Skip to main content
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 6.4.18a
Chapter 6, Problem 6.4.18a

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.
Volleyball The numbers of service aces scored by 15 teams randomly selected from the top 50 NCAA Division I Women’s Volleyball teams for the 2021 season have a sample standard deviation of 26.1. Use an 80% level of confidence. (Source: National Collegiate Athletic Association)

Verified step by step guidance
1
Step 1: Understand the problem. We are tasked with constructing an 80% confidence interval for the population variance (σ²) based on a sample standard deviation (s = 26.1) and a sample size (n = 15). 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 given by: \( \left( \frac{(n-1)s^2}{\chi^2_{\text{right}}}, \frac{(n-1)s^2}{\chi^2_{\text{left}}} \right) \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) are the critical values of the chi-square distribution for the given confidence level.
Step 3: Calculate the sample variance \( s^2 \). The sample variance is the square of the sample standard deviation: \( s^2 = (26.1)^2 \).
Step 4: Determine the degrees of freedom and the critical chi-square values. The degrees of freedom (df) are \( n-1 = 15-1 = 14 \). For an 80% confidence level, find the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for df = 14. These correspond to the upper and lower tails of the distribution, with 10% in each tail (since 80% is the middle area).
Step 5: Plug the values into the confidence interval formula. Substitute \( n-1 = 14 \), \( s^2 \), and the critical chi-square values into the formula \( \left( \frac{(n-1)s^2}{\chi^2_{\text{right}}}, \frac{(n-1)s^2}{\chi^2_{\text{left}}} \right) \). Simplify the expressions to find the confidence interval for the population variance. Finally, interpret the interval in the context of the problem.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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 true population parameter with a specified level of confidence. For example, an 80% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 80% of those intervals would contain the true population parameter.
Recommended video:
06:33
Introduction to Confidence Intervals

Population Variance

Population variance (σ²) measures the spread 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 the question, constructing a confidence interval for the population variance allows us to estimate the range within which the true variance of service aces scored by all teams lies.
Recommended video:
04:48
Population Standard Deviation Known

Sample Standard Deviation

Sample standard deviation (s) 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 26.1 is crucial for estimating the population variance and constructing the confidence interval, as it reflects the variability of service aces among the selected teams.
Recommended video:
Guided course
08:45
Calculating Standard Deviation
Related Practice
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.

[IMAGE]

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

[IMAGE]

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.

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.

[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)

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.

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

Textbook Question

Soccer Balls A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.15 inch.

a. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.5 inch

Textbook Question

Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed.

a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.75 ounce.

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.

[IMAGE]

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.