z^* \) is the critical value corresponding to the desired confidence level, \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, and \( n \) is the sample size.
Note that an increase in the confidence level leads to a larger critical value \( z^* \), as higher confidence levels correspond to capturing more of the standard normal distribution's area under the curve.
Recognize that a larger \( z^* \) directly increases the margin of error \( z^* \frac{s}{\sqrt{n}} \), which in turn increases the width of the confidence interval.
Conclude that increasing the level of confidence results in a wider confidence interval, as the interval must expand to ensure a higher probability of containing the true population parameter.

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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, typically 95% or 99%, indicating the degree of certainty that the parameter lies within the interval. The width of the interval reflects the precision of the estimate; narrower intervals suggest more precise estimates.
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Level of Confidence

The level of confidence represents the probability that the confidence interval will contain the true population parameter if the same sampling method is repeated multiple times. Common levels of confidence are 90%, 95%, and 99%. Increasing the level of confidence results in a wider confidence interval, as it requires a broader range to ensure that the true parameter is captured within the interval.
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Margin of Error

The margin of error is the amount of error that is allowed in the estimation of a population parameter. It is influenced by the level of confidence and the variability in the data. A higher margin of error leads to a wider confidence interval, as it accounts for greater uncertainty in the estimate, which is particularly relevant when the level of confidence is increased.
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Related Practice
Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

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Textbook Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

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Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

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Textbook Question

Juice Dispensing Machine A beverage company uses a machine to fill half-gallon bottles with fruit juice (see figure). The company wants to estimate the mean volume of water the machine is putting in the bottles within 0.25 fluid ounce.

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a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 1 fluid ounce.

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Textbook Question

Cholesterol Contents of Cheese A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.75 milligram of the population mean.

a. Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.10 milligrams.

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Textbook Question

Senate Filibuster You wish to estimate, with 99% confidence, the population proportion of U.S. adults who disapprove of the U.S Senate’s use of the filibuster. Your estimate must be accurate within 2% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

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