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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 6.1.53a
Chapter 6, Problem 6.1.53a

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

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1
Step 1: Identify the formula for determining the minimum sample size for estimating a population mean. The formula is: n=zσ2E2, where n is the sample size, z is the z-score corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error.
Step 2: Determine the z-score for a 99% confidence level. For a 99% confidence interval, the z-score corresponds to the critical value where the area under the standard normal curve is 0.995 (since 99% confidence level leaves 0.5% in each tail). The z-score can be found using a z-table or statistical software.
Step 3: Substitute the given values into the formula. The population standard deviation σ is 0.5 inches, and the margin of error E is 0.15 inches. Replace these values along with the z-score into the formula: n=z⋅0.520.152.
Step 4: Simplify the numerator by squaring the product of the z-score and the population standard deviation. Then, simplify the denominator by squaring the margin of error.
Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. If the result is not a whole number, always round up to the nearest whole number, as sample size must be an integer.

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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 population parameter with a specified level of confidence. In this case, a 99% confidence interval means that if we were to take many samples and construct intervals, approximately 99% of those intervals would contain the true mean circumference of soccer balls.
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Introduction to Confidence Intervals

Sample Size Determination

Sample size determination involves calculating the number of observations needed to achieve a desired level of precision in estimating a population parameter. The formula for determining sample size for estimating a mean includes the desired margin of error, the population standard deviation, and the critical value corresponding to the confidence level.
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Population Standard Deviation

The population standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, it is given as 0.5 inches, which indicates how much individual soccer ball circumferences are expected to deviate from the mean circumference. This value is crucial for calculating the required sample size to ensure the estimate is within the specified margin of error.
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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.

[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

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

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.