Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.18b

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)

Verified step by step guidance

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 >= \frac{z^{2}}{p} E^{2}

, 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 >= \frac{{2.576}^{2}}{(} {0.02}^{2}

Simplify the numerator by calculating

{2.576}^{2}

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.

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.

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

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Textbook Question

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