### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Introductory Statistics I STA 2023

USF

GPA 3.53

### View Full Document

## 27

## 0

## Popular in Course

## Popular in Statistics

This 81 page Class Notes was uploaded by Daphnee Quitzon DVM on Wednesday September 23, 2015. The Class Notes belongs to STA 2023 at University of South Florida taught by Ling Wu in Fall. Since its upload, it has received 27 views. For similar materials see /class/212691/sta-2023-university-of-south-florida in Statistics at University of South Florida.

## Similar to STA 2023 at USF

## Reviews for Introductory Statistics I

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/23/15

Introduction to Statistics I CHAPTER 4 Introduction to Set Theory J English Logic Set Theory pqr PQR And 19 q P Q Or 19 v q P U Q Not p P39 or PC Implies p gtq PQQ Probability 3 Events can be named with capital letters A B C PA means the probability of A occurring RA is read P of A o s PA s 1 Introduction to Probability Probability is a numerical measure between 0 and 1 that describes the chances or likelihood that an event will occur Probabilities closer to 1 indicate that the event is highly likely to occur Probabilities greater than 05 indicates that the event is more than likely to occur Probabilities closer to 0 indicate that the event is unlikely to occur Probability is a language of inferential statistics Certainities ancl Improbable If the probability of an event is 1 then the event is certain to occur If the probability of the event is 0 then the event is imgrobabe that is the event is certain not to occur UNotation P E 3 ME nS 1 05 for quotProbabilityquot for quotofquot for thequoteventquot also used A B C denotes the end of the event quotisquot for thequotnumberquot of outcomes decribed by theevent for the quotsample sp acequot the entire set of p ossible outcomes Formulation or Translation A probability assignment based on egualz likey outcomes uses the formula 05 the number of outcomes favorable to event n5 total number of outcomes Alternative Formulation A probability assignment based on relative eauency uses the formula PE i n where f is the frequency of the event occurrence in a sample of n observations Example 1 Henry guesses a truefalse question What is the probability of getting right 3 A random sample of 500 students at USF were surveyed and it was 375 wore glasses or contact lenses What is the probability that a USF student selected at random wears corrective lenses Law of Large Numbers LLN lt gt In general As the sample size increase the statistic approaches the parameter In terms of Means As the sample size increase the sample mean approaches the population mean lt32 In terms of Proportions As the sample size increase the sample proportion relative frequency approaches the population proportion true probability Q In terms of Variance As the sample size increase the sample variance approaches the population vanance Q EI39C 10 Law of Large Numbers L In the long run as the sample size increases the relative frequency will get closer and closer to the theoretical probability Example We repeat the penny experiment and the relative frequency gets closer and closer to Phead 050 Relative Frequency 052 0518 0495 0503 04996 f number of heads 104 259 495 1006 2498 n number of flips 200 500 1000 2000 5000 11 LLN in Shorthand Let 6 rep resent the p aramet er and 6 rep resent the statistic then In general as n gt 00 6 gt 6 ForMeansas n gt00c gt u For Proportions as n gt 0015 i gt p n 2 2 For Varlance as n gt 003 gt 039 12 An application of Law of Large Numbers Eoulette 13 Terminology as A Statistical Experiment or Statistical Observation as any random activity that results in a definite outcome An Event is a collection of one or more outcomes of a statistical experiment A Simple Event is an outcome of a statistical experiment that consists of one and only one of the outcomes The set of all simple events constitutes the sample Space of an experiment Two events are Mutualy Exclusive if there intersection is empty Q 14 Principles of Probability The probability of the sample space is one MathematicallyPS 1 