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Confidence Intervals about a Population Mean in Practice Page 1 The Student t Distribution The goal of this activity is to use various methods to find values associated with the Student tdistribution. Using Table B Table B from that AP Statistics Exam provides values for the Student t cumulative distribution function. Specifically, the table provides the values of t for given values of p and the degrees of freedom (df). The value of t is the point to the right of zero (the mean of the t-distribution) for which p is the area under the curve for values greater than t. This is illustrated in the figure on the right (and at the top of the AP Table page) with the shaded region, labeled probability. The figure below demonstrates how to use the table to find the value of t given a p-value of .05 and the degrees of freedom, df = 15. Notice that the value 1.753 is the intersection of the .05 column and the 15 row. Thus, P(T > 1.753) = .05. In addition, the bottom of the table indicates a confidence level C or 90%. This indicates that the area under the Student t probability distribution function between -1.753 and 1.753 is .90. Use Table B to determine the following values of t. 1. The area to the right of t is .025 and df = 20. 2. The value of p is .01 and the value of n is 30. 3. The degrees of freedom are 12 for a 99% confidence interval. Robert A. Powers University of Northern Colorado Confidence Intervals about a Population Mean in Practice Page 2 Using Technology The TI graphing calculator provides a built in function for determining the cumulative distribution function for the Student t distribution. It is located in the DISTR menu above the VARS key. The syntax of the function is given below. tcdf(lower,upper,df) For example, to find the probability that t > 1.75 for df = 20, the following command would be entered. tcdf(1.75,1E99,20) Notice that 1E99 is used to represent ∞. 1. Find the area under the t-distribution curve with df = 43 that lies to the right of t = 1.31. 2. Find the following probability: P(1.07 < t < 3.29), when n = 55. 3. Find the area under the t-distribution curve with df = 28 that lies to the left of t = -0.935. Robert A. Powers University of Northern Colorado