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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 6.3.19a
Chapter 6, Problem 6.3.19a

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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1
Step 1: Recall the formula for determining the minimum sample size for estimating a population proportion when no preliminary estimate is available: n = (Z^2 * 0.25) / E^2. Here, Z is the critical value corresponding to the desired confidence level, and E is the margin of error.
Step 2: Identify the given values. The confidence level is 90%, so the critical value Z can be found using a Z-table or standard normal distribution. For a 90% confidence level, Z ≈ 1.645. The margin of error E is given as 0.03 (3%).
Step 3: Substitute the values into the formula. Use Z = 1.645, E = 0.03, and the maximum variability for the population proportion (p = 0.5, q = 1 - p = 0.5, so p * q = 0.25). The formula becomes n = (1.645^2 * 0.25) / 0.03^2.
Step 4: Simplify the numerator by squaring the Z-value and multiplying it by 0.25. Then, simplify the denominator by squaring the margin of error (E).
Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. 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.

Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, U.S. families who eat fast food at least once per week. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a defined group.
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Constructing Confidence Intervals for Proportions

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to achieve a desired level of precision in estimates. In this scenario, it involves using the desired confidence level (90%) and margin of error (3%) to ensure that the sample accurately reflects the population proportion.
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Sampling Distribution of Sample Proportion

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. For this question, a 90% confidence interval means that if the same sampling method were repeated multiple times, 90% of the intervals would contain the true population proportion.
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Introduction to Confidence Intervals
Related Practice
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

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)

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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.

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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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.

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.

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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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