lt9 The probability of an event is at least zero and at most one Mathematically0 g PE g 1 a The sum of all simple events that is mutually exclusive events which in union form the sample space is one Mathematically El mEj Qj 72 j EIUEsz UEH S gt PE1PE2PEn1 1 5 Mutually Exclusive Events J 16 Set Theow Venn Diagrams 17 Complement The Complement of an event E is the set of outcomes NOT in E denoted EC or E39 18 Complementaw Events EUE39zS E EmE Where Q the empty set and PE PE39 1 19 Examples Probabilities Given the three events ElE2E3 are mutually exclusive and are the simple events which form the sample space determine which of the following probability distributions are plausible if the probability distribution is implausible state why PltE1gt PE2gt PltE3gt 3 PE1 0145PE2 0521PE3 0333 oxIH PE19PE2gtPE3 20 pmers Probabilities Given the three even are mutually exclusive and are the simple events which form the sample space determine which of the follow probability distributions are plausible if the probability distribution is implausible state Why PltE1PltE2gtPltE3gt PE1 o 145 1952 O521PE3 0333 220999 lt Limplausible PltE1gt3PltE2gtPltE3gt 21 RECAP r Probability is a measure between 0 and 1 inclusive which describe the chance of an event occurring The probability of the sample space is 1 The probability of any event is at least 0 and at most 1 The sum of the probabilities of all simple events is 1 Mutually exclusive events are such that their intersection is the empty set The sum of the probability of an event and its complement is one lt Q 22 Some Probability Rules for Comgound Events rl Two events are independent if the occurrence or nonoccurrence of one event does not change the probability that the other event will occur Otherwise if there is a implied condition between the occurrence of the two events then the events are dependent 23 Independent Events Two events are independent if the probability of both A and B happening is simply the probability of A times the probability of B NOTE In general when there are two events AND means MULTIPLY PA and B PA x 103 24 Dependent Events Two events are dependent if the probability of both A and B happening is the probability of A times the probability of B GIVEN A has occurred AND still means MULTIPLY but a condition has been set forth PA and B PA gtlt PB given A PA gtlt PB A PB and A PB gtlt PA given B PB gtlt PA B 25 rl Example Independent Dependent Two cards are selected from a standard deck of 52 cards WITH REPLACMENT what is the probability that the two cards selected are Hearts INDEPENDENT PH and H 52 52 E Two cards are selected from a standard deck of 52 cards WITHOUT REPLACMENT what is the probability that the two cards selected are Hearts DEPEN DENT 13Xg 1 1 17 PHandH gtlt 26 Multiplication Rule for Independent Events PA and B PA 39 PB General Multiplication Rule For all events independent or not PA and B PA PB A PA and B PB PA B 27 Conditional Probability Two events are dependent if the probability of both A and B happening is the probability of A times the probability of B GIVEN A has occurred hence solving for the conditional statement we have PA B P1B PBA 2m P04 28 nArB 715 PB quot3 725 PA B nA B X nS nA B nS nB nB 29 Examples Conditional Probabilities E Eye Color Female Male TOTAL Brown 20 5 25 Blue 15 30 45 Green 30 50 80 TOTAL 65 85 150 30 Examples Eye Color Female Male TOTAL Brown 20 5 Blue 15 30 45 Green 30 50 80 TOTAL 65 85 150 PBr0wn 12750 OIt K 31 Examples Given Female Eye Color Female Male TOTAL Brown 5 25 Blue 15 3O 45 Green 30 50 80 TOTAL 65 85 150 amp i PBr0wn Female 65 13 32 Examples Eye Color Female Male TOTAL Brown 20 5 25 Blue 15 3O 45 7 Green 30 50 80 TOTAL i 65 gt 85 150 Q 2 PFemale 150 3O 33 Examples Given Brown Eye Color Female Male TOTAL Brown lt 20 5 25 Blue 1 5 30 45 Green 30 50 80 TOTAL 65 85 150 PFemale Brown 2 g 08 34 Ouick review of Multiplication rules 7 One of the multiplication rules can be used any time we are trying to find the probability of two events happening together Pictorially we are looking for the probability of the shaded region in Figure PA and B PA PB A PA and B PB PA B K Note that and is