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# 235 Class Note for STAT 30100 with Professor Zhao at Purdue

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Section 13 The Normal Distribution Normal curves are Np 039 o Bellshaped unimodal symmetric 0 Mean u always center of the graph 0 Standard deviation 6 controls the spread of the graph where curve changes direction 0 Probabilities are just the area under the curve integral between the points of interest 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 Nu 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 standard Normal distribution Convert using the formula x I 039 Z scores are what you need in order to use Table A in the front cover of your book Z Z scores 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 0 PZ gt zscore l 7 PZS zscore o PZ zscore 0 Therefore PZ S 2 score PZ lt z score and PZ Z z score PZ gt z score 0 PaltZltbPZltb7PZlta Z scores tell you how far measured in standard deviations the original observations fall from the mean To nd a probability if you have X Np 039 and a sample score to work with x I 039 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 Practice using the Normal table with some generic problems where we already know Z instead of starting from X Find the probabilities a P Z lt 124 b P Z gt 124 c P Z lt 124 d P Z gt 124 e P Z 124 f P124ltZlt124 g P Z lt 45 h P Z lt 12 Example Now practice using the Normal table with some generic problems where we staIt withX and then have to nd Z before going to the table For all the problems below use u 2 and 6 12 a PXlt1 b PXgt4 c P0ltXlt34 d PX24 e PXlt12 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 6 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 if necessary by doing the l 7 trick 3 If you have a 2sided probability use Pzo lt Z lt 20 2 PZ lt zscore 7 l 4 Convert the zscore to X by converting with X p Z6 Example Practice using the normal table with some generic backwardsnormal problems where we know the probability and are just looking for zo instead of going all the way to X0 Find the probabilities a b c d e t g PZltzo05000 PZltzo09953 PZltzo02206 PZltzo095 PZgtzo00129 PZgtzo05025 PzoltZltzo020 Example Now practice using the normal table with some generic backwardsnormal problems where we know the probabilities and are now going all the way to x0 by way of 20 For all the problems below use u 2 and 6 12 a b c PXltX002206 PXgtX0095 Find the 2 values between which 20 of the data lies 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 What is the account balance X0 such that the probability of a balance being more than it is 05 Hint you should be able to do this one without math d Between what 2 central values do 40 of the balances fall General situations for independent random variables Sum of 2 independent random variables x y Mean w 11 5 Variance to standard deviation Difference of 2 independent random variables x y Mean ay 1x y Variance to standard deviation Other linear combinations of 2 independent and normally distributed random variables aX i bY i c where a b and c are constants and X and Y are the variables Mean yam cw ibyy to 2 2 2 2 2 0thth a 0x b Oy Variance to standard dev1at10n 2 O 01th m athYtL Any linear combination of independent normal random variables is also normally distributed If you need to nd a probability for a sum or a difference of two independent normal random variables just make sure to use the mean and standard deviation for the sum or the difference rst Then the problem will work just like a regular normal distribution problem Example adapted from p 303 ofMampM Scores on the math part of the SAT exam in a recent year had an average of 519 and a standard deviation of 115 The verbal part had an average of 507 and a standard deviation of 1 11 Both portions of the exam were approximately normally distributed and we will assume that the verbal and math scores are independent from each other a What are the mean and standard deviation of the total SAT score math verbal What is the probability that a randomly selected student s total SAT score will be less than 1000 What was the cutoff for the top 10 of total SAT scores What is the mean and standard deviation of the difference between a student s math and verbal scores math verbal What is the probability that a randomly selected student will score at least 20 points higher on the verbal part of the exam than on the math part Do you think that our assumption that the verbal and math scores are independent is reasonable Why or why not mmtu Table enlry or z u Lllc area nnxler um nandmd mm unvc m um IcEI m 1 TABLE smnmm nmmnl pmbabi uus an u ur n39 22 no r 2 4 mm mm mm um mm mm 2 2 nuns nuns unm nDN Irma am 2 2 mum nuns unns nms unus nuns 2 x mm mung anus mm mm man 22 um2 um2 unH nun mm mm 722 mm 2212 ms ums uuu mp 722 uuzs 2224 any nun mm mm 727 un2s un22 mm mm nu27 W22 72 5 man man uu2g nmx uu27 nuza 723 ms nuso nusz nus nmg mm 2 4 uu22 um um um muss any 2 2 mm mu2 mum uuxu um any 22 212 um qu um mm mm 2 EH79 mm um msu nus BM 22 2222 um um 2192 ms m2 42 2227 2274 n2sn uzu 2222 x2222 4 2 u2sg 2224 n214 u2nv 2294 47 Ma 2427 n222 my ma 4 s 2542 nszs mg 42425 2455 71 5 mm 2642 n59 2522 mac 7 x 4 man Um um um mm 1 2 new 2924 am 2252 2222 42 Mix m2 mm mm was 11 2227 2 222 mm mm x n 2527 2529 ms 2222 mu rug 2222 1722 was 1st J H 722 zuo 2252 law 2922 m 727 mu 2252 2226 22m 2242 725 2742 1676 2525 2514 145 Va 5 2m 2m 2277 2242 2m 24 2446 2272 2222 2192 2m 2 2 222 2m 2524 2as 2222 22 42m 4 29 29 2922 22w 2 x 4622 2522 4254 2225 222 Em suuu wzn nm 272 AM 979999999 Table unry 99 9 s m 999 under 09 999999 99mm curve 99 um Ian 99 z TABLE A Smndmd 999mm 9999999199959979199271 99 9 9 92 w 95 99 99 9 9 5999 5999 5999 5 929 599 52 9 5259 9 x 5299 5429 5479 55 9 5595 57H 572 9 2 5792 5922 5979 59m 5997 5992 5m 92 5979 9297 9255 5292 5259 9999 557 9 4 5 959 9929 5594 5725 5974 5979 95 5995 9995 7919 7999 7999 95 7297 7229 7257 7922 7512 9 7 7599 2992 7572 7724 7922 9 9 7999 7929 7997 9922 9999 9929 99 9959 9292 9229 299 9295 9299 x 9 9m 9459 9995 952 9599 952 9 x 9549 9925 9799 9749 9999 9929 x 2 9949 9999 999 9922 999 9995 2 9922 9992 9 95 9992 997 H 9992 9229 9255 9299 9279 95 9222 9257 9279 9299 9429 994 95 9452 9979 9494 9595 9525 9595 x 7 9554 9572 9592 9599 9925 9522 9 9 9599 9955 9594 9579 9999 9795 x 9 9792 9725 9722 9794 979 9757 2 9 9772 9792 9799 9799 99 92 99w 2 x 992 9929 9924 9942 9954 9957 2 2 9959 9959 997 9979 999 9999 2 2 9992 9999 9999 9995 99v 9995 2 4 99 x 9 9922 9925 9929 9924 9925 2 5 9929 9999 9922 9995 995 9952 2 5 9952 9955 9957 9959 9992 9999 2 7 9955 9957 9999 9979 9922 9979 2 9 9979 9975 997 9979 9999 9999 2 9 9999 9992 9992 9999 9999 9995 29 9997 9997 9999 9999 9999 9999 2 x 9999 9999 9999 9992 9992 9992 22 9992 9999 9994 9994 9995 9995 2 2 9995 9995 9999 9995 9999 9997 2 9 9997 9997 9997 9997 999 9999

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