Crash Test Results The following data represent the repair...

Crash Test Results The following data represent the repair cost for a low-impact collision in a simple random sample of miniand micro-vehicles (such as the Chevrolet Aveo or Mini Cooper). In Problem 33 from Section $9.2,$ it was verified that the data come from a population that is normally distributed with no outliers and = 1007.4542 dollar . Construct and interpret a $90 \%$ confidence interval for the standard deviation repair cost of a low-impact collision involving mini- and micro-vehicles. $$ \begin{array}{lrlrr} \hline \$ 3148 & \$ 1758 & \$ 1071 & \$ 3345 & \$ 743 \\ \hline \$ 2057 & \$ 663 & \$ 2637 & \$ 773 & \$ 1370 \\ \hline \end{array} $$

Crash Test Results The following data represent the repair cost for a low-impact collision in a simple random sample of miniand micro-vehicles (such as the Chevrolet Aveo or Mini Cooper). In Problem 33 from Section $9.2,$ it was verified that the data come from a population that is normally distributed with no outliers and = 1007.4542 dollar . Construct and interpret a $90 \%$ confidence interval for the standard deviation repair cost of a low-impact collision involving mini- and micro-vehicles.

$$
\begin{array}{lrlrr}
\hline \$ 3148 & \$ 1758 & \$ 1071 & \$ 3345 & \$ 743 \\
\hline \$ 2057 & \$ 663 & \$ 2637 & \$ 773 & \$ 1370 \\
\hline
\end{array}
$$

Key Concepts

Standard Deviation and Variance

Standard deviation is a measure of the amount of dispersion or spread in a set of values, while variance is the square of the standard deviation. Estimating these measures accurately is crucial in understanding the reliability and variability inherent in the data. Confidence intervals provide a statistical method to assess the precision of these estimates from sample data.

Normal Distribution

The assumption of normality is critical when constructing confidence intervals for variance and standard deviation using the chi-square distribution. The properties of the normal distribution guarantee that the sample variance, under proper standardization, follows a chi-square distribution, enabling the derivation of valid interval estimates for the population standard deviation.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. In the context of standard deviation, it quantifies the uncertainty around the estimate, providing an interval within which the true standard deviation is expected to lie with a certain level of confidence (90% in this instance). This interval is computed under the assumption that the data come from a known distribution.

Chi-Square Distribution

The chi-square distribution is used when constructing confidence intervals for the variance or standard deviation of a normally distributed population. When the sample variance is scaled by the true variance and multiplied by the degrees of freedom, the resulting variable follows a chi-square distribution. This property is essential for deriving the confidence interval formulas associated with the variability measures.

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