Week 5 MAT 121 Notes!
Week 5 MAT 121 Notes! MAT 121
Popular in Probability and Statistics for the Liberal Arts I
Popular in Mathematics (M)
This 11 page Class Notes was uploaded by Aria Sivick on Saturday October 3, 2015. The Class Notes belongs to MAT 121 at Syracuse University taught by in Fall 2015. Since its upload, it has received 40 views. For similar materials see Probability and Statistics for the Liberal Arts I in Mathematics (M) at Syracuse University.
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Date Created: 10/03/15
925 Empirical Rule If data is approximately normal then about 68 of it is within 1 standard deviation of the mean 95 is within 2 standard deviations of mean 997 is within 3 st devs 0quot 11713 aw xlafrqa 1 I i of 0 14 1qu A II 16 quot n If l J l V I quot I 397 t quot 39 51 g b 6 A I r l 4quot J I i r39 ILfl Aquot w j i t I I KN 3ft 39 I E 1 J J quot I 7 J g 39g 3 w l t l39 AJ J A S i quot N Jquot yk 7 y r V lt 7 t in g 4 rt 6 1 M H 397 Weight 110140170 200 230 Height 63 66 69 72 75 IQ 7O 75 115 130 2 1 O 1 2 The water melons grown on some form have weights with a mean of 148 pounds and a st dev of 31 funds and they are normally distributed Complete the following statements 1 About 95 of the watermelons weigh between 86 and 210 lbs 2 About 68 of watermelons weight between 117 and 179 pounds Measures of Position Percentiles There are 99 percentiles P1 P2 P99 in general Px Px tries to divide the data into the lower k and the upper 100 k o P50 median Quartiles Q1 P25 Q2 P50 03 P75 Deciles D1 P10 D2 P20 D5 Q2 P50 D9 P90 Suppose n data are ordered lowest to highest To compute Px 1 Compute its Position ion the list Compute L k100 If L is an integer add 05 If L is not an integer round up to an integer 2 Look in that Position on the list If 1 wields an integer look in that Position If the result of 1 ends in 05 average the Lth and the Lth 1 things in the list 2587162 students took the SAT You scored better than 1822035 What is your percentile percentile ranking of lower observations x 100 round to the nearest integer of total observations MX 100 70th percentile 2587162 Another Measure of Position ZScore is a number of st devs from the mean positive H above mean negative 6 below mean The mean has a z score of 0 JJ 7 7 2mm Z 2 3 X 1 928 5 Number Summary of data The 5 number summary for the number of chocolate chips in are chips ahoy is 19225 The 5 number summary is often pictured in a Box Plot box and whisker diagram A zscore is an observation expressed as a signed number of standard deviations from the mean with positive meaning above the mean and a negative sign meaning below the mean If x is an observation X s z score is 2 signs are our most common way of deciding what s unusual z score 2 1 1 2 squared 75 75 within 2 st devs of the mean At most 25 is more than 2 st dev from the mean For approximately normal data about 95 of data is within 2 st devs of the mean About 5 is moe than 2 st dev from the mean Deciding what s unusual 5 probability is the default The apples and oranges grown at an orchard are approximately normal weights apples m 121 oz st dev 17 oz oranges m 152 oz st dev 21 oz What s more unusual a 17 oz apple or a 20 oz orange 2 score for 17 oz apple is z 1721 288 17 2 score for 20 oz orange is z 20 152 229 21 2 score for 95 orange z 95 152 271 21 Rule for rounding round 2 scores of 239 what does it weigh 121 162 17 93 oz The probability that the coin is HEADS is 12 5050 chance Likelihood Probability is the fraction of the time that an event is expected to happen 930 A probability experiment is a procedure designed to produce an outcome A simple outcome is outcome that can t be further broken down A sample space is a collection of all simple outcomes of an experiment An event is a collection of simple outcomes Le a subset of the sample space We ll use capital letter to name events for example E and PCE to mean the probability that E happens 1 Toss can observe H or T Sample space H T Events not equal to zero H T HT 1 If an experiment has n equally likely outcomes the probability of any of them is 1n 2 If an experiments has n equally likely outcomes and event E happens m k of the outcomes than PE kn Ex Urn contains 5 red and 3 blue balls Pget red ball 58 625 2 Roll die observe the number of spots on top 88 1 23456 Pt P2 P3 P4 P5 P6 16 P at least 5 spots 5 6 PA 26 13 3 Roll 2 dice observe the total number of spots on top 88 23456789101112 Roll two dice and observe the number of spots on each die 4 Roll 2 dice observe the total number of spots on top 88 23456789101112 6 536 P10336 P7 636 P11236 336 P8536 P12136 P9 The sum of the probabilities is 1 A event more than zero spots 23456789101112 PA 3636 1 If A is an event then A bar is the event hat A does not happen the complementary event Complimentary rule PA PA bar 1 PA bar 1 PA PA 1 PA bar PA at least one 1 Pnone Ex Count bags of random family with 3 kids 88 01 23 PO 18 P2 38 P1 38 P3 18 Observe sequence of B s and G s in a random family 0 3 kids 88 see BG probs There are 8 equally likely outcomes each with probability 18 Event 2 boys BBG BGB GBB Toss 3 coins count H s 0 1 2 3 PO 18 P2 38 P 1 38 P3 18 1 02 Toss coin until HEADS happens Observe the number of tosses it took 88 1234567 infinite discrete but not finite P1 toss 12 P2 toss 14 P3 tosses 12 n 1 121418116132 12quotn 1 Complement Rule PA PA bar 1 P A bar 1 PA PA 1 PA bar PA at least one 1 Pnone OLA x vcm atness D 9u u9 4 Y N 5N Events A B 21 outcomes equally likely each with probability of 121 Definitions If A and B are vents than 1 A and B is the event that A happens and B happens 2 A or B is the event that A happens for B happens or they can both happen PA 621 PB 721 PA and B 221 PA or B 1121 The General Addition Rule PA or B PA PB PA and B Contingency Table PA or B PA PB PA and B 621 721 221 1121 Special Addition Rule lf PA and B 0 then PA or B PA PB PA and B 0 means 1 A and B can not both happen 2 A and B are disioint 3 A and B are mutually exclusive In A MAT 121 class 68 of the students are female 11 of the students are athletes 7 of the students are female athletes If a student is picked at ransom what s the probability that the student picked is a female gran athlete Let F be the event that the chosen student is female Let A be the event that the chosen student is an athlete PF orA PF PA PF and A 68 11 07 72