1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 6 Technology

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves Use a TI-83/84 Plus calculator to find the indicated area under the appropriate normal curve. a.Area between z 1 = −1.23 and z 2 = 2.04. b.Area to the left of z = 2.37. c.Area to the right of z = −1.09. d.Area under a normal curve with a mean of 64.0 and a standard deviation of 5.3 to the left of x = 58.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) Solution a.Press and then to access the DISTR menu and then scroll down to option 2. Since this is the area under the standard normal curve, we do not need to enter  or . Enter normalcdf(-1.23,2.04), as shown in the screenshot. The area is approximately 0.8700.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) b.Press and then to access the menu and then scroll down to option 2. Since this is the area under the standard normal curve, we do not need to enter  or . We want the area to the left, so the lower bound would be  ∞. We cannot enter  ∞, so we will enter a very small value for the lower endpoint, such as  10 99. This number appears as -1 E 99 when entered correctly into the calculator.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) To enter -1û99, press. Enter normalcdf ( -1û99, 2.37 ), as shown in the screenshot. The area is approximately 0.9911.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) c.Press and then to access the menu and then scroll down to option 2. Again, this is the area under the standard normal curve, so we do not need to enter  or . We want the area to the right, so the upper bound would be ∞. We cannot enter ∞, so we will enter a very large value for the upper endpoint, such as 10 99. This number appears as 1û99 when entered correctly into the calculator. To enter 1û99, press.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) Enter normalcdf(-1.09,1û99), as shown in the screenshot. The area is approximately 0.8621.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) d.Press and then to access the menu and then scroll down to option 2. This is the area under a normal curve, but not the standard normal curve, so we will enter  = 64.0 and  = 5.3. We want the area to the left, so we will enter a very small value for the lower endpoint, such as −10 99.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.1: Using a TI-83/84 Plus Calculator to Find Areas under Normal Curves (cont.) Enter normalcdf(-1û99,58,64.0,5.3), as shown in the screenshot. The area is approximately 0.1288.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables Use a TI-83/84 Plus calculator to find the indicated value. a.The value of z with an area of 0.6781 to its left. b.The value of z with an area of 0.2500 to its right. c.The value of z such that 90% of the area is between z and  z. d.A normal distribution has a mean of 25.00 and a standard deviation of 2.45. What value of x has an area of 0.3761 to its left?

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) Solution a.Press and then to access the DISTR menu and then scroll down to option 3. Enter invNorm(0.6781), as shown in the screenshot. The value of z is approximately 0.46.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) b.Since the area given is the area to the right, we must first determine the area to the left. The area to the left is 1  0.2500 = 0.7500. Press and then to access the menu and then scroll down to option 3. Enter invNorm(0.7500), as shown in the screenshot. The value of z is approximately 0.67.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) c.Since the area given is between two values of z, 1  0.90 = 0.10 is the area in the two tails. Thus, half of that area, or 0.05, is the area to the left of  z. Press and then to access the menu and then scroll down to option 3. Enter invNorm(0.05), as shown in the screenshot on next slide. The value returned by the calculator is approximately  1.644854.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) Note that the value we obtained earlier in this chapter using the normal distribution table was 1.645.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) d.This is not the standard normal distribution, so we will need to enter  = 25.00 and  = 2.45. The area given is to the left, so we can enter it directly. Press and then to access the menu and then scroll down to option 3.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.2: Using a TI-83/84 Plus Calculator to Find Values of Normally Distributed Random Variables (cont.) Enter invNorm(0.3761,25.00,2.45), as shown in the screenshot. The value of x is approximately 24.23.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.3: Using Microsoft Excel to Find a Value of a Normally Distributed Random Variable The body temperatures of adults are normally distributed with a mean of 98.60  F and a standard deviation of 0.73  F. What temperature represents the 90 th percentile? Solution The 90 th percentile is the temperature x for which 90% of the area under the normal curve is to the left of x. So, we need to find the value of the normally distributed random variable with a mean of 98.60 and a standard deviation of 0.73 that has an area of 0.90 to its left.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.3: Using Microsoft Excel to Find a Value of a Normally Distributed Random Variable (cont.) Thus, we enter the following formula: =NORM.INV(0.90, 98.60, 0.73). This formula returns the value 99.53553264. Thus, a temperature of approximately 99.54  F represents the 90 th percentile.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.4: Using M INITAB to Find the z-Value with a Given Area to Its Left Find the value of z given that the area to the left of z is 0.9265. Solution Enter 0.9265 in the first column and row and then go to Calc ► Probability Distributions ► Normal. Select Inverse cumulative probability, and make sure the Mean is 0.0 and the Standard deviation is 1.0. Enter C1 as the Input column and click OK.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.4: Using M INITAB to Find the z-Value with a Given Area to Its Left (cont.) The Normal Distribution dialog box is shown in the following screenshot.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example T.4: Using M INITAB to Find the z-Value with a Given Area to Its Left (cont.) The answer, which appears in the Session window, is z ≈ 1.45, as shown in the following screenshot.