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# 65 Class Note for STAT 30100 with Professor Gundlach at Purdue

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Date Created: 02/06/15
Section 13 The Normal Distribution Learning goals for this chapter I Know when and how to use the empirical 6895997 rule 0 Understand what the standard Normal distribution is and how it is related to other Normal distributions 0 Calculate both forwards and backwards Normal distribution problems 0 Draw a Normal distribution curve appropriate for a story Normal curves No 6 Bellshaped unimodal symmetric Mean u always in the center of the curve the peak Standard deviation 6 controls the spread of the graph wide or narrow Probabilities are just the area under the curve integral between the points of interest Total area under the Normal curve l or 100 Curve stretches from w to 00 but the area under the curve gets very small the farther you go from the mean In the last set of Chapter 1 notes we discussed bellshaped distributions standardization and the 6895997 rule Standard Normal Distribution What if you need different probabilities for X Np 6 Do we have to use Calculus No We have a great shortcutithe Normal table Table A in the front cover of your book You must convert X Nu 6 to Z N0 l where Z has the stande Normal distribution Convert using the formula Zx7 039 Zscores are what you need in order to use Table A in the front cover of your book Zscores also let you compare 2 values from different Normal distributions to see their probabilities on the same scale P Z lt z score is what you will nd on the Normal table What if you want to know something else I PZ gt zscore l 7 PZS zscore PZ zscore 0 Therefore PZ 2 z score PZ z score PaltZltbPZltb7PZlta Zscores tell you how far measured in standard deviations the original observations fall from the mean To find a probability if you have X NL 039 and a sample score to work with x 0 2 Rearrange if necessary the inequality so that it uses lt or S Remember that PZ gt zscore 1 7 PZ S zscore 1 Convert X to Z Z 3 Look up the probability for your zscore on Table A 4 If z score is between 2 table values either pick the closer one or average the two closest values Example Checking account balances are N N1325 25 Bill has a balance of 1270 a What is the probability an account will have less money than Bill s b What is the probability an account balance will be more than 1380 c What is the probability an account balance will be exactly 1380 d What is the probability an account will have less than 1325 the mean e What is the probability that an account will have between 1310 and 1390 1 What is the probability an account will have less than 10 Backwards Normal Problems If you are given the probability and know X Nu S but you don t know the sample s score backwards from the previous problems 1 Treat it as PZ lt zscore the probability Work backward from the probability in Table A to a corresponding zscore 2 Adjust to lt ifnecessary by doing the 1 7 trick 3 If you have a 2sided probability use Pzo lt Z lt 20 2 PZ lt zscore 7 1 4 Convert the zscore to X by converting with X u 26 Example In the checking account example where the balances are N N 1325 25 a What is the account balance X0 such that the percentage of balances less than it is 23 b What is the account balance X0 such that the probability of a balance being more than it is 015 c Between what 2 central values do 40 of the balances fall Mixed forwards and backwards problems IQ Scores are Normally distributed with a mean of 100 and a standard deviation of 15 1 What is the IQ range for the top 10 of people 2 What percent of the population scores between 100 and 120 3 Between what two scores does the central 20 of the population have 4 Ifa person is selected at random what is the chance he scores below an 85 Mr er var um um um cm rzr 9m 5m 2m am zvsr m assz m m um um m u m mass 155 1m 5m st m m 9m 2on vm an nnri rm ma mz m1 sz 5m mm 5 r vsrz m2 HSZ m our 739 7 ma m vuzz 352 ram LUV war my 2m wnz umz Eur vm 5m nwr Mr 2sz NSF an an mm m 5m om Zysv39 n 7 um um am 201 rm 5er H mu mm nzm may am mV39 17 ma mu mm mm min vson i 1 mm man39 mu ran39 mm mm 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