1. Section 6.5 Finding t-Values Using the Student t-Distribution with TI-84 info added by D.R.S. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

2. Similar to the normal distribution in shape but with more area under the tails and is defined by the number of degrees of freedom. HAWKES LEARNING SYSTEMS math courseware specialists Student t-Distribution: Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution 1.A t-curve is symmetric and bell-shaped, centered about 0. 2.A t-curve is completely defined by its number of degrees of freedom, d.f. 3.The total area under a t-curve equals 1. 4.The x-axis is a horizontal asymptote for a t-curve. Properties of a Student t-Distribution:

3. HAWKES LEARNING SYSTEMS math courseware specialists Comparison of the Normal and Student t-Distributions: Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution

4. HAWKES LEARNING SYSTEMS math courseware specialists Student t-Distribution Table: Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 13.0786.31412.70631.82163.657 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution TI-84 (but not available on TI-83/Plus): invT(area to left, degrees of freedom) Example: invT(0.025, 3)=-3.182446305 and fix up the sign - or use invT(0.975, 3)

5. HAWKES LEARNING SYSTEMS math courseware specialists Student t-Distribution Table (continued): Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution When calculating the t-values, round your answers to three decimal places. 1.The numbers across the top row represent an area to the right of t, known as . 2.The numbers down the first column represent the degrees of freedom, d.f.  n – 1. 3.Where the appropriate row and column intersect, we find the t-value associated with the particular area and degrees of freedom.

6. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t 0.025 with 25 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution t 0.025  2.060 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779 TI-84 invT(0.025, 25)=-2.059538532

7. HAWKES LEARNING SYSTEMS math courseware specialists How many degrees of freedom make t 0.010 0.005  4.604? Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 13.0786.31412.70631.82163.657 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 13.0786.31412.70631.82163.657 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 d.f.  4 TI-84 can’t do this – must use the table

8. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area to the right is 0.1 for 17 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution t 0.100  1.333 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 151.3411.7532.1312.6022.947 161.3371.7462.1202.5832.921 171.3331.7402.1102.5672.898 181.3301.7342.1012.5522.878 191.3281.7292.0932.5392.861 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 151.3411.7532.1312.6022.947 161.3371.7462.1202.5832.921 171.3331.7402.1102.5672.898 181.3301.7342.1012.5522.878 191.3281.7292.0932.5392.861 TI-84 invT(0.1, 17)=-1.33337939

9. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area to the left is 0.05 for 11 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution t 0.050   1.796 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 91.3831.8332.2622.8213.250 101.3721.8122.2282.7643.169 111.3631.7962.2012.7183.106 121.3561.7822.1792.6813.055 131.3501.7712.1602.6503.012 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 91.3831.8332.2622.8213.250 101.3721.8122.2282.7643.169 111.3631.7962.2012.7183.106 121.3561.7822.1792.6813.055 131.3501.7712.1602.6503.012 t 0.050  1.796, however the table assumes that the area is to the right of t. Since the t-curve is symmetric at t  0, we can simply change the sign of the t-value to obtain the correct answer.

10. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area to the left is 0.05 for 11 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution TI-84 invT(0.05, 11)=-1.795884781

11. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area in the tails is 0.02. Assume there are 7 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution t 0.010  2.998 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 71.4151.8952.3652.9983.499 81.3971.8602.3062.8963.355 91.3831.8332.2622.8213.250 101.3721.8122.2282.7643.169 111.3631.7962.2012.7183.106 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 71.4151.8952.3652.9983.499 81.3971.8602.3062.8963.355 91.3831.8332.2622.8213.250 101.3721.8122.2282.7643.169 111.3631.7962.2012.7183.106 This type of problem is called two-tailed. If the area in both tails is 0.02, then the area in one tail would be 0.01.

12. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area in the tails is 0.02. Assume there are 7 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution This type of problem is called two-tailed. If the area in both tails is 0.02, then the area in one tail would be 0.01. TI-84 invT(0.01, 7)=-2.997951566

13. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area between –t and t is 99%. Assume 24 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution t  2.797. Since 99% of the area of the curve is in the middle, that leaves 1%, or 0.01 of the area on the outside. Because of symmetry each tail will only have half of 0.01 in its area, 0.005. Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779 Student t-Distribution Table d.f.0.1000.0500.0250.0100.005 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779

14. HAWKES LEARNING SYSTEMS math courseware specialists Find the value of t such that the shaded area between –t and t is 99%. Assume 24 degrees of freedom. Continuous Random Variables 6.5 Finding t-Values Using the Student t-Distribution Since 99% of the area of the curve is in the middle, that leaves 1%, or 0.01 of the area on the outside. Because of symmetry each tail will only have half of 0.01 in its area, 0.005. TI-84 invT(0.005, 24)=-2.796939498