repaced by Intersection Q 35 Introducing Addition rules Another way to combine events is to consider the possibility of one event ganother or both occurring Pictorially the shaded portion below represents the outcomes satisfying the or condition Notice that the condition A or B is satisfied by any one of the following conditions 1 Any outcome in A occurs K 2 Any outcome in B occurs 3 Any outcome in both A and B occurs 36 Addition Rules Given two events A and B are mutually exclusive PA u B PA PB Given two events A and B are NOT mutually exclusive PA u B PA PB PA m B Overlap Illustrated PA u B PA 103 PA m B 38 Example Mutually Exclusive Given two events A and B are mutually excusive PA K B PUD B 131 PM 0 25 PHUB 0 65 0 35 39 rl x Example NOT Exclusive Given two events A and B are NOT mutually excusive HA u B PM PB PA m B S 0 65 PA0 25 PB05 PAmB005 PH UBO35 40 Exa m p I e DISCUSS in Class 13 If LFLH LilLL le l3 What is yam es Bright Electric 39 it Either an elem is the sample 51 assigned to the CD or D PD PD 1 00599 09401 1 a Relative frequency 19317 00599 or 599 b 1 00599 09401 or approximately 94 c Defective not defective the probabilities add up to one 41 Example a You draw hall befur that the E h Repeat pa drawing ti a With replacement PRed first and Green second 312712 748 or 0146 INDEPENDENT b Without replacement PRed first and Green second 312711 21132 or 0159 DEPENDENT 42 Example Very Satis ed Total Assume the sample r Find the pmbabilityl Not satisfied PNS 35360 Not satisfied and walkin PNS and WI 21360 Not satisfied given referred PNSR 5149 Very satisfied PVS 107360 Very satisfied given referred PVSR 48149 Very satisfied and TV ad PVS and TV 31360 Are the events satisfied and referred independent or not Explain your answer No Preferred 149360 is not equal to Preferred given satisfied 59138 43 Summary A statistical experiment or statistical observation is any random activity that results in a recordable outcome The sample space is the set of all simple events An event A is any subset of the sample space The probability of an event A is denoted by PA i Pentire sample space1 ii For any event A D g PA g 1 iii The complement of an event A is denoted by A PAC l PA iv Event A and B are independent events it PA PAlB P A a d B v Conditional Probability PAB 8 vi Multiplication Rules General PA and B PM PBlA Independent event PA and B PA PB vii Events A and B are mutually exclusive if PA and BO viii Addition rules General PA or B PA PB PA and B Mutually exclusive events PA or B PA rl Trees and Counting Techniques fair Multiplication Principle AND means MULTIPLY Summation Principle OR means ADD 45 Multiplication Rule for Counting g Multiplication rule of counting If there are 11 possible outcomes for event E1 and m possible outcomes for event E3 then there are a total of n X m or rm possible outcomes for the series of events E1 followed by E2 This rule extends to outcomes involving three four or more series of events 46 Example Multiplication Principle Given there are three choices for salad five choices for the entr e and two choices for dessert how many dinning selections are there for this three course meal ANSWER 3X5X23O 47 Tree Diagrams 7 Displays the outcomes of an experiment consisting of a sequence of activities The total number of branches equals the total number of outcomes Each unique outcome is represented by following a branch from start to finish Tree Diagrams Wit7 Probability We can also label each branch of the tree with its respective probability Q To obtain the probability of the events we can multiply the probabilities as we work down a particular branch rlw 48 Urn Example 32 Suppose there are five balls in an urn Three are red and two are blue We will select a ball note the color and without replacing the J first ball select a second ball prelimxiii e Tree Diagram for Color of Color oi Um Experiment rst SECOHCI E piFL 9W8 R s 4 Red Pm givequot Bi 3 4 3 PM give 5 1 Blue U given 8 l 4 There are four possible outcomes Red Red Red Blue Blue Red Blue Blue We can find the probabilities of the outcomes by using the multiplication rule for dependent events 49 A Tree Diagram