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The chi-square distribution is a fundamental concept when dealing with the variability of data, particularly in estimating variances. It’s used in various statistical tests and confidence interval calculations, especially when the data is assumed to be normally distributed. The chi-square distribution is asymmetrical, ranging from 0 to infinity, and it varies depending on the degrees of freedom, which is determined by the sample size \((n)\). For instance, in our exercise with a sample of 15 automobile repairs, the degrees of freedom are 14 (since it’s \(n-1\)). These degrees of freedom are crucial because they affect the shape of the chi-square distribution curve, which in turn influences the critical values we look up in the chi-square table. These critical values, \(\chi^2_{0.05, 14}\) and \(\chi^2_{0.95, 14}\), determine the confidence interval bounds for the variance and standard deviation. Understanding this distribution helps in estimating how spread out the repair costs could be within a certain confidence level, giving us a better grasp of the variability in costs.

Variance measures the spread or dispersion of a set of data points. When estimating the variance using a sample, we utilize confidence intervals to gauge how accurate our sample variance is in representing the true population variance. In this context, variance estimation becomes essential, as we need to know how much variability there is in the cost of automobile repairs. This helps in understanding the average distance between each data point from the mean. Using our calculated sample standard deviation of 120.82 and applying the chi-square critical values, we computed the confidence interval for the variance.The formula used for the variance confidence interval is straightforward:

• The numerator involves squaring the sample standard deviation and multiplying by the degrees of freedom \((n-1)\).
• The denominator consists of the chi-square critical values.

This method allows us to state that we are 90% confident that the true variance of repair costs lies between 102550.67 and 370423.71, which provides a quantified measure of the variability.

The standard deviation is an extension of the variance, serving as a measure of the spread of data, but expressed in the same units as the data itself. Calculating the standard deviation gives a direct sense of average deviation from the mean. When dealing with confidence intervals for the standard deviation, we follow a specific process involving the variance. For our given exercise, once we have derived the confidence interval for the variance, calculating the interval for the standard deviation is simple. You just take the square roots of the lower and upper bounds of the variance interval. This step transforms the variance interval from squared units back into the original units, making it more interpretable. Thus, from our calculations, the confidence interval for the standard deviation of automobile repair costs is about (320.21, 608.62). This tells us there’s a 90% chance that the true standard deviation of repair costs falls within this range, offering a meaningful insight into how dispersed the repair costs are on average.