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The concept of 'degrees of freedom' can often confuse students, but it is crucial for the proper use of the Student's t-distribution. Essentially, the degrees of freedom (df) in a statistical context refer to the number of values in a calculation that are free to vary. In other words, these are the number of independent pieces of information that go into the estimation of a parameter.

When working with sample data, one degree of freedom is lost for every parameter that is estimated. For example, when estimating the mean of a sample, one value is constrained by the requirement for the sample mean to equal the sum of the observations divided by the number of observations. Hence, for a sample of size 'n', the degrees of freedom for the estimation of the mean would be 'n-1'. This concept is directly tied to the variability in the data and the estimation of the population parameter. As the degrees of freedom increase, the t-distribution comes closer to the normal distribution, which is why higher degrees of freedom typically result in a narrower, more peaked distribution.

Next, we have the t-distribution table, which is a handy reference tool for determining critical values in the Student's t-distribution. This table lists 't' values, which are the threshold values for certain areas in the two tails of the t-distribution curve. These values are organized by probability (the area under the curve) and the degrees of freedom.

To use this table effectively, you must first identify your significance level (often denoted as alpha, \( \alpha \) ), which corresponds to the probability in the tails beyond the t-values in question. Then, locate your degrees of freedom, which usually form the rows of the table. The intersection of the column and row gives the critical t-value. Understanding how to read this table is fundamental when you are performing hypothesis tests or constructing confidence intervals, as it provides the cutoff points for determining whether a test statistic is in the rejection region or not.

Finally, the two-tailed test is an inferential statistical test that is used when the direction of the effect (either positive or negative) is not specified in the null hypothesis. It is ideal for situations where an effect can occur in both directions, meaning you are looking for differences that could be greater than or less than a certain value.

In a two-tailed test, you split your alpha level between the two tails of the distribution. This is why the original exercise mentioned 5% in each tail for a 90% confidence interval. The critical region, where the null hypothesis is rejected, is located at both ends of the t-distribution. If your test statistic falls into either tail beyond the critical t-values (which you can find using the t-distribution table), you reject the null hypothesis. When analyzing our t-distribution example, we use the table to find the degrees of freedom that correspond to the critical t-value for our given confidence level, but we must consider both tails of the distribution, effectively doubling the significance level for a one-tailed test to account for the two tails of interest.