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## Chapter 3: Normal Distributions

by: Rachel Kasashima

17

0

22

# Chapter 3: Normal Distributions MATH 243

Marketplace > University of Oregon > Mathematics (M) > MATH 243 > Chapter 3 Normal Distributions
Rachel Kasashima
UO
GPA 3.95
Intro Probability and Statistics
Harker H

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Chapter 3 notes on powerpoint.
COURSE
Intro Probability and Statistics
PROF.
Harker H
TYPE
Class Notes
PAGES
22
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

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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