Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.20a

Chapter 6, Problem 6.3.20a

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

Verified step by step guidance

Determine the formula for the minimum sample size required to estimate a population proportion with a given margin of error. The formula is:

n_{\min} = \frac{z^{2}}{E ²} × 0.25, where z is the z-score corresponding to the confidence level, E is the margin of error, and 0.25 is used when no preliminary estimate is available.

Identify the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. This value corresponds to the critical value of the standard normal distribution for a two-tailed test.

Substitute the margin of error (E) into the formula. The problem specifies that the estimate must be accurate within 5%, so

E = 0.05 .

Substitute the values of

z

and

E

into the formula:

n_{\min} = \frac{{1.96}^{2}}{0.05 ²} × 0.25 .

Simplify the expression to calculate the minimum sample size. Ensure that the result is rounded up to the nearest whole number, as sample size must be a whole number.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, the proportion of motor vehicle fatalities caused by alcohol-impaired driving. Understanding this concept is crucial for estimating how widespread an issue is within a larger group, which informs decisions and policies.

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed to ensure that the sample accurately reflects the population. In this context, it involves using statistical formulas to find the minimum number of fatalities required to estimate the population proportion with a specified level of confidence and margin of error.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, such as 95%. This concept is essential for understanding the reliability of the estimate, as it indicates how much uncertainty is associated with the sample data.

Recommended video:

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Introduction to Confidence Intervals

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.

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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

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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.

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

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.

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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.

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.

SAT Scores The SAT scores of 12 randomly selected high school seniors

Textbook Question

Ages of College Students An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages 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 1.6 years.