Anthony J Greene1 Computing Probabilities From the Standard Normal Distribution

Table B.1 p. 687

Anthony J Greene3 Table B.1 A Closer Look

Anthony J Greene4 The Normal Distribution: why use a table?

Anthony J Greene5 From x or z to P To determine a percentage or probability for a normally distributed variable Step 1Sketch the normal curve associated with the variable Step 2Shade the region of interest and mark the delimiting x- values Step 3 Compute the z-scores for the delimiting x-values found in Step 2 Step 4Use Table B.1 to obtain the area under the standard normal curve delimited by the z-scores found in Step 3 Use Geometry and remember that the total area under the curve is always 1.00.

Anthony J Greene6 From x or z to P Finding percentages for a normally distributed variable from areas under the standard normal curve

Anthony J Greene7 Finding percentages for a normally distributed variable from areas under the standard normal curve 1. ,  are given. 2.a and b are any two values of the variable x. 3.Compute z-scores for a and b. 4.Consult table B-1 5.Use geometry to find desired area.

Anthony J Greene8 Given that a quiz has a mean score of 14 and an s.d. of 3, what proportion of the class will score between 9 & 16? 1.  = 14 and  = 3. 2.a = 9 and b = z a = -5/3 = -1.67, z b = 2/5 = In table B.1, we see that the area to the left of a is and that the area to the right of b is The area between a and b is therefore 1 – ( ) = or 60.79%

Anthony J Greene9 Finding the area under the standard normal curve to the left of z = 1.23

Anthony J Greene10 What if you start with x instead of z? z = 1.50: Use Column C; P = What is the probability of selecting a random student who scored above 650 on the SAT?

Anthony J Greene11 Finding the area under the standard normal curve to the right of z = 0.76 The easiest way would be to use Column C, but lets use Column B instead

Anthony J Greene12 Finding the area under the standard normal curve that lies between z = –0.68 and z = 1.82 One Strategy: Start with the area to the left of 1.82, then subtract the area to the right of P = 1 – – = Second Strategy: Start with 1.00 and subtract off the two tails

Anthony J Greene13 Determination of the percentage of people having IQs between 115 and 140

Anthony J Greene14 From x or z to P Review of Table B.1 thus far Using Table B.1 to find the area under the standard normal curve that lies (a)to the left of a specified z-score, (b) to the right of a specified z-score, (c)between two specified z-scores Then if x is asked for, convert from z to x

Anthony J Greene15 From P to z or x Now the other way around To determine the observations corresponding to a specified percentage or probability for a normally distributed variable Step 1Sketch the normal curve associated the the variable Step 2Shade the region of interest (given as a probability or area Step 3Use Table B.1 to obtain the z-scores delimiting the region in Step 2 Step 4Obtain the x-values having the z-scores found in Step 3

Anthony J Greene16 From P to z or x Finding z- or x-scores corresponding to a given region. Finding the z-score having area 0.04 to its left Use Column C: The z corresponding to 0.04 in the left tail is x = σ × z + μ If μ is 242 σ is 100, then x = 100 × x = 67

Anthony J Greene17 The z  Notation The symbol z α is used to denote the z-score having area α (alpha) to its right under the standard normal curve. We read “z α ” as “z sub α” or more simply as “z α.”

Anthony J Greene18 The z  notation : P(X>x) = α This is the z-score that demarks an area under the curve with P(X>x)= α P(X>x)= α

Anthony J Greene19 The z  notation : P(X<x) = α This is the z-score that demarks an area under the curve with P(X<x)= α P(X<x)= α Z

Anthony J Greene20 The z  notation : P(|X|>|x|) = α This is the z-score that demarks an area under the curve with P(|X|>|x|)= α P(|X|>|x|)= α α/2 1- α

Anthony J Greene21 Finding z Use Column C: The z corresponding to in the right tail is 1.96

Anthony J Greene22 Finding z 0.05 Use Column C: The z corresponding to 0.05 in the right tail is 1.64

Anthony J Greene23 Finding the two z-scores dividing the area under the standard normal curve into a middle 0.95 area and two outside areas Use Column C: The z corresponding to in both tails is ±1.96

Anthony J Greene24 Finding the 90th percentile for IQs z 0.10 = 1.28 z = (x-μ)/σ 1.28 = (x – 100)/ = x

Anthony J Greene25 What you should be able to do 1.Start with z-or x-scores and compute regions 2.Start with regions and compute z- or x-scores