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.

3. Kinds of Problems 1.Given area = ____ to left of some point, what z value makes it happen? And x value? 2.Given some percentile or quartile, what z makes the split at that point? And x value? 3.Given area = ____ to the right of some point, what z value makes it happen? And x value? 4.Given symmetric area = ____ in the middle, what z and –z values make it happen? And x values? Can you draw pictures to represent these situations?

4. 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? Solution In order to sketch a graph of where the area under the curve lies, we need to shade under more than half of the curve because 0.7357 > 0.5. Therefore, we know that the z-score will be a positive number.

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 (cont.)

6. 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 (cont.) Scan through the interior of Table B from Appendix A for an area of 0.7357. Look to the left and up to find the corresponding z-value. Looking at the row and column titles, you will see that z = 0.63.

7. 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 (cont.) 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.

8. TI-84 Commands  Remember from before: normalcdf(-1E99,z) = area to the left of that z  To go the other way, invNorm(area to left) = the z value that makes the split between area to left and to right.  Note that invNorm only talks in terms of area to the left. (This will cause extra work for some other problems we will look at soon.)

9. 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 (cont.) Let’s also demonstrate how to find the solution by using a TI-83/84 Plus calculator. Begin by pressing and then. Choose option 3:invNorm(. This option requires you to enter the area to the left of the z-value that you are interested in finding. For problems where the area to the left is given, such as this one, knowing the value to enter in the calculator is easy. This will become more challenging with the other types of problems. The area to the left of the z-score we are interested in is 0.7357.

10. 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 (cont.) Enter invNorm(0.7357), as shown in the screenshot, and press. The answer is z ≈ 0.63.

11. Example 6.17 with Excel What z-value has an area of 0.7357 to its left?  RECALL: NORM.S.DIST(z, TRUE) gives area to left  NOW: NORM.S.INV(area to left) gives you the z

12. 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 Find the value of z such that the area to the left of z is 0.2000. Solution First, sketching a graph of the curve shows us that since 0.2000 < 0.5, our z-score will be a negative value.

13. 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 (cont.)

14. 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 (cont.) We can then begin to find 0.2000 in the body of the cumulative normal distribution table for negative z-values, Table A from Appendix A. Notice that none of the areas is exactly 0.2000. Our area falls between the areas of 0.2005 and 0.1977 in the table. One method for finding the z-value for an area not in the table is interpolation; however, for our purposes, we will use the area closest to the one we want. The value of 0.2005 is the closest one to 0.2000. Its z-value is  0.84. So, the approximate z-score for an area of 0.2000 is z =  0.84.

15. 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 (cont.) z0.070.060.050.040.03  1.2 0.10200.10380.10560.10750.1093  1.1 0.12100.12300.12510.12710.1292  1.0 0.14230.14460.14690.14920.1515  0.9 0.16600.16850.17110.17360.1762  0.8 0.19220.19490.19770.20050.2033  0.7 0.22060.22360.22660.22960.2327  0.6 0.25140.25460.25780.26110.2643

16. 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 (cont.) Since the area we are given is the area to the left of the unknown z-value, using a TI-83/84 Plus calculator is straightforward. Begin by pressing and then. Choose option 3:invNorm(. The area to the left of the z-score we are interested in is 0.2000.

17. 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 (cont.) Enter invNorm(0.2000), as shown in the screenshot, and press. The answer is z ≈  0.84.

18. 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? Solution The 90 th percentile is the z-value for which 90% of the area under the standard normal curve is to the left of z. So, we need to find the value of z that has an area of 0.9000 to its left. Should be an EASY problem! Just a matter of knowing that by definition, “the 90 th percentile” means “90% of the values are lower than mine”.

19. 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 (cont.) This drawing seems to be a bit off, doesn’t it? Can you choose a better spot for the z?