Two fair coins are tossed a nickel and a dime J 2 x 2 4 different outcomes Nickel Dime Sample Space Two Coins S 2 HH HT TH TT Since these were fair coins they are equally likely to occur and the probabilities can be taken from the sample space above that is PHH PHT PTH PTT 51 Tree Diagram Two unfair coins are tossed a nickel and a dime Aw Nickel Dime 02 H H lt 08 T 02 06 H T 08 T 52 1 Sample Space Two Coins I S 2 HH HT TH TT Since these were unfair coins they are not equally likely to occur that is PHH 04x 02 008 PHT 04 x 08 032 PTH 06x 02 012 PTT 06x 08 048 53 Factorials Factorials count the number of ways to order n objects without repetition and is denoted nngtltn 1gtltn 2gtltgtlt2gtlt1 11 O1 Soforexample5 5432 1 120 Note that i is read five factorialquot NOTE Factorials are not de ned for negative numbers 54 Example Factorials How many ways can someone rearrange 6 books on a shelf a 6 6X5X4X3x2x1720 55 KIPermutations Permutation ordered grouping of objects 39Fhisistiientrmbe1eeways to arrange in order n distinct objects taken rat a time without repetition 56 Counting Rule Counting rule for permutations The number of ways to orange in order n distinct objects taking them 339 at a time is where n and r are whole numbers and n E t Another commonly used notation for permutations is nPr 57 Permutations Extension JAND means MULTIPLY Pn rnP ngtlt n 1gtlt gtlt T T T st 1 rlh PnrnPr n 11gtltn 21gtltgtltn r1 T T T 1st 2n 6 58 Example Permutations How many ways can the first three horses out of five cross the finish line 5P35X4X36O 59 Combinations A combination is a grouping that pays no attention to order This is the number of ways to combine without order from n distinct objects taken r at a time without repetition 60 Counting rule f Counting rule for combinations The number of combirmtions of 11 objects taken r at a time is n T n r run r where n and 1quot are Whole numbers and n 2 r Other commonly used 7 1 notatlmls tor c0111b11 1at10ns 111clucle IICI and lt r 10 61 Combinations Formulated tit Since there are r ways to order the r objects selected using permutations n objects permute robjects That is Pn r r Cnr 62 Permutation Permutation means arrangement of things The word arrangementis used if the order of things is considered 7 Conbination Combination means selection of things The word selection is used when the order of things has no importance Example Suppose we have to form a number consisting of three digits using the digits 1234 To form this number the digits have to be arranged Different numbers will get formed depending upon the orc er in which we arrange the digits This is an example of Permutation Now suppose that we have to make a team of 11 players out of 20 players This is an example of combination because the order of players in the team will not result in a change in the team No matter in which order we list out the players the team will remain the same For a different team to be formed at least one player will have to be changed 63 Example How many ways can you permute 3 out of 4 letters Letters LABCD 4X3X2 24 ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA ACD ADC CAD CDA DAC DCA BCD BDC CBD CDB DBC DCB 64 Example How many ways can you combine 3 out of 4 letters Letters LABCD 4 3x2X1 PERMUTATIONS 9 COMBINATIONS ABC ACB BAC BCA CAB CBA 9 ABC ABD ADB BAD BDA DAB DBA 9 ABD ACD ADC CAD CDA DAC DCA 9 ACD BCD BDC CBD CDB DBC DCB 9 BCD Odds The odds in favor of an event A is the ratio P04 P04 P04 Pn0tA PAc PA39 M no M LA nltA39 nS 66 rl x Example Workbook Part 4 7 nE x and PE i y gt 715 y since by de nition PE 2 LE nS 2 quotE39 y x since nE nE 2 715 2 PE39 since PE 1PE quotE39 V nS and OExy x or OE39y xx0r NOTE Odds can not be written as a mixed fraction x y x x x or decimal that is Odds must be a ratio 67 Example Workbook Part 4 8 a 61 11EaandPE 11E 115 gt 115 2 a 9 since by de nition PE 2 11E39 9 since 11E 11E39 2 115 since PE 1 PE quotE3 SPOT ab 115 and 0E a b or OE baor NOTE Odds can not be written as a mixed fraction or decimal that is Odds must be a ratio 68 Example There are eight words to be matched to their respective definitions How many different ways can