1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 8.2 Student’s t-Distribution With the usual enthralling extra content you’ve come to expect, by D.R.S., University of Cordele

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Student’s t-Distribution Properties of a t-Distribution 1.A t-distribution curve is symmetric and bell-shaped, centered about 0. 2.A t-distribution curve is completely defined by its number of degrees of freedom, df. 3. The total area under a t-distribution curve equals 1. 4. The x-axis is a horizontal asymptote for a t-distribution curve.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 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 A t-distribution is pretty much the same as a normal distribution! There’s this additional little wrinkle of “d.f.”, “degrees of freedom”. Slightly different t distributions for different d.f.; higher d.f. is closer & closer to the normal distribution.

4. Observations about the t distribution In the middle, at the mean, the t distribution peaks { lower or higher } than the normal distribution.. In the tails, the t distribution is { lower or higher } than the normal distribution. The differences between the t and the normal distributions are because there is more un__________ty built into the t distribution. As the sample size n gets larger, the degrees of freedom d.f. gets larger, the t distribution gets { closer to, farther from } the normal distribution bell curve we use in z problems.

5. Why bother with t ? If you don’t know the population standard deviation, σ, but you still want to use a sample to find a confidence interval. t builds in a little more uncertainty based on the lack of a trustworthy σ. The plan: 1.This lesson – learn about t and areas and critical values, much like we have done with z. 2.Next lesson – doing confidence intervals with t.

6. The t and the z tables – the same but different z, Standard Normal Distribution Lookup z in row label and column header Read inward to find the area.

7. The t and the z tables – the same but different z, Standard Normal Distribution The t Distribution 1 tail0.1000.0500.0250.010 df2 tails0.2000.1000.0500.020 n – 1#.### Lookup z in row label and column header Read inward to find the area. Read inward to find the t value. Choose which header row and column for Area: Choose the df row = sample size n, minus 1. area in 1 tail or in 2 tails

8. The t and the z calculator functions z, Standard Normal Distribution The t Distribution If I know AREA and want to work backward to find z 1) What z has area =.0500 to its left? 2) What z has area.9500 to its left? If I know AREA and want to work backward to find t Find the t and –t values such that the area in two tails is.1000 for df = 20. Area in one tail is ________; Answer: t = _____ (positive) If I know two z values and I want the area between them What is the area between z = -1.645 and z 1.645 ? If I know two t values and I want the area between them (Less common, use tcdf(low t, high t, df) if it comes up.)

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.9: Finding the Value of t α Find the value of t 0.025 for the t-distribution with 25 degrees of freedom. t α means “what t value has area α to its right? (And because of symmetry, α is also the area to the left of –t α ) For finding critical values of t, using the table is the quickest and easiest way to find the needed value. Even though the TI-84 has an invT function, we’ll emphasize the table method for t problems. Also - invT is not available with the TI-83 family. Answer: t = ____________

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.9: Finding the Value of t α (cont.) * df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.0500.0200.010 23 1.3191.7142.0692.5002.807 24 1.3181.7112.0642.4922.797 25 1.3161.7082.0602.4852.787 26 1.3151.7062.0562.4792.779 27 1.3141.7032.0522.4732.771 28 1.3131.7012.0482.4672.763 Seeking t 0.025 … Note TWO sets of headings! One Tail & Two Tails USE THIS ONE TI-84 only – not available on TI-83: invT(area to the left, degrees of freedom) it’s at 2 ND DISTR 4:invT( invT(area to the left, degrees of freedom) invT(0.025,25) gives -2.059538532 then you have to fix up the sign & round

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.10: Finding the Value of t Given the Area to the Right Find the value of t for a t-distribution with 17 degrees of freedom such that the area under the curve to the right of t is 0.10. Answer: t = ____________

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.10: Finding the Value of t Given the Area to the Right (cont.) * df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.500.0200.010 151.3411.7532.1312.6022.947 161.3371.7462.122.5832.921 171.3331.7402.1102.5672.898 181.3301.7342.1012.5522.878 191.3281.7292.0932.5392.861 201.3251.7252.0862.5282.845 invT(area to the left, degrees of freedom) invT(0.100,17) gives -1.33337939 then you have to fix up the sign & round

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.11: Finding the Value of t Given the Area to the Left Find the value of t for a t-distribution with 11 degrees of freedom such that the area under the curve to the left of t is 0.05. Answer: (careful of sign!!!) t = ____________

