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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 6.4.22a
Chapter 6, Problem 6.4.22a

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.
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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Step 1: Understand the problem. We are tasked with constructing a confidence interval for the population variance (σ²) using the sample standard deviation (s = 6.46), the sample size (n = 61), and a 98% confidence level. The population is assumed to be normally distributed.
Step 2: Recall the formula for the confidence interval of the population variance. The confidence interval is based on the chi-square distribution and is given by: [ (n-1)s² χ² upper , (n-1)s² χ² lower ] where n is the sample size, s² is the sample variance, and χ² values correspond to the chi-square critical values for the given confidence level.
Step 3: Calculate the sample variance (s²). The sample variance is the square of the sample standard deviation: 6.46 2
Step 4: Determine the degrees of freedom (df) and the chi-square critical values. The degrees of freedom are calculated as: df = n - 1 For a 98% confidence level, find the chi-square critical values (χ² lower and χ² upper) using a chi-square table or statistical software.
Step 5: Plug the values into the confidence interval formula. Substitute (n - 1), s², and the chi-square critical values into the formula to compute the lower and upper bounds of the confidence interval for the population variance. Interpret the results by explaining the range within which the true population variance is likely to fall with 98% confidence.

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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 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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Introduction to Confidence Intervals

Population Variance

Population variance (σ²) measures the dispersion of a set of values in a population. It is calculated as the average of the squared differences from the mean. In the context of confidence intervals, estimating the population variance helps in understanding the variability of the data and is crucial for constructing accurate intervals.
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Population Standard Deviation Known

Sample Standard Deviation

The sample standard deviation is a statistic that quantifies the amount of variation or dispersion in a sample data set. It is calculated as the square root of the sample variance. In this exercise, the sample standard deviation of 6.46 is used to estimate the population variance and to construct the confidence interval for the annual precipitation amounts.
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Calculating Standard Deviation
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 (a) the population variance σ^2. Interpret the results.

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

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.

Textbook Question

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

Textbook Question

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.

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

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.

Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

Use technology to find a 95% confidence interval for the population proportion of adults who

a. view foreign trade as an economic opportunity.