Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.4.21a

Chapter 6, Problem 6.4.21a

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.
Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

Verified step by step guidance

Step 1: Identify the given information. The sample size (n) is 18, the sample standard deviation (s) is 0.25 hours, 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 \( df = 18 - 1 = 17 \).

Step 4: Determine the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for \( df = 17 \) 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: Plug the values into the confidence interval formula. Use \( s^2 = (0.25)^2 \) 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 interval in the context of the problem.

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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 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.

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. Understanding population variance is crucial for constructing confidence intervals, as it helps quantify the uncertainty around the estimate of the population parameter.

Sample Standard Deviation

Sample standard deviation is a statistic that measures the dispersion of a sample data set around its mean. It is calculated by taking the square root of the sample variance. In the context of confidence intervals, the sample standard deviation is used to estimate the variability of the population, which is essential for determining the width of the confidence interval.

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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.

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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.

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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)

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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.

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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.

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Textbook Question

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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