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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 6.1.43c
Chapter 6, Problem 6.1.43c

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

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Understand the formula for the confidence interval: The width of a confidence interval is determined by the formula: \( \text{Width} = 2 \cdot Z \cdot \frac{\sigma}{\sqrt{n}} \), where \( Z \) is the critical value, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
Identify the role of the population standard deviation (\( \sigma \)): The population standard deviation is in the numerator of the formula for the margin of error, which directly affects the width of the confidence interval.
Analyze the effect of increasing \( \sigma \): When \( \sigma \) increases, the value of \( \frac{\sigma}{\sqrt{n}} \) also increases, leading to a larger margin of error.
Relate the margin of error to the width: Since the width of the confidence interval is twice the margin of error, an increase in \( \sigma \) will result in a wider confidence interval.
Conclude the relationship: Therefore, an increase in the population standard deviation will increase the width of the confidence interval, assuming all other factors remain constant.

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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%. The width of the interval reflects the uncertainty associated with estimating the parameter; a wider interval indicates more uncertainty.
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Population Standard Deviation

The population standard deviation is a measure of the dispersion or spread of a set of values in a population. It quantifies how much individual data points deviate from the population mean. A larger standard deviation indicates greater variability in the data, which can lead to wider confidence intervals when estimating population parameters.
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Effect of Standard Deviation on Confidence Interval Width

When the population standard deviation increases, the width of the confidence interval also increases, assuming all other factors remain constant. This is because a larger standard deviation indicates more variability in the data, necessitating a broader range to maintain the same level of confidence in capturing the true population parameter.
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Related Practice
Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

d. Find the minimum sample size needed to estimate the population proportion at the 99% confidence level to ensure that the estimate is accurate within 3% of the population proportion.

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Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.

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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.

c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.

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Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population 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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