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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 6.4.23a
Chapter 6, Problem 6.4.23a

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.

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Step 1: Identify the given information. The sample size (n) is 28, the sample standard deviation (s) is 56.1, and the confidence level is 98%. The population is assumed to be normally distributed.
Step 2: Recall the formula for the confidence interval for the population variance (σ²). The confidence interval is given by: \( \left( \frac{(n-1)s^2}{\chi^2_{\text{right}}}, \frac{(n-1)s^2}{\chi^2_{\text{left}}} \right) \), where \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) are the critical values of the chi-square distribution for the given confidence level.
Step 3: Calculate the degrees of freedom (df). The degrees of freedom is \( n-1 \), so \( df = 28 - 1 = 27 \).
Step 4: Determine the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for \( df = 27 \) and a 98% confidence level. The critical values correspond to the upper and lower tails of the distribution, with \( \alpha/2 = 0.01 \) in each tail.
Step 5: Substitute the values into the confidence interval formula. Use \( s^2 = (56.1)^2 \), \( n-1 = 27 \), and the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) to compute the lower and upper bounds of the confidence interval for the population variance. Finally, interpret the results by stating that you are 98% confident the true population variance lies within the calculated interval.

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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 statistic, 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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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. Understanding population variance is crucial for constructing confidence intervals, as it helps quantify the uncertainty around the sample estimate.
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Population Standard Deviation Known

Sample Standard Deviation

Sample standard deviation (s) is a statistic that measures the amount of variation or dispersion in a sample data set. It is calculated as the square root of the sample variance. In the context of constructing confidence intervals, the sample standard deviation is used to estimate the population standard deviation, which is essential for determining the width of the confidence interval.
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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 (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

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

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.

b. What was the greatest value you obtained for p^?

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

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)

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.