The Wall Street Journal reported that automobile crashes...

The Wall Street Journal reported that automobile crashes cost the United States $\$ 162$ billion annually (The Wall Street Joumal, March $5,2008$ ). The average cost per person for crashes in the Tampa, Flonida, 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 fora 95$\%$ confidence interval? What would you recommend if the study required a margin of enror of $\$ 150$ or less?

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of a population parameter. It is constructed using a specified confidence level (such as 95%), which indicates the probability that the interval calculated from random samples will capture the parameter.

Margin of Error

The margin of error quantifies the uncertainty in a sample estimate by representing the maximum expected difference between the true population parameter and the sample statistic. It is computed as the product of the critical value and the standard error, determining the width of the confidence interval.

Critical Value (z-Score)

The critical value, especially in scenarios where the population standard deviation is known, is the z-score corresponding to the desired confidence level. It specifies how many standard deviations from the mean the endpoints of the confidence interval are set, for example, approximately 1.96 for a 95% confidence level.

Standard Error

The standard error measures the variability of the sample mean around the population mean and is calculated by dividing the population standard deviation by the square root of the sample size. It reflects the precision of the sample estimate in representing the population parameter.

Sample Size Determination

Sample size determination involves calculating the number of observations required to ensure that the margin of error of an estimate does not exceed a specified threshold. This is typically done by rearranging the margin of error formula to solve for the sample size, ensuring sufficient precision in the results.

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