Final answer:
The question seeks the critical t-value for a .99 probability within a t-distribution with 9 degrees of freedom. Using R's qt function with a probability adjusted for a two-tailed test, we find this t-value to determine the missing number labeled as 't'.
Step-by-step explanation:
The student is asking to find the value of t such that P(-t < T < t) = .99 for a t-distribution with 9 degrees of freedom.
To find this value, you can use the qt function in R, which is the inverse of the cumulative distribution function for a t-distribution. The qt function will give us the critical value of t that corresponds to a cumulative probability. Given that the question asks for a two-tailed critical value—where the middle 99% of the distribution is captured—we're interested in the critical values that cut off the outer 0.5% on each side. Since R's qt function works with one-tailed probabilities, we need to adjust the probability to 0.995 (99.5%) to find the positive critical value t for the upper bound, as the question asks for symmetry about zero.
To calculate this in R, the command is qt(0.995, 9). Executing this function will give us the value of t.