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# Chapter 3: Normal Distributions MATH 243

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This 22 page Class Notes was uploaded by Rachel Kasashima on Monday October 19, 2015. The Class Notes belongs to MATH 243 at University of Oregon taught by Harker H in Fall 2015. Since its upload, it has received 17 views. For similar materials see Intro Probability and Statistics in Mathematics (M) at University of Oregon.

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Date Created: 10/19/15

Chapter 3 Normal Distributions Suppose you have a histogram and want to approximate it using a curveHasmsPeciioshape 447 M De nition density curve is a curve that o is alway on or above the horizontal aXis and c has area exactly 1 underneath it It describes the overall pattern of the distribution Normal curves am cxamples vf otensi rv curves 39 quot I I 2 l Rather than enumerate the data for a histogram we can estimate the num ber of observations in a certain range by looking at the area under the curve in that range s to 20 mo on CANNOT bointeqm rcd If you consider a value x on the hori zontal aXis we have The area to the left of x is the proportion of scores or values less than or equal to x mm 09 09 of ne population it hm W m m 507 of the Population Recall that the area under the entire curve is 1 so the proportion Will be a number between 0 and l Density curves areidealized descriptions of data o The median of a density curve is the equal areas point oThe mean of a density curve is the balance point m The mean is denoted by g A g The standard deviation is denoted by G Urinal Distrim Normal Gaussian density curve is o mmetrieLsingle peaked and bell shaped completely described by giving its mean and its standard deviation 0 N040 Whm concavity changed 0 Changing Without Changing 6 moves the curve along the horizontal axis 0 Curves With larger 6 are more spread out N small c5 lamb A Normal distribution is described by a Normal density curve It is completely speci ed by and G We write 0 Example Heights of women are approximately Normal with 64 inches and G 27 inches 6427 Men s heights have 693 inches and 6 28 inches 69328 How far ways from the mean is a 70 inch woman 70 inch man Woman 6 inches taller 4 wnsiow standard elmin ow marr 07 inches taller Consider the same distributions Women s heights have distribution 64 27 and Men s heights have distribution 693 28 Consider a 75 inch tall man and a 72 inch tall woman Who is taller for their M s M b 1139 7mm 16 39m The woman is ller for her gender 71x 36 we bb39J We M 72 M W In the previous examples we measured distance from the mean by multi ples of the standard deviation 6 If x is an observation from a distribu tion 6 then x 6 z is the standardized value of x or its Z SCOI BLmualle of standard deviation x z6 OR z Returning to the last example we would say 0 man of height 75 inches has zseore 2 O 4 2 75643 23 o woman of height 72 inches has z 7254 score 296 z 27 Th6 68 95 997 Rule 68 of data 95 of data 7 997 of data 41 u In 3 2 I 1 0 M 36 M Zb 40 M These an approximan39ons 1 MW 2 3 M39Zb 14336 In the Normal distribution With mean and standard divination 6 o pprox 68 of the observations are Within 6 of o pprox 95 of the observations are Within 26 of o pprox 997 of the observations are Within 36 of Example Scores of students on the S T follow the Normal distribution 1426 150 pproXimately What percent of seniors score between 1276 and 1576 W pproXimately What percent of students score less than 1726 3 179 Pevoenf less than 39le z InaNu z 2 IBD 20 39 Mppmx l W igngO gigrgg ezj 0 lessfln w t l me Mb um mt De nitions The standard Normal distribution is the Normal distribution 0 l The cumulative proportion for a value x is the proportion of observations in the distribution less than or equal to x normsdistz score1cumulative normdistxmeanstandarddevcumulative I 12 446 M M b Example Scores of students on the S T follow the Normal distribution 1026209 What percent of seniors score at least 820 In Excel normdisxu l Lcumulafive s 4 100l novmdis1XM 6 I I l l T I I w 608 3r me 3923 um quot53 m z a q Look up 20 in Table A Table A giver cumulative Proportion 0J5quot molt 0J6quot Example other direction Find the number z such that the propor tion of observations that are less than z in a standard Normal distribution is 08 NW 0 2 g 5 Find cumulative pvoporh39on 08 in TableA zscoveeso39 Example percent to zseore Find the number z such that 35 of all observations from O 1 are greater than z lntableAz is batwun 038 and 034 S T scores Scores on one part of the S T follow 504 111 HOW high must a student score in order to place in the top 10 of all students M Excel CPO 0D39 zscom wormsmvccp 4 I I I u H 287 3ln 3933 bl39z39x 776 857 SMSGDW Z3930 159 s Flv 15 cP0 I O 70 IT A6AIN Table A gives 2 Li usmg GPor 08447 movm inv LP M 0 X39u 1 zoxu X904I39Llu 008 at 647 will mm m you39re in wrap 039 Quartiles The thorax lengths in a population of male fruit ies follows a Normal dis tribution 8 0078 measured in mil limeters What are the median and rst and third quartiles of thoraX length

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