(a) For a variable that has a t distribution with 10 degrees of freedom, the percentage of the time it will fall between -3.17 and 3.17 can be found by using the t-distribution table or a calculator.
To find this percentage, we look at the area under the t-distribution curve between -3.17 and 3.17. This area represents the probability of the variable falling within this range.
Using a t-distribution table or a calculator, we can find that the percentage is approximately 95.0%.
(b) For a variable that has a t distribution with 21 degrees of freedom, the percentage of the time it will fall between -1.72 and 1.72 can be calculated similarly.
Using the t-distribution table or a calculator, we can find that the percentage is approximately 90.0%.
(c) To find the percentage of the time a variable that has a t distribution with 21 degrees of freedom will fall to the right of 1.72, we need to calculate the area under the t-distribution curve to the right of 1.72.
Using the t-distribution table or a calculator, we can find that the percentage is approximately 5.0%.
In summary:
(a) 10 df, between -3.17 and 3.17: Approximately 95.0%
(b) 21 df, between -1.72 and 1.72: Approximately 90.0%
(c) 21 df, to the right of 1.72: Approximately 5.0%
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