Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problema 6.3.18a

Capitolo 6, Problema 6.3.18a

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.
a. No preliminary estimate is available. Find the minimum sample size needed.

Guida verificata passo dopo passo

Step 1: Identify the formula for determining the minimum sample size for estimating a population proportion. The formula is:

n >= \frac{z^{2}}{} 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.

Step 2: Since no preliminary estimate of the population proportion is available, use

p = \frac{1}{2}

(i.e., 0.5) to maximize the required sample size. This is a conservative approach that ensures the sample size is large enough.

Step 3: Determine the z-score for a 99% confidence level. For a 99% confidence level, the z-score is approximately

2.576

. This value corresponds to the critical value in a standard normal distribution.

Step 4: Substitute the values into the formula. Use

z = 2.576

p = 0.5

, and

E = 0.02

(2% margin of error). The formula becomes:

n >= \frac{{2.576}^{2}}{} {0.02}^{2}

Step 5: Simplify the expression to calculate the minimum sample size. First, calculate the numerator

{2.576}^{2} * 0.5 * 0.5

, then divide by the square of the margin of error

{0.02}^{2}

. Round up the result to the nearest whole number, as sample size must be an integer.

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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. Understanding this concept is crucial for estimating how widespread an opinion is within a larger group, which is essential for statistical analysis and survey design.

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05:45

Constructing Confidence Intervals for Proportions

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 scenario, it involves using the desired confidence level and margin of error to find the minimum sample size required to estimate the population proportion with a specified accuracy.

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06:14

Coefficient of Determination

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 level indicates that if the same sampling method were repeated multiple times, 99% of the calculated intervals would contain the true population proportion, providing a measure of reliability for the estimate.

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06:33

Introduction to Confidence Intervals

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