Thirty small communities in Connecticut (population near...

Thirty small communities in Connecticut (population near 10,000 each) gave an average of $\bar{x}=138.5$ reported cases of larceny per year. Assume that $\sigma$ is known to be $42.6$ cases per year (Reference: Crime in the United States, Federal Bureau of Investigation). (a) Find a $90 \%$ confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (b) Find a $95 \%$ confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?

Thirty small communities in Connecticut (population near 10,000 each) gave an average of $\bar{x}=138.5$ reported cases of larceny per year. Assume that $\sigma$ is known to be $42.6$ cases per year (Reference: Crime in the United States, Federal Bureau of Investigation).
(a) Find a $90 \%$ confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?
(b) Find a $95 \%$ confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error?

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. In the context of estimating the population mean, it accounts for the variability in the data and the confidence level chosen, providing a probabilistic estimate of where the true mean lies rather than a single point estimate.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and the sample estimate. It is calculated by multiplying the critical value (from the z-distribution when the population standard deviation is known) by the standard error. This margin determines the width of the confidence interval and reflects the precision of the estimate.

z-Score and Critical Value

When the population standard deviation is known and the sample size is moderately large, the z-score (or standard normal value) is used to determine the critical values for the confidence interval. These values correspond to the desired confidence level (e.g., 90% or 95%) and indicate the number of standard errors that one must extend from the sample mean to capture the true population mean.

Standard Error

The standard error measures the variability of the sample mean estimate relative to the population mean. It is calculated by dividing the known population standard deviation by the square root of the sample size. The smaller the standard error, the more precise the sample mean is as an estimate of the population mean.

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