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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 6.3.18b
Chapter 6, Problem 6.3.18b

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.
b. Find the minimum sample size needed, using a prior survey that found that 34% of U.S. adults disapprove of the U.S Senate’s use of the filibuster. (Source: Monmouth University)

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1
Identify the formula for determining the minimum sample size for estimating a population proportion with a given confidence level and margin of error. The formula is: n >= z2p(1-p)E2, where z is the z-score corresponding to the confidence level, p is the estimated population proportion, and E is the margin of error.
Determine the values for the variables in the formula. The confidence level is 99%, so the z-score corresponding to this level is approximately 2.576. The estimated population proportion p is 0.34 (from the prior survey), and the margin of error E is 0.02 (2%).
Substitute the values into the formula: n >= 2.5762(0.34)(1-0.34)0.022.
Simplify the numerator by calculating 2.5762 and (0.34)(1-0.34). Then simplify the denominator by squaring 0.02.
Divide the simplified numerator by the simplified denominator to find the minimum sample size n. 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.

Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, U.S. adults who disapprove of the Senate's use of the filibuster. It is denoted as 'p' and is crucial for estimating how representative a sample will be of the entire population. Understanding this concept helps in determining the accuracy and reliability of survey results.
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Constructing Confidence Intervals for Proportions

Sample Size Calculation

Sample size calculation is the process of determining the number of observations or replicates needed to ensure that a statistical estimate is reliable and meets specified criteria, such as confidence level and margin of error. In this scenario, the formula incorporates the desired confidence level (99%) and the margin of error (2%) to find the minimum sample size required to estimate the population proportion accurately.
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Sampling Distribution of Sample Proportion

Confidence Level

The confidence level indicates the degree of certainty that the population parameter lies within a specified interval around the sample estimate. A 99% confidence level means that if the same sampling method were repeated multiple times, 99% of the calculated intervals would contain the true population proportion. This concept is essential for understanding the reliability of the estimate derived from the sample.
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Introduction to Confidence Intervals
Related Practice
Textbook Question

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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 (b) the population standard deviation σ. Interpret the results.

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Textbook Question

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Textbook Question

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Textbook Question

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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 (b) the population standard deviation σ. Interpret the results.

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