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Understanding the concept of 'degrees of freedom' (df) is crucial when working with statistics, especially in hypothesis testing and estimation problems. In simple terms, degrees of freedom refer to the number of values in a statistical calculation that are free to vary. To illustrate, imagine you have a dataset of five test scores and you know their average. If you have four of the tests scores, you can calculate the fifth one because it is not 'free'—it is constrained by the known average.

• When dealing with the t-distribution, degrees of freedom are determined by the sample size minus one (-df = n-1-); where -n- is the number of observations or sample size.
• The degrees of freedom are used to determine the shape of the t-distribution curve. As the degrees of freedom increase, the t-distribution approaches a normal distribution.
• In practical terms, higher degrees of freedom generally result in a t-curve that is flatter and more spread out. This means that the critical values will get closer to those of the normal distribution as the degrees of freedom increase.

The exercise provided asks for a calculation using 35 degrees of freedom. As the given df is relatively high, the shape of the t-distribution would be expected to closely resemble a standard normal distribution, though still with slightly heavier tails.

The 'left-tail area' in a t-distribution table represents the probability that a t-value will fall to the left of a specified value. It is a critical concept when performing left-tailed hypothesis tests, which typically involve testing whether a mean is less than a certain value.

• The left-tail area is denoted by 'a' and in our exercise, it is 0.10. It is often associated with the significance level of a test. In our case, having a left-tail area of 0.10 means that there is a 10% probability that the t-value will be less than the critical value.
• This area is important because it helps us to find the critical value of t, which determines the threshold for rejecting the null hypothesis. If the calculated test statistic is less than this critical value, the null hypothesis is rejected in favor of the alternative hypothesis.

Locating the left-tail area in the t-distribution table is as simple as finding the appropriate column titled with the desired probability, which corresponds directly to the level of significance for the test.

The 'critical value of t-distribution' is the t-value that separates the region where the null hypothesis is not rejected from where it is rejected. This value is found in the t-distribution table, given a specific left-tail area (or significance level) and degrees of freedom associated with the statistical test.

• To locate the critical value, one must intersect the row corresponding to the degrees of freedom with the column for the desired left-tail area in the t-distribution table.
• The critical value represents a threshold: if the calculated t-value from a test statistic falls beyond this value in the left tail, we have sufficient evidence to reject the null hypothesis at the given significance level.
• The closer the left-tail area is to 0.5, the smaller the absolute value of the critical value, indicating a larger area where the null hypothesis would be rejected.

For the exercise problem, the critical t-value at 35 degrees of freedom and a left-tail area of 0.10 would be found at the intersection given in the t-distribution table. This is the value required to determine the outcome of the statistical test based on the t-distribution.