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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 6.4.15a
Chapter 6, Problem 6.4.15a

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)
Table displaying annual earnings (in thousands) of 21 computer hardware engineers, with values ranging from 52.4 to 95.5.

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Step 1: Calculate the sample variance (s^2). First, compute the sample mean (x̄) by summing all the earnings values and dividing by the sample size (n = 21). Then, use the formula for variance: s^2 = (Σ(xi - x̄)^2) / (n - 1), where xi represents each individual data point.
Step 2: Identify the degrees of freedom (df). For this problem, df = n - 1, where n is the sample size. Since n = 21, df = 20.
Step 3: Use the chi-square distribution to find the critical values for the 99% confidence interval. The chi-square critical values are determined using the chi-square table or a statistical calculator for df = 20 and the desired confidence level (99%). The lower critical value corresponds to α/2, and the upper critical value corresponds to 1 - α/2, where α = 0.01.
Step 4: Construct the confidence interval for the population variance (σ^2) using the formula: [(df * s^2) / χ²_upper, (df * s^2) / χ²_lower], where χ²_upper and χ²_lower are the chi-square critical values.
Step 5: Interpret the results. The confidence interval provides a range of values within which the true population variance is likely to fall with 99% confidence. This interval reflects the variability in annual earnings for level 1 computer hardware engineers.

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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 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population parameter.
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Population Variance

Population variance is a measure of the dispersion of a set of values in a population, indicating how much the values deviate from the population mean. It is denoted by σ² and is calculated as the average of the squared differences from the mean. Understanding population variance is crucial for constructing confidence intervals for variance.
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Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Many statistical methods, including confidence interval calculations, assume that the underlying data follows a normal distribution, especially when sample sizes are small.
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Related Practice
Textbook Question

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

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

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.

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)

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.

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

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (a) find the sample mean

[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

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.