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The normal distribution, often called the bell curve, is one of the most important probability distributions in statistics. It describes how the values of a variable are distributed. In a normal distribution, data points are symmetrically distributed around the mean, with most values clustering around a central region. This property makes the normal distribution extremely useful for various statistical analyses.
For our travel tax data, it was already verified that the data are normally distributed. This means that we can use statistical methods, such as calculating confidence intervals, which rely on normal distribution assumptions. Knowing our data follows a normal distribution allows us to make accurate inferences about the population standard deviation using sample data.

The chi-square distribution is crucial when dealing with sample variances, especially for constructing confidence intervals for the standard deviation. This distribution is used to test hypotheses about how the sample variance compares to the true population variance.
The shape of the chi-square distribution changes depending on the degrees of freedom (df), which is usually the sample size minus one ( - 1).
In our exercise, we used the chi-square distribution with 7 degrees of freedom (df = 8 - 1 = 7). To compute the confidence interval, we found the critical values of the chi-square distribution at the 90% confidence level. These values played a key role in determining the range within which the population variance, and consequently the standard deviation, lies with a given level of certainty.

Sample variance measures how data points in a sample are spread out around the mean. It is the average of the squared differences between each data point and the mean of the sample.
Mathematically, the sample variance ( s^2ds) is calculated as:
$$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} $$,
where x_i denotes each individual value in the sample, \bar{x} is the sample mean, and n is the sample size.
In our solved problem, given that the sample standard deviation (s) is 12.324 dollars, the sample variance is calculated as:
$$ s^2 = (12.324)^2 = 151.85 $$. This sample variance was then used along with the chi-square critical values to determine the confidence interval for the travel tax standard deviation.

Degrees of freedom (df) refer to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of variance and standard deviation, the degrees of freedom are often the sample size minus one (-1).
This term is significant because it affects the shape and properties of the chi-square distribution we use to create confidence intervals.
For our exercise, with a sample size of 8 cities, the degrees of freedom were:
= 8 - 1 = 7.
These 7 degrees of freedom helped us determine the appropriate critical values from the chi-square distribution table, which were essential to calculate the 90% confidence interval for the standard deviation.