1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 6.4 Finding Values of a Normally Distributed Random Variable With a few swell additions by D.R.S., University of Cordele

2. Forward and Backwards, a Mathematical Pattern 1.Addition and Subtraction are inverse operations. Each one undoes the other. 2.Multiplication and Division are inverse operations. Each one undoes the other. 3.Squaring and Square Rooting are inverse operations. Each one undoes the other.  In Statistics, we go from x value to z value & back.  Also: z values lead us to areas, and now, we work backward from areas to z values, and also backward from areas to x values.

3. Problems We’ve Done Before – “Find the area”. x z Area = ??? 70.55 0 1.37 We knew the x values and we knew the z values

4. The New Problems. x z Area =.0853 ??? 0 Find the x and z values that make it happen

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.17: Finding the z-Value with a Given Area to Its Left What z-value has an area of 0.7357 to its left? Sketch a picture, first. Be aware of proportion, like 0.7357 is _____ than half of the area, and we expect to get a __________ sign for the z value.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. TABLE METHOD – look in the INTERIOR Of the table for the area, read outward to find z. z00.010.020.030.04 0.00.50000.50400.50800.5120.516 0.10.53980.54380.54780.55170.5557 0.20.57930.58320.58710.59100.5948 0.30.61790.62170.62550.62930.6331 0.40.65540.65910.66280.66640.6700 0.50.69150.69500.69850.70190.7054 0.60.72570.72910.73240.73570.7389 0.70.75800.76110.76420.76730.7704 Search the INTERIOR of the table and read OUTWARD to find the z.

7. TI-84 Commands for z problems. Previously – normalcdf Tell it the z values. It gives you area. New – invNorm Tell it the area to the left. It gives you the z value. The area to the left of z = 1.23 is 0.8907. What z value has area to the left = 0.8907? z = 1.23. z=1.23 area.8907

8. TI-84 Commands  Back to the earlier problem.  The z value that has area = _________ to its ______ is z = ______.  The Excel version is =NORM.S.INV(area to left)

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.18: Finding the z-Value with a Given Area to Its Left - Table Method Find the value of z such that the area to the left of z is 0.2000. Always draw a picture first ! (next slide) TABLE METHOD – your area value is not in the table! Circle the closest value to area = 0.2000 instead. and read outward to get the answer, z = _______ has area of.2000 to its left.

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

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.18: Finding the z-Value with a Given Area to Its Left TI-84 invNorm method Find the value of z such that the area to the left of z is 0.2000. Always draw a picture first ! (previous slide) TI-84 METHOD – invNorm( _______ ) gives result of z = ____________________. Compare to the answer we got by using the table.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.19: Finding the z-Value That Represents a Given Percentile What z-value represents the 90 th percentile? Make a sketch (next slide). “The 90 th percentile” means “90% of the values are lower than mine”, so we are looking for the z value that has area _______ to its _______ TABLE METHOD – the closest area to the value 0.9000 in the table is the area value ______ and reading out, we find the z = ______. TI-84 METHOD – Command: Answer: z = ______________

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: What z value has area = 0.0096 to its right? If 0.0096 is to the right, then 1 – 0.0096 = ________ is to the left.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: What z value has area = 0.0096 to its right? We have drawn a picture (previous slide) and determined that we need to find the z value that has area = _______ to its _______. TABLE METHOD – the closest area to the value ______ in the table is the area value ______ and reading out, we find the z = ______. TI-84 METHOD – Command: Answer: z = ______________ Another TI-84 way to do it is to give the command invNorm( ____ - ______ )

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example: Find the z value such that the area between –z and z is 0.9000 1)They told us 0.9000 in the middle. 2)So think 1 – _____ = _____ in two tails total. 3)So _____ ÷ 2 = _____ is the area in each tail. 4)Find the z that has area ______ on its left to get the value of –z. 5)By symmetry, you instantly know the z, too.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.21: What –z and z values delineate the middle 0.9000 of the area? We have drawn a picture (previous slide) and determined that we need to find the z value that has area = _______ to its _______. TABLE METHOD – the closest area to the value ______ in the table is the area value ______ and reading out, we find the z = ______. TI-84 METHOD – Command: Answer: z = ______________ ANSWERING THE QUESTION – (applies to both Table and TI-84 Methods) “The value of z such that 0.90 of the area is between –z and z is z = ____________”

