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In statistics, the term 'significance level' is used to describe the probability of rejecting the null hypothesis when it is actually true. This is denoted by the symbol \( \alpha \) and is a crucial part of hypothesis testing. For instance, in the exercise above, a 90% confidence interval implies a significance level of 10% (i.e., \( \alpha = 0.10 \)). In simpler terms, the significance level represents the risk you are willing to take of making a Type I error, which is incorrectly rejecting the null hypothesis. By carefully choosing \( \alpha \), researchers can control the probability of making such errors.

Degrees of freedom (df) represent the number of independent values or quantities which can be assigned to a statistical distribution. In the context of the chi-square distribution, the degrees of freedom are typically calculated as the sample size (n) minus one \( (df = n - 1) \). This adjustment accounts for the number of parameters estimated from the data. For the given problem, with a sample size of 20, the degrees of freedom are 19 (i.e., \( df = 20 - 1 \)). This determines the appropriate chi-square distribution to use when finding critical values.

The chi-square distribution is a statistical distribution often used for hypothesis testing and constructing confidence intervals. It is particularly useful in tests of independence and goodness-of-fit tests. The shape of the chi-square distribution is determined by the degrees of freedom, making it skewed to the right, especially for lower values of df. As df increases, the distribution becomes more symmetric. In our problem, using a table for the chi-square distribution (or a relevant software tool), we locate the critical values for our specified degrees of freedom and levels of significance (0.05 and 0.95). These values are crucial for determining the thresholds within which we expect the true population parameter to lie.

A confidence interval provides a range of values that is likely to contain a population parameter with a certain level of confidence. In the context of the chi-square distribution, a 90% confidence interval means that we are 90% confident that the true population parameter lies within this range. For the problem given, using the chi-square table, we find the critical values at the respective tails (i.e., \( \chi_{1-\frac{\alpha}{2}}^2 \) and \( \chi_{\frac{\alpha}{2}}^2 \)). For a confidence level of 90%, the critical values calculated—approximately 30.144 and 10.117—mark the boundaries of the confidence interval. This interval helps make informed decisions about the population parameter based on sample data.