20. 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 (cont.) Looking for 0.9000 (or an area extremely close to it) in the interior of the cumulative normal tables, we find 0.8997, which corresponds to a z-value of 1.28. Thus z ≈ 1.28 represents the 90 th percentile. 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

21. 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 (cont.) Using a TI-83/84 Plus calculator, we can find a value of z with a given area to its left. As noted previously, the 90 th percentile is the z-value that has an area of 0.9000 to its left. Enter invNorm(0.9000), as shown in the screenshot, and press. The answer is z ≈ 1.28.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: Finding the z-Value with a Given Area to Its Right What z-value has an area of 0.0096 to its right? Solution First, sketch a graph of the curve. Because 0.0096 is such a small area on the right, we know we should sketch the z-score on the positive side of the graph. Since we need the area to the left, not the area to the right, in order to use the tables, subtract the area to the right from 1. This gives us an area to the left of z equal to 0.9904.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: Finding the z-Value with a Given Area to Its Right (cont.) You have to change the problem from an “area to the right” problem into an “area to the left” problem. If 0.0096 is to the right, then 1 – 0.0096 = 0.9904 is to the left.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: Finding the z-Value with a Given Area to Its Right (cont.) Scan through the interior of the table for an area of 0.9904. Look to the left and up to find the corresponding z-value. What we find is that z = 2.34. z00.010.020.030.04 20.97720.97780.97830.97880.9793 2.10.98210.98260.98300.98340.9838 2.20.98610.98640.98680.98710.9875 2.30.98930.98960.98980.99010.9904 2.40.99180.99200.99220.99250.9927 2.50.99380.99400.99410.99430.9945

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: Finding the z-Value with a Given Area to Its Right (cont.) Since the area we are given is area to the right of the unknown z-value, using a TI ‑ 83/84 Plus calculator requires that we first find the area to the left of z. We have already calculated that the area to the left is 0.9904, but we can have the calculator find this area at the same time that it finds the z-score.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20: Finding the z-Value with a Given Area to Its Right (cont.) Enter invNorm(1Þ0.0096), as shown in the screenshot, and press. The answer is z ≈ 2.34.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.20 with Excel What z-value has an area of 0.0096 to its right?  Use the area to the left = 1 – area to right trick again

28. 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 Find the value of z such that the area between  z and z is 0.90. Solution If the area between  z and z is 0.90, then the total area in the two tails must be 1  0.90 = 0.10. In our sketch, the shaded area will take up most of the area under the curve. Because the curve is symmetric, the area left after subtracting from 1 is divided equally between the two tails. So, half of that area, which is is in the left tail.

29. 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.) 1.They told us 0.9000 in the middle. 2.So think 1 – 0.9000 = 0.1000 in two tails total. 3.So 0.1000 ÷ 2 = 0.0500 is the area in each tail. 4.Find the z that has area 0.0500 on its left to get the value of –z. 5.By symmetry, you instantly know the z, too.

30. 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.) Because the cumulative normal tables give area to the left, we can look up this tail area and find the corresponding z-value. Since the probabilities in the normal tables are given to four decimal places, we look for the probability that is closest to the left tail area of 0.0500.

31. 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  2.0 0.01920.01970.02020.02070.0212  1.9 0.02440.02500.02560.02620.0268  1.8 0.03070.03140.03220.03290.0336  1.7 0.03840.03920.04010.04090.0418  1.6 0.04750.04850.04950.05050.0516  1.5 0.05820.05940.06060.06180.0630 We have a tie !

32. 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.) In the interior of the table, we find two areas that are equally close to 0.0500, so use the z-value exactly halfway between the z-scores corresponding to those two areas. The z-value halfway between z =  1.64 and z =  1.65 is  1.645. Thus the positive z-value such that the area between  z and z is 0.90 is z = 1.645. Since the area we are given is the area between the unknown z-values, using a TI-83/84 calculator requires that we first find the area to the left of  z. We have already calculated that the area to the left of  z is 0.05.

33. 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.) Enter invNorm(0.05), as shown in the screenshot, and press. The answer is  z ≈  1.644854. Rounded to three decimal places, the positive z-value is 1.645. Although if you go with two digits, the TI-84 reveals that it’s actually closer to z=-1.64

34. 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. Solution If the total area in both tails is 0.1616, then the area in one of the tails must be half of that or

35. 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 (cont.)

36. 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 (cont.) Scan through the interior of the cumulative normal table for an area of 0.0808. The corresponding z-value is  z =  1.40. Thus, the z-value such that the area to the left of  z plus the area to the right of z is 0.1616 is z = 1.40. z0.040.030.020.010.00  1.6 0.05050.05160.05260.05370.0548  1.5 0.06180.06300.06430.06550.0668  1.4 0.07490.07640.07780.07930.0808  1.3 0.09010.09180.09340.09510.0968  1.2 0.10750.10930.11120.11310.1151  1.1 0.12710.12920.13140.13350.1357

37. 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 (cont.) Since the area we are given is the total area in the two tails, we must first find the area to the left of  z. We have already calculated that the area to the left of  z is 0.0808, but we can have the calculator find this area at the same time that it finds the z-score.

