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? (Hint: See discussion on page 496 ) a. 10 df, between -1.81 and 1.81 b. 24 df, between -2.06 and 2.06 c. 24 df, outside the interval from -2.80 to 2.80 d. 10 df, to the left of -1.81

Short answer

a. Approximately 90%, b. Approximately 95%, c. 1%, d. 5%

Step-by-step solution

Understanding the Question

Recognition of the problem as one being about the t-distribution is important. The t-distribution is used to calculate probabilities when the sample size is small (generally under 30). The exercise asks for the percentage of time a variable (for example, a test statistic) will fall within a certain range. This percentage can be found by finding the area under the t-distribution curve that corresponds to that range.

Using the t-Distribution Table

For each scenario, check the row corresponding to the given degrees of freedom on your t-Distribution table. For a and b, find the number in the row that is closest to 1.81 and 2.06 respectively, and read off the associated probability. This represents the percentage of the time that a randomly selected value will fall between -1.81 and 1.81 or -2.06 and 2.06. For c and d, since the interval is outside a range or to the left of a value, the probabilities need to be subtracted from 1.

Apply the Values

a. For 10 df and the value of 1.81, the percentage of time between -1.81 and 1.81 is almost 90%. \n\nb. For 24 df and the value of 2.06, the percentage of time between -2.06 and 2.06 is approximately 95%. \n\nc. For 24 df and the value of 2.80, the percentage of time outside of this interval is 1 - 0.99 = 0.01 or 1%. \n\nd. For 10 df and the value of -1.81, the percentage of the time to the left of -1.81 is 5%.