The Wall Street Journal reported that automobile crashes cost the United States \(\$ 162\) billion annually (The Wall Street Journal, March 5,2008 ). The average cost per person for crashes in the Tampa, Florida, area was reported to be \(\$ 1599 .\) Suppose this average cost was based on a sample of 50 persons who had been involved in car crashes and that the population standard deviation is \(\sigma=\$ 600 .\) What is the margin of error for a \(95 \%\) confidence interval? What would you recommend if the study required a margin of error of \(\$ 150\) or less?

Understanding the confidence interval (CI) is essential when analyzing data and making inferences about a population. A CI is a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter.

In our example, we're focusing on estimating the average cost of car crashes per person in Tampa, Florida. The CI gives us a range that we are 95% confident contains the actual average cost for the entire population. It is important to note that '95%' refers to the confidence level, which indicates how sure we are that the interval includes the true value. If we were to take many samples and create a CI from each one, we expect about 95% of those intervals to contain the true population parameter. The actual average cost is not a range, but the CI provides a way to express the uncertainty inherent in our estimate.

The standard error (SE) measures the amount of variability or dispersion of a sample mean in relation to the true population mean.

In this context, the SE tells us how far the sample mean (The standard error is calculated from the population standard deviation and the sample size, as seen in the exercise, where we use the formula
\[SE = \frac{\sigma}{\sqrt{n}} \]. By knowing the SE, we can gauge the accuracy of our sample mean in representing the population mean—the smaller the SE, the more reliable the sample mean. However, it's worth noting that the SE alone doesn't provide a direct answer to our final question; it's an intermediate step towards calculating the confidence interval's margin of error.

The population standard deviation (\(\sigma\)), a key component of our formula, is a measure of the dispersion or variation in a set of data values.

A small standard deviation indicates that the values tend to be close to the mean of the set, while a larger one shows that the values are spread out over a wider range. In the given exercise, the population standard deviation is \(\sigma=\$600\), signifying the variability in individual costs of car crashes in the population under study. This figure is crucial for calculating the standard error, which in turn is vital for determining the margin of error for the confidence interval.

Sample size (\(n\)) plays a pivotal role in statistical analysis. It refers to the number of observations or data points included in a sample drawn from a population.

As demonstrated in the step-by-step solution, increased sample size generally results in a lower standard error and thereby a more narrow margin of error, leading to a more precise confidence interval. If the survey of Tampa's car crash costs aimed for a margin of error less than \(\$150\), then we would recommend a larger sample size than the initial 50 participants. This change would provide more reliability and precision in the results.

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean.

In the context of our confidence interval, the z-score corresponds to the desired level of confidence and is used to capture the tail areas of the normal distribution needed for our estimation. For a 95% confidence interval, the appropriate z-score is approximately 1.96. This means that our margin of error extends 1.96 standard deviations from the sample mean on both sides, establishing the range where we are 95% confident the true mean lies.