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Section 6.4

Section 6.4. Finding Values of a Normally Distributed Random Variable. With a few swell additions by D.R.S., University of Cordele. Forward and Backwards, a Mathematical Pattern. Addition and Subtraction are inverse operations. Each one undoes the other.

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Section 6.4

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  1. 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 Addition and Subtraction are inverse operations. Each one undoes the other. Multiplication and Division are inverse operations. Each one undoes the other. 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. Kinds of Problems – and pictures Given area = .#### to left of some point, what z value makes it happen? And x value? Given some percentile or quartile, what z makes the split at that point? And x value? Given area = .#### to the right of some point, what z value makes it happen? And x value? Given symmetric area = .#### in the middle, what z and –z values make it happen? And x values?

  4. 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. Beaware of proportion, like 0.7357 is _____ than halfof the area, and we expectto get a __________ signfor the z value.

  5. TABLE METHOD – look in the INTERIOR Of the table for the area, read outward to find z. Search the INTERIOR of the table and read OUTWARD to find the z.

  6. 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.) FOR THIS EXAMPLE: area to the left of z = _______ is found by invNorm( _______ ) and the result is z = ________

  7. 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

  8. 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. Always draw a picture first ! TABLE METHOD – your area value is not in the table! Find the closest value to area = 0.2000 instead.

  9. 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. Always draw a picture first ! TI-84 METHOD – invNorm( _______ ) gives result of z = ____________________. Compare to the answer we got by using the table.

  10. Example 6.19: Finding the z-Value That Represents a Given Percentile What z-value represents the 90th percentile? Should be an EASY problem! Just a matter of knowing that by definition, “the 90th percentile” means “90% of the values are lower than mine”, so we are looking for the z value that has area _______ to its _______ TI-84 METHOD – Command: Answer: z = ______________ TABLE METHOD – the closest area to the value 0.9000 in the table is the area value ______ and reading out, we find the z = ______.

  11. Example 6.20: What z value has area = 0.0096 to its right? 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.

  12. 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( ____ - ______ )

  13. 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

  14. 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 Draw a picture first ! Then reason out the area that’s in the middle the area that’s in the two tails the area that’s in each of the tails

  15. Example 6.21: Finding the z-Value with a Given Area between -z and z (cont.) Find the z that has area______ on its left to get thevalue of –z. By symmetry,you instantlyknow the z, too. They told us 0.9000 in the middle. So think 1 – _____ = _____ in two tails total. So _____ ÷ 2 = _____ is the area in each tail.

  16. 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 = ____________”

  17. Example 6.21: Finding the z-Value with a Given Area between -z and z (cont.) We have a tie ! Even though we’ve been rounding z values to ____ places, we can resolve this case by using z = ___________

  18. 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!!!

  19. 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 = ____________”

  20. NEXT – moving up from backwards z problems to do backwards x problems • 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? • 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? (The problems will of course give you extra information, the mean and standard deviation.)

  21. 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! 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 for stdev.

  22. 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? Doing it with TI-84 entirely in x language, and letting the TI-84 handle the z stuff transparently: We want the x value that has area to its LEFT = ____________. You could do this with tables, get a z answer and then convert to an x answer using the formula

  23. 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? We want the x value that has area to its LEFT = ____________. OLD: TI-84 invNorm in z language was invNorm(area) = z value that has that area to its left. NEW: TI-84 invNorm in x language invNorm(area to the left, mean, stdev) = x value that has that area to its left. For this problem: invNorm( ____, ____, ___ ) gives the answer _______ so we say: “The value of X that has area 0.6700 to its right is X = ______”

  24. Summary – TI-84 talking in z language,and also in x language if you give it ____ and ____ normalcdf(-1E99,z) = area to the left of that zandnormalcdf(-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,andinvNorm(area to left,μ,σ)= the x value

  25. 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 90th percentile? We want the x value that has area = _______ to its _______, so we use the TI-84 command ___________ ( _______, ______, ______ ) which gives the answer _________. Response: “The temperature ______oF represents the 90th percentile.” Draw the picture (in proportion, and label the mean and stdev too.)

  26. 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 90th percentile? Recall NORM.DIST(x value, mean, stdev) = area to left Now NORM.INV(area to left, mean, stdev) = x value

  27. 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 75th percentile, or Q3.

  28. 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? 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.” Draw the picture (in proportion, and label the mean and stdev too.)

  29. 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.)

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