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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 6.4.22b
Chapter 6, Problem 6.4.22b

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (b) the population standard deviation σ. Interpret the results.
Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)

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1
Identify the key components of the problem: The sample size (n) is 61, the sample standard deviation (s) is 6.46, and the confidence level is 98%. The goal is to construct a confidence interval for the population standard deviation (σ).
Recognize that the confidence interval for the population standard deviation is based on the chi-square distribution. The formula for the confidence interval is: \( \left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{upper}}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{lower}}}} \right) \), where \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) are the critical values of the chi-square distribution.
Determine the degrees of freedom (df), which is \( n-1 \). In this case, \( df = 61 - 1 = 60 \).
Find the critical values \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) for a 98% confidence level. This involves splitting the remaining 2% (1% in each tail) and using a chi-square distribution table or calculator to find the critical values for \( df = 60 \).
Substitute the values into the confidence interval formula. Use \( n-1 = 60 \), \( s = 6.46 \), and the critical values \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) to calculate the lower and upper bounds of the confidence interval for \( \sigma \). Finally, interpret the interval in the context of the problem, explaining that it provides a range of plausible values for the population standard deviation of annual precipitation in Chicago.

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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 a sample, that is likely to contain the population parameter with a specified level of confidence. For example, a 98% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 98% of those intervals would contain the true population parameter.
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Population Standard Deviation (σ)

The population standard deviation (σ) 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. In the context of confidence intervals, estimating σ is crucial for determining the width of the interval and understanding the variability of the data.
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Sample Standard Deviation

The sample standard deviation is an estimate of the population standard deviation based on a sample from the population. It reflects the variability of the sample data and is calculated using the formula that divides the sum of squared deviations from the sample mean by the sample size minus one. This adjustment (Bessel's correction) helps provide an unbiased estimate of the population standard deviation.
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Related Practice
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 (b) the population standard deviation σ. Interpret the results.

[APPLET] Earnings The annual earnings (in thousands of dollars) of 21 randomly selected level 1 computer hardware engineers are listed. Use a 99% level of confidence. (Adapted from Salary.com)

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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 (b) the population standard deviation σ. Interpret the results.

Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

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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.

b. Find the minimum sample size needed, using a prior survey that found that 34% of U.S. adults disapprove of the U.S Senate’s use of the filibuster. (Source: Monmouth University)

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Textbook Question

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.

b. Find the minimum sample size needed, using a prior survey that found that 21% of likely U.S. voters think Congress is doing a good or excellent job. (Source: Rasmussen Reports)

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Textbook Question

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

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Textbook Question

Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. Your estimate must be accurate within 5% of the population proportion.

b. Find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)

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