38. 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 (cont.) Enter invNorm(0.1616/2), as shown in the screenshot, and press. The answer is  z ≈  1.399711. Rounded to two decimal places, the positive value of z is 1.40.

39. 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? Solution Remember that we need to use the steps in reverse. Begin by using the given area to find the value of z. We will use a TI-83/84 Plus calculator to find the solution to this question.

40. 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 (cont.)

41. 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 (cont.) Since we know that the area to the right of x is 0.6700, enter invNorm(1Þ0.6700), as shown in the screenshot, and press. The answer is z ≈  0.439913. But we will see a better way in just a moment…

42. 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 (cont.) The final step is to use this z-score, along with the given values of the mean and standard deviation, to solve for x. Substituting these values into the formula for x we have the following. But we will see a better way in just a moment…

43. 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 (cont.) Alternate Calculator Method – THE BETTER WAY It is also possible to have the calculator solve this problem directly without doing any intermediate calculations. Instead of finding the z-score first and then solving for x, we can have the calculator find the value of the normally distributed value in one step. The only difference when working with a normal distribution that is not the standard normal distribution is that we must enter the mean and standard deviation in addition to the area to the left of the value in which we are interested.

44. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. invNorm talks x language, too  Remember from before: normalcdf(-1E99,z) = area to the left of that z and normalcdf(-1E99,x,μ,σ) = area to the left of that x  invNorm(area to left) = the z value that makes the split between area to left and to right, and invNorm(area to left,μ,σ) = the x value

45. 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 (cont.) The function syntax is invNorm( area, ,  ). Enter invNorm(1Þ0.6700,28.0,2.5), as shown in the screenshot, and press. The answer is x ≈ 26.9.

46. 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? Solution In order to determine the temperature that represents the 90 th percentile, we first need to find the z-value that represents the 90 th percentile. In Example 6.19, we found this value to be z ≈ 1.281552. Once we have the value of z, we can substitute z, , and  into the formula for x that we have from earlier in this section.

47. 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 (cont.) Thus, a temperature of approximately 99.54  F represents the 90 th percentile. BUT DO IT DIRECTLY IN x LANGUAGE, SEE NEXT SLIDE

48. 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 (cont.) Alternate Calculator Method When using a TI-83/84 Plus calculator to solve this problem, it is not necessary to first find the z-value that represents the 90 th percentile. The temperature that represents the 90 th percentile can be found in one step by entering invNorm(0.90,98.60,0.73), as shown in the screenshot. SUPERB SHORTCUT !!!

49. Example 6.24 with Excel 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? Recall NORM.DIST(x value, mean, stdev) = area to left Now NORM.INV(area to left, mean, stdev) = x value

50. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile Let’s 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 a baby could be and still have a length that is amongst the top 25% of baby lengths? Solution If we want to find the minimum length for the top 25% of baby lengths, we can simply find the value of the 75 th percentile, or Q 3.

51. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile (cont.)

52. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile (cont.) A percentage of 75% is the same as an area of 0.75 under the normal distribution. We will begin by finding the z-score that has an area of 0.75 to its left. Enter invNorm(0.75), as shown in the screenshot, and press. The answer is z ≈ 0.674490. BUT DO IT DIRECTLY IN x LANGUAGE, SEE SECOND SLIDE AHEAD >>>>>>>>>>>

53. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile (cont.) Now that we know the value of z, we must use it along with the mean and standard deviation to calculate the corresponding value of the random variable X that represents the length of the baby. We can use the formula for x that we saw previously. BUT DO IT DIRECTLY IN x LANGUAGE, SEE NEXT SLIDE

54. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile (cont.) Rounding this to one decimal place (the same as in the standard deviation), we see that the minimum length a baby can be and still be in the top 25% of lengths of full-term newborn babies is approximately 20.8 inches. Alternate Calculator Method When using a TI-83/84 Plus calculator to solve this problem, it is not necessary to first find the z-score that has an area of 0.75 to its left.

55. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.25: Finding the Value of a Normally Distributed Random Variable That Represents a Given Quartile (cont.) The length that represents the 75 th percentile can be found in one step by entering invNorm(0.75, 20.0,1.2), as shown in the screenshot. MUCH BETTER, IN ONE STEP LIKE THIS.