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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 6.4.18b
Chapter 6, Problem 6.4.18b

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)

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Step 1: Identify the given information. The sample size (n) is 15, the sample standard deviation (s) is 26.1, and the confidence level is 80%. The goal is to construct a confidence interval for the population variance (σ²).
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 \( \chi^2_{\text{right}} \) 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 \( df = 15 - 1 = 14 \).
Step 4: Determine the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for \( df = 14 \) and an 80% confidence level. The confidence level splits the remaining 20% into two tails, so the left tail has 10% (0.10) and the right tail has 10% (0.90).
Step 5: Substitute the values into the confidence interval formula. Use \( s^2 = (26.1)^2 \) for the sample variance, \( df = 14 \), 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.

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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 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.
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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.
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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.
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Calculating Standard Deviation
Related Practice
Textbook Question

When all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? Explain.

b. Increase in the error tolerance

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.

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

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)

Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

Textbook Question

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

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

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.

b. Find the minimum sample size needed, using a prior study that found that 83% of U.S. families eat fast food at least once per week. (Source: The Barbecue Lab)