In the Gallup poll described in Exercise 2,49 % of those...

In the Gallup poll described in Exercise $2,49 \%$ of those polled considered themselves to be "thriving." a) Calculate the margin of error for the proportion of all American adults who were rated as "thriving" for $99 \%$ confidence. b) Would the margin of error be larger or smaller for $95 \%$ confidence? Explain.

In the Gallup poll described in Exercise $2,49 \%$ of those polled considered themselves to be "thriving."
a) Calculate the margin of error for the proportion of all American adults who were rated as "thriving" for $99 \%$ confidence.
b) Would the margin of error be larger or smaller for $95 \%$ confidence? Explain.

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Key Concepts

Margin of Error

Margin of error is a statistic that quantifies the range in which the true population parameter is expected to fall, given a particular level of confidence. It accounts for sampling variability and is calculated using the critical value from the relevant probability distribution along with measures of variability from the sample.

Confidence Level

Confidence level indicates the probability that the calculated confidence interval includes the true population parameter. Common confidence levels are 95% or 99%, with a higher percentage implying greater certainty but also resulting in a wider interval, which means a larger margin of error.

Critical Value

The critical value is a factor based on the chosen confidence level from the statistical distribution (typically the standard normal distribution in large samples for proportions). It scales the standard error during the calculation of the margin of error. Higher confidence levels yield larger critical values, which in turn increase the margin of error.

Sample Proportion

The sample proportion is an estimate of the probability of occurrence of an event within a population, derived from a sample. It is essential in constructing confidence intervals for population proportions and directly affects the calculation of the margin of error.

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