1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 6.2 Finding Area under a Normal Distribution IMPORTANT: “Area” is “_____________” IMPORTANT: “Probability” is “_________”. With a few little improvements and extras 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. Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table Find the area under the standard normal curve to the left of z = 1.37. Use the printed table. z0.050.060.070.080.09 1.00.85310.85540.85770.85990.8621 1.10.87490.87700.87900.88100.8830 1.20.89440.89620.89800.89970.9015 1.30.91150.91310.91470.91620.9177 1.40.92650.92790.92920.93060.9319 1.50.93940.94060.94180.94290.9441

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table (cont.) TI-84 also has ShadeNorm(-1E99,1.37) 2 ND DISTR right arrow to DRAW 1:ShadeNorm etc. 2 ND DRAW (on the PRGM key) 1:ClrDraw gets rid of unwanted leftover drawings.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator Find the area under the standard normal curve to the left of z = −2.03. z0.040.030.020.010  2.2 0.01250.01290.01320.01360.0139  2.1 0.01620.01660.01700.01740.0179  2.0 0.02070.02120.02170.02220.0228  1.9 0.02620.02680.02740.02810.0287  1.8 0.03290.03360.03440.03510.0359  1.7 0.04090.04180.04270.04360.0446

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative z­ Value Using a Table or a TI-83/84 Plus Calculator (cont.) Try TI-84: normalcdf(-1E99,-2.03) and optionally ShadeNorm(-1E99,-2.03) (remember 2 ND DRAW 1:ClrDraw if you need to clear out previous drawing)

6. Excel: Area to the left of z = -2.03 =NORM.S.DIST(z value, TRUE) The “TRUE” tells it to give you Cumulative, from -∞ to z

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table Find the area under the standard normal curve to the right of z = 1.37. Table Method: Total area under curve is _______, Use Subtraction: Total area ________ Minus area to the left of z = 1.37, which is ________ Equals area to the right of z = 1.37, which is ________ TI-84 Method: normalcdf(left endpoint, right endpoint) normalcdf(1.37, 1E99); the result is _______________

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.)

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.) Alternative Table Method: Because of symmetry, the area to the right of z = 1.37 is the same as the area to the _______ of z = ________ z0.090.080.070.060.05  1.6 0.04550.04650.04750.04850.0495  1.5 0.05590.05710.05820.05940.0606  1.4 0.06810.06940.07080.07210.0735  1.3 0.08230.08380.08530.08690.0885  1.2 0.09850.10030.10200.10380.1056  1.1 0.11700.11900.12100.12300.1251

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator Find the area under the standard normal curve to the right of z =  0.90. Table Method: TI-84 method: (show details) (show the command and the result)

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative z­ Value Using a Table or a TI-83/84 Plus Calculator (cont.)

12. Excel: Area to the right of z = -0.90 =1-NORM.S.DIST(z value, TRUE) Like the printed table, NORM.S.DIST only gives you area to the left. It doesn’t do area “between” two z values like the TI-84’s normalcdf() does. So the “1 minus area to the left” technique is needed.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator Find the area under the standard normal curve between z 1 =  1.68 and z 2 = 2.00. Table Method: Area to the left of z = _______ is ________ Subtract: _______ - _______ = ________ TI-84 Method: normalcdf ( _____, _____) = _________________

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator (cont.) Area to the left minus equals of the area to the left the area right endpoint of the between left endpoint the two endpoints

15. Excel: Area between z = -1.68 and z=2.00 =NORM.S.DIST(high z,TRUE)-NORM.S.DIST(low z, TRUE) Like the printed table, NORM.S.DIST only gives you area to the left. It doesn’t do area “between” two z values like the TI-84’s normalcdf() does. So the subtraction of two areas technique is needed.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.7: Finding Area between Two z-Values Using a TI-83/84 Plus Calculator Find the area under the standard normal curve between z 1 = 1.50 and z 2 = 2.75. Solution – show your table and/or TI-84 details

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator Find the total of the areas under the standard normal curve to the left of z 1 = −2.50 and to the right of z 2 = 3.00. Solution There are two areas that we must find. (Show details here)

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.)

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Note an alternative method for finding this area that is particularly clever. By definition, we know that the total area under the curve equals 1. Using this fact, the area in the tails can be obtained by finding the area between z 1 = −2.50 and z 2 = 3.00 and then subtracting that area from 1.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator Find the total of the areas under the standard normal curve to the left of z 1 =  1.23 and to the right of z 2 = 1.23. Use SYMMETRY. Solution Area to the left of z = -1.23 times 2. (Show details)

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Thus, (0.109349)(2)  0.2187. So, the total area in the two tails is approximately 0.2187.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI ‑ 83/84 Plus Calculator Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator. Write details and draw sketches. a.P(z < 1.45) b.P(z  −1.37)

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI ‑ 83/84 Plus Calculator Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator. Write details and draw sketches. c.P(1.25 2.5)

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI ‑ 83/84 Plus Calculator Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator. Write details and draw sketches. e.P(z < −4.01) f. P(z  3.98)