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.21: Finding the z-Value with a Given Area between  z and z (cont.) z0.070.060.050.040.03 -2.00.01920.01970.02020.02070.0212 -1.90.02440.02500.02560.02620.0268 -1.80.03070.03140.03220.03290.0336 -1.70.03840.03920.04010.04090.0418 -1.60.04750.04850.04950.05050.0516 -1.50.05820.05940.06060.06180.0630 We have a tie ! Even though we’ve been rounding z values to ____ places, we can resolve this case by using z = ___________

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.22: Finding the z-Value with a Given Area in the Tails to the Left of  z and to the Right of z Find the value of z such that the area to the left of  z plus the area to the right of z is 0.1616. Got to have a picture!!! Got to read carefully!!! How do they get 0.0808? _____________________

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.22: Area to left of -z + area to right of z is 0.1616 Area in two tails total is _________ and because of symmetry, the area in each individual tail is _________. TABLE METHOD – the closest area to the value ______ in the table is the area value ______ and reading out, we find the z = ______. TI-84 METHOD – Command: Answer: z = ______________ ANSWERING THE QUESTION – (applies to both Table and TI-84 Methods) “the value of z such that the area to the left of  z plus the area to the right of z is 0.1616 is z = ____________”

25. NEXT – moving up from backwards z problems to do backwards x problems So far, we’ve done a lot backwards z problems: What z has area = #### to its left? What z has area = #### to its right? What –z and z have area = #### between them, in the middle? What –z and z have area = #### in the two tails? Building on these skills, do backwards x problems: What x has area = #### to its left? What x has area = #### to its right? What two x values have area = #### between them, in the middle? What two x values have area = #### in the two tails? (The problems will of course give you extra information, the mean and standard deviation.)

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.23: Finding the Value of a Normally Distributed Random Variable with a Given Area to Its Right If a normal distribution has a mean of 28.0 and a standard deviation of 2.5, what is the value of the random variable X that has an area to its right equal to 0.6700? Always draw a picture! (Next slide.) A sense of proportion: 0.6700 is about where, since total area is 1 ? Mark mean in middle. Just under mean is a good place to jot down the standard deviation..

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

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.23: Finding the Value of a Normally Distributed Random Variable with a Given Area to Its Right Fill in the TI-84 command. Then run it. Given that the mean is _______ and the standard deviation is ____, the x value that has area.6700 to its right is x = ________

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.24: Finding the Value of a Normally Distributed Random Variable That Represents a Given Percentile 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? Draw the picture (in proportion, and label the mean and stdev too.) (Use next slide.) We want the x value that has area = _______ to its _______, so we use the TI-84 command ___________ ( _______, ______, ______ ) which gives the answer _________. Response: “The temperature ______ o F represents the 90 th percentile.”

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.24: Finding the Value of a Normally Distributed Random Variable That Represents a Given Percentile Assume that the lengths of newborn full-term babies in the United States are normally distributed with a mean length of 20.0 inches and a standard deviation of 1.2 inches. What is the minimum length that gets your baby into the third quartile? Draw the picture (in proportion, and label the mean and stdev too.) (Next slide has a place to sketch.) We want the x value that has area = _______ to its _______, so we use the TI-84 command ___________ ( _______, ______, ______ ) which gives the answer _________. Response: “The third quartile begins at ______ inches in length.”

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

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.24 extended: Finding the Value of a Normally Distributed Random Variable That Represents a Given Percentile Assume that the lengths of newborn full-term babies in the United States are normally distributed with a mean length of 20.0 inches and a standard deviation of 1.2 inches. What lengths are the boundaries of the middle 30%? Draw the picture (in proportion, and label the mean and stdev too.) (Sketch on the next slide.)

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