Question

Given a variable that has a \(t\) distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? a. \(10 \mathrm{df}\), between \(-1.81\) and \(1.81\) b. \(10 \mathrm{df}\), between \(-2.23\) and \(2.23\) c. \(24 \mathrm{df}\), between \(-2.06\) and \(2.06\) d. \(24 \mathrm{df}\), between \(-2.80\) and \(2.80\) e. 24 df, outside the interval from \(-2.80\) to \(2.80\) f. \(24 \mathrm{df}\), to the right of \(2.80\) g. \(10 \mathrm{df}\), to the left of \(-1.81\)

Short answer

The percentages corresponding to the t-values at the specified degrees of freedom are approximately: a) 80%, b) 95%, c) 95%, d) 99%, e) 1%, f) 0.5%, g) 10%.

Step-by-step solution

Understand t-distribution and t-table

A t-distribution is a type of probability distribution that is symmetrical and bell-shaped, similar to the normal distribution but is used when the sample size is small and/or when the population standard deviation is unknown. A t-table is a table that shows probabilities for different t-scores or areas under the curve of the t-distribution. The first step is understanding how to use this table.

Identify degrees of freedom and t-scores

In each sub-question, identify the degrees of freedom (df) and the t-scores. The degree of freedom is the number mentioned with df and the t-score can be positive or negative values given for each problem.

Locate the t-scores in the t-table

Using the degrees of freedom and t-scores from step 2, find the corresponding probability or area in the t-table. Remember that for a negative t-score, the percentage of time it falls in the indicated region is just the probability we find from the table. For a positive t-score, since the t-distribution is symmetrical, subtract the probability we find from the table from 1 to get the percentage of time.

Calculate the percentages for each sub-question

For each sub-question, the percentage of the time will depend on the interpretation of the question. For questions asking for the area between two t-scores, compute that by calculating the area for the positive t-score (step 3) and subtracting the area for the negative t-score. For questions asking for the area outside a range, subtract the area between the range from 1. For questions about areas to the left or right of a t-score, refer to steps outlined in step 3.

Identify results

After performing these steps for each sub-question, the resulting percentages correspond to the percentage of the time that the variable's value will fall in the specified region for each situation.