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.11: Finding the Value of t Given the Area to the Left (cont.) * df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.0500.0200.010 91.3831.8332.2622.8213.25 101.3721.8122.2282.7643.169 111.3631.7962.2012.7183.106 121.3561.7822.1792.6813.055 131.3501.7712.1602.653.012 141.3451.7612.1452.6242.977 invT(area to the left, degrees of freedom) invT(0.050,11) gives -1.795884781 then round (in this case, it is negative t)

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.12: Finding the Value of t Given the Area in Two Tails Find the value of t for a t-distribution with 7 degrees of freedom such that the area to the left of  t plus the area to the right of t is 0.02, as shown in the picture. Answer: t = ____________

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.12: Finding the Value of t Given the Area in Two Tails (cont.). df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.0500.0200.010 51.4762.0152.5713.3654.032 61.4401.9432.4473.1433.707 71.4151.8952.3652.9983.499 81.3971.8602.3062.8963.355 91.3831.8332.2622.8213.250 101.3721.8122.2282.7643.169 invT(area to the left, degrees of freedom) invT(______,7) gives -2.997951566 then round and use the positive value If using the table, just go directly to 0.020 in the Two Tails heading. But if using the TI-84 invT(), you must divide 0.020 ÷ 2 = ______ area in one tail first, and then….

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.13: Finding the Value of t Given Area between  t and t Find the critical value of t for a t-distribution with 29 degrees of freedom such that the area between −t and t is 99%. This is a two-tail problem. The area in two tails is _____ and the table has a two tails heading. If you’re using TI-84 invT instead of the table, you’ll need to divide by 2 and work with the area in one tail. Area in middle = ______ Area in tails = ______ Answer: t = ____________

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.13: Finding the Value of t Given Area between  t and t (cont.) @ df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.0500.0200.010 271.3141.7032.0522.4732.771 281.3131.7012.0482.4672.763 291.3111.6992.0452.4622.756 301.3101.6972.0422.4572.750 311.3091.6962.0402.4532.744 321.3091.6942.0372.4492.738 invT(area to the left, degrees of freedom) invT(______,29) gives ______________ then round and use the positive value

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.14: Finding the Critical t-Value for a Confidence Interval Find the critical t-value for a 95% confidence interval using a t-distribution with 24 degrees of freedom. Solution This is a two-tail problem. The area in two tails is ____ and if using TI-84 invT, you need to compute the area in one tail which is ____ Area in middle = ______ Area in tails = ______ Answer: t = ____________

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.14: Finding the Critical t-Value for a Confidence Interval (cont.) # df Area in One Tail 0.1000.0500.0250.0100.005 Area in Two Tails 0.2000.1000.0500.0200.010 211.3231.7212.0802.5182.831 221.3211.7172.0742.5082.819 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779 invT(area to the left, degrees of freedom) invT(______,____) gives _____________ then round and use the positive value

21. 8.2 Practice and Certify Problems They come in pairs. You are given some area. You find the t value. The first part is easy – multiple choice from four pictures: easy – left tail or right tail or two tails or between two tails. The second part asks for the “critical value of t”, what t value separates out the prescribed area. But the problem presentation is awkward and you need to be prepared for it.

22. 8.2 Practice and Certify Problems The second part asks for the “critical value of t”, what t value separates out the prescribed area. The way it’s set up, you have to click in the table. It doesn’t let you type in the box. There’s only one heading and it isn’t labeled. You just have to know that it’s for area in one tail, so if you have a 95% level of confidence, that means there is area ________ in two tails, so there is area ________ in one tail, and that’s the heading you need for these problems.

23. 8.2 Practice and Certify Problems The table the problem only has one header row and the row isn’t labeled. It’s an “Area In One Tail”. Area in One Tail scroll bar to get the proper df

24. 8.2 Practice and Certify Problems When you click in the table, the t value is copied to the answer box. If you need to change it to a negative t value, click the +/- button. Also – “scroll bar” is barra di scorrimento in Italian.

25. Example 8.12: Finding the Value of t Given the Area in Two Tails (cont.) – with Excel Recall: we seek t and –t such that two tails total area 0.02, d.f. = 7 Excel with convenient =T.INV.2T(total area, d.f.) Or Excel with one-tailed version, manually divide area by 2: = T.INV(one tailed area, d.f.)

26. Example 8.11: Finding the Value of t Given the Area to the Left, with Excel Excel: T.INV(area to the left of t, df), same thing. Excel special if you know area in two tails total: =T.INV.2T(area in two tails total, df)