Final answer:
To find the areas or probabilities associated with specific t-values for a t distribution with 20 degrees of freedom, one refers to a t-distribution table or statistical software. The specific values are typically in a cumulative format from the left and require subtraction for tail probabilities or calculations of the area between two values.
Step-by-step explanation:
To find the requested probabilities for a t distribution with 20 degrees of freedom, you can use a t-distribution table or statistical software. Since we don't have the precise values, I'll explain the process.
- (a) To find the area to the right of 2.086, you would look up the t-distribution value for 20 degrees of freedom and identify the two-tailed area that corresponds to the given t-value. Then subtract the area from 0.975 (or use the one-tail value if available).
- (b) For the area to the left of 2.528, refer to the table, and the cumulative area up to 2.528 will give you the desired probability.
- (c) The area to the left of −1.725 can be found similarly, as t-distributions are symmetric about zero. Look up the positive 1.725 and subtract from 1 to get the area to the left of the negative value.
- (d) To find the area to the right of 1.325, look up the cumulative area to the left for the corresponding positive t-value and subtract it from 1.
- (e) For the area between −2.086 and 2.086, find the cumulative areas to the left of these values and subtract the smaller from the larger.
- (f) The same process applies to find the area between −1.725 and 1.725.
Remember to round your answers to three decimal places as requested.