The county finance department in Exercise 7.90 also wants...

The county finance department in Exercise 7.90 also wants information about renewals of disabled parking placards. Suppose that in a sample of 350 transactions for disabled parking placards, it was found that 250 were paid electronically. a. What is the margin of error for a $99 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically? b. Without calculating, is the margin of error for a $95 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically larger, smaller, or the same as that found in part a for a $99 \%$ confidence interval?

The county finance department in Exercise 7.90 also wants information about renewals of disabled parking placards. Suppose that in a sample of 350 transactions for disabled parking placards, it was found that 250 were paid electronically.
a. What is the margin of error for a $99 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically?
b. Without calculating, is the margin of error for a $95 \%$ confidence interval estimate of the population proportion of disabled renewal transactions paid electronically larger, smaller, or the same as that found in part a for a $99 \%$ confidence interval?

Key Concepts

Margin of Error

The margin of error quantifies the range within which the true parameter, such as a population proportion, is expected to fall with a certain level of confidence. It is calculated by multiplying a critical value by the standard error of the estimate, and it reflects the uncertainty inherent in the sampling process.

Confidence Interval

A confidence interval is a range of values, derived from a sample statistic, that is likely to contain the true population parameter. The width of the interval depends on the sample data variability and the chosen level of confidence, with higher confidence levels generally leading to wider intervals.

Population Proportion

The population proportion represents the fraction or percentage of the entire population that exhibits a particular characteristic. In the context of confidence intervals, an estimate of the population proportion is obtained from a sample, and the interval provides a range of plausible values for the true proportion in the entire population.

Standard Error

The standard error measures the variability or dispersion of a sample statistic, such as the sample proportion, from the true population parameter. It is an essential component in constructing confidence intervals and calculating the margin of error, as it reflects the precision of the sample estimate.

Critical Value

The critical value, often derived from a statistical distribution like the standard normal distribution, defines the number of standard errors to extend beyond the sample estimate to achieve a specific confidence level. Higher confidence levels correspond to larger critical values, which in turn increase the margin of error.

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