Question

What percentage of the time will a variable that has a \(t\) distribution with the specified degrees of freedom fall in the indicated region? a. \(10 \mathrm{df}\), between -2.23 and 2.23 b. 24 df, between -2.80 and 2.80 c. \(24 \mathrm{df}\), to the right of 2.80

Short answer

a. The percentage of the time the variable falls between -2.23 and 2.23 is 7.68%.\nb. The percentage of the time the variable falls between -2.80 and 2.80 is 1.42%.\nc. The percentage of the time the variable falls to the right of 2.80 is 0.71%.

Step-by-step solution

Solve for part a

In part a, the \(t\) distribution has 10 degrees of freedom and the region of interest is between -2.23 and 2.23. We consult a \(t\) distribution table and we find that the probability that a variable will fall in this region is approximately 0.0384. Because we are interested in both sides of the distribution, we should double this number, so 0.0384 * 2 = 0.0768 or 7.68%.

Solve for part b

In part b, the degrees of freedom have increased to 24, and the region of interest ranges from -2.80 to 2.80. Consulting the \(t\) distribution table with 24 df, the probability corresponding to this t-value is approximately 0.0071. To get the total probability for the entire region, we should double this number, so 0.0071 * 2 = 0.0142 or 1.42%.

Solve for part c

In part c, like in part b, the \(t\) distribution has 24 degrees of freedom but this time we are interested in the region to the right of 2.80. Consulting the \(t\) distribution table again for 24 df, we find the probability that a variable will fall on the right of 2.80 is approximately 0.0071. However, because this time we are only interested in one side of the distribution, we won't need to double this number. So the probability is 0.0071 or 0.71%.