you match the words to a definition not necessarily correctly so Word 1 can be one of eight definitions 2nd word therefore would only be one of seven etc Hence the various number of distinct orderings is 8X7X6X5X4X3X2X140320 69 Example There are eight words to be matched to ve respective definitions How many different ways can you match the words to a definition not necessarily correctly so Word 1 can be one of eight definitions 2nd word therefore would only be one of seven etc but only up to five questions Hence the various number of distinct orderings is 8x7x6x5x4 6720 70 Example 2 P Pro t in the First Year 2 PA 065 PPr0 t in the Second Year 2 PB 071 PB A 087 Then aPA 065 bPB 071 cPB A 087 dPA m B PA gtlt PB A O65gtlt 087 057 ePA U B PA PB PA m B 065 071 057 079 fPAu 13 1 PAUB 1 079 021 71 Example Given seven women and ve men a how many six member committees can be formed 12x11gtlt10gtlt9gtlt8gtlt7 5712gt12C6 924nS 6x5x4x3x2x1 b how many six member committees consisting entirely of women 7gtlt6gtlt5gtlt4gtlt3gtlt2 7 nltEgt 7 6 6gtltSgtlt4gtlt3gtlt2gtlt1 c What is the probability of a committee consisting entirely of women 7 72 Ex ple Additional Question Given seven women and five men a how many siX member committees can be formed 57 2123156 szm n5 6x5x4x3x2x1 b how many siX member committees consisting of two women and four men 7x6x5gtlt4gtlt3gtlt2 2105nE 2x1 4x3gtlt2gtlt1 c what is the probability of a committee consisting of 7C2X5 C4 two women and four men ME 2 m 2 i 924 44 73 Example T One Visit Two Visit Three Visit Four Visits Five Visits Six or more TOTAL 1824 E35 58 12 5 4 4 148 25 39 38E 239 59 22 1 15 739 4049 219 151 35 13 9 5 434 5064 185 192 35 14 F 5 349 ES and over 115 59 1B 12 7 3 224 TOTAL 962 B29 1 GE 44 32 One Visit Two Visit Three Visit Four Visits Five Visits Six or more TOTAL 18 24 148 2539 3913 T39 4949 219 434 59434 185 349 65 and over 115 224 TOTAL 962 620 1T1 66 44 32 a Pat least two visitsi 932 1894 74 r Example cont One Visit Two 1Visit Three Visit Four Visits Five Visits Sis or more TOTAL 15 24 55 55 12 5 4 4 148 2539 1 114 11 39 41149 215 151 3 13 9 5 434 51154 155 152 35 14 5 349 65 and over 115 59 15 12 F 3 224 TOTAL 962 520 170 55 44 32 h Fiat least two visitsl2539 353 739 One Visit Two Visit Three Visit Four Visits Five Visits Six or more TOTAL 1524 55 58 12 148 2539 355 235 59 T39 41149 215 151 35 434 5064 155 152 35 349 ES and over 115 59 15 224 TOTAL 962 52 170 65 44 32 c1 Piat more than three visits 142 1394 75 Example cont One Visit Two Visit Three Visit Four Visits Five Visits Six or more TOTAL 1824 55 58 12 5 4 4 148 2539 385 230 59 22 1 15 T39 4049 21B 1131 38 13 9 5 434 5064 188 102 35 14 3139 5 349 55 and over 115 139 18 f 1 39 i 39 TOTAL 962 E20 17 BE 44 32 d Fiat more than three visitsl and over 22 11 224 112 One Visit Two Visit Three Visit Four Visits Five Visits Sir or mare TOTAL 1824 55 58 12 5 4 4 I 148 2539 239 41149 434 50 64 gt 349 65 and over 224 TDTAL 170 e PHI or ulder 101 1894 76 Example cont One Visit Two Visit Three Four Visits Five 1iar39isits Six or mare TOTAL 18 24 55 58 12 5 4 4 148 2539 3813 230 BB 22 1 15 739 4049 21B 181 38 434 5064 391 102 35 349 65 and over 115 59 1B 224 TOTAL 962 620 1m 56 44 32 f PHI or olderlfour visits or more 1395 142 g PBS39 and more than once a week1 353 1894 P25391 739 PiMnre than once a week1 932 1894 1894 111863II39BU4 not equal to 019199963 77 Example Randomly to answer three trueor false question T Tlt F T Flt F T Tlt F T Flt F Since each question is equally likely there is only one correct sequence out of a total of eight PCorrect Answers 18 78 Example With TWO decks of cards and drawing ONE FROM EACH What is the probability they are one ace and one king INDEPENDENT why 52 x 52 2704 Hence having an ACE first and a KING second is 4 x 4 16 i 1 PAK 2704 169 1 PAK u K4 1 169 79 Llrr1rlrrlrrlrr1rr lIrrIIJiIIJI IJn Ilrrllrrlrrrrlrrlrr 1 1 1 1 1 1 36 4J 63 XX5 6392 SEW nn eeS mvv deeO 0mm wooM nnP 80 END OF CHAPTER4 81

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I made $350 in just two days after posting my first study guide."

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.