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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 r x 7 65m mu zxsu um mu 55w 5 55m 59m 5m mu gzsn39 LISU 739 7 um um rxm m m Law 791339 r mm mm mm mu rem mm 239 7 ma em mu mu nzn39 a 7 mu am am mu uzu a mu mu usm vvm uLm vz mm mm mm zm mn39 mm at mun mu own mm mm rum39 z Nun 99m xvuu 5mm mu uxuu v z mu mu xsun Lsnn csnu uvuu s 2 mm Linn xmn mu mn39 smu39 v 27 am LZDD39 xzun Zmn mu39 rm L smu39 nznu mm mm mu mu 539 7 Hum Nun 5mm um amn gym 5 r mun mun nun zmu Uuu39 mu i mun mun mm mm mun 5er x 1 sum snnu 5mm vmu yuuu mu r mun mun mm mm sum 5er i 1 mm mm mm mm mm mm r r on 2n su zu m sanqrqvqmd nuunu pJEvums V mm ume u 393 0 u 9quot nv mm mm plwplll39u aquot mum a n no my mu rummmy Table entry m z 5 me am under me nanhid nm nnl curve I m 5152 52 TABLE A smmtmd nmnm pmbnbxlmes can mled mu at Hz 2 Eu 5 us 22 an 3mm mu 5m 525a u x 5557 5595 57m 5752 u2 5542 5957 522 am a 2 522 5252 mu 55w 2 2 ma 5725 5222 mg n 5 3252 7222 man 7222 25 1229 7422 7517 2529 n 7 my 7722 3222 7552 n 2 1225 say 2122 5122 n 2 5252 2222 2225 2222 x n 2522 232 my 5522 x l 5722 not 22m 5522 x 2 2925 2922 252 om5 2 w l5 9122 9177 m 5255 02m 2212 x 5 22422 22w 1 5 9525 2525 n 9525 2522 1 2 9929 m5 2 9 2m 2157 2 u 92 2 9m 2 2 2252 5257 2 2 222 mu 2 2 5912 out 5 2 2 9922 9925 2 5 22251 2952 2 5 22922 2252 2 7 3212 921 2 a 5922 3922 2 9 5925 9225 2 u own 999m 5 2 mm 9992 2 2 9995 9995 3 2 22922 9957 2 4 255 9902

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