Probability and Statistics
Probability and Statistics MATH 2600
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MATH 2600 Probability and Statistics Chapter 2 Section 21 Overview Important Characteristics of Data 1 A representative or avemge value that indicates Where the set is located 2 A of the amount that the values among themselves 3 3 The am or 7 Ufa 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr distribution of such as bellshaped uniform or skewed 4 Sample values that lie from the vast majority of other sample values 5 Time Changing characteristics of the data over time Section 22 Freguengx Distributions Key Concept 7 When Working With large data sets it is o en helpful to organize and summarize data by constructing a table called a frequency distribution de ned later Because computer so ware and calculators can genemte frequency distributions the details of constructing them are not as important as What they tell us about data sets or Frequency Table data values either or by of intervals along With their corresponding or counts Example Frequency Distribution Ages of Best Actresses Original Data mu 21 Academy Awards Age of east Table 22 in ovder begmmng va Frequency Distribution mum Ages of Best Actresses 37 s 32 26 1 27 27 28 302529243a2529n 3015 A f 35 33 29 3x 54 24 2 45 M 2a 960 Au 3 29 27 3 33 29 25 35 an Actress Frequency 433531u2737414i3632 33 3174 33 5a 33 m u 4 25 so 42 29 33 35 45 49 9 4 2130 28 26 25 33 z 35 23 Wm 31 40 30 4 Av 62 51 AI 34 4 52 41 37 4150 12 53 3 143554 394957 a is 42 52 SI 35 m 39 u 4 5150 2 49 as 47 n 47 37 57 42 5 oz u 52 u 42 4a 49 55 as m an 5170 2 4o 42 35 75 39 5 45 as 62 4 51 31 42 so 52 37 as 32 is an 7130 2 a a 7 29 z Frequency Distributions Defmintions Class Limits are the numbers that can actually belong to different classes Class Limits are the numbers that can actually belong to different classes Class are the numbers used to separate classes but the gaps created by class limits Class can be found by adding the lower class limit to the upper class limit and dividing the sum by two Class is the difference between two consecutive lower class limits or two consecutive lower class boundaries Reasons for Constructing Frequency Distributions 1 data sets can be 2 We can gain some into the nature of data 3 We have a basis for constructing important graphs Constructing A Frequency Distribution 1 Decide on the number of classes should be between 5 and 20 N Calculate round up Class Width m Starting point Begin by choosing a limit of the class Using the lower limit of the first class and class width proceed to list the class limits List the lower class limits in a vertical column and proceed to enter the upper class limits Go through the data set putting a tally in the appropriate class for each data value 9939er Example Let s create a frequency distribution for number of hours worked last semester from the survey taken on the rst day of class Relative Frequency Distribution I c1 de the same class limits as afrequency distribution but frequencies are used instead of actual frequencies Relative Frequency Example Table 22 Table 23 Frequency Dismbution Relative Frequency Ages of Best Adresse Distribution of Best Actress Ages Age 0 Actress Frequency Age 039 Relative Anns Frequenty Z140 28 ztzu 37 4 3 3 4t 39 50 2 a su 16 5139 2 51450 3 70 1 61770 3 714 2 7Hm 3 Example Let s create a frequency distribution for number of hours worked last semester from the survey taken on the rst day of class Cumulative Frequen 39y Distribution a Table 2392 Cumulative Frequency Frequency Distributton Dismbuuon of Beg Ages of Best Actresses Actress Ages Age of Age of Cumulauve w Actress Frequency 21730 28 Less than 31 28 3 0 30 Less than 41 58 22 rrr New coo NNN EE I 3 2 is 71 80 2 Less than 81 4 ox Critical Thinking Interpreting Frequency Distributions In later chapters there 39 be frequent reference to data With a distribution One key characteristic of anorrnal distribution is that it has a shape u The frequencies start low then increase to some maximum frequency then decrease to a low frequency v The distribution should be approximately symmetric Section 23 Histograms Key Concept Ahistogram is an important type of graph that portrays the distribution Histogram A bar graph in which the represents the of data values and the represents the Relative Frequency Histogram Has the same shape and horizontal scale as a histogram but the vertical scale is marked with frequencies instead of frequencies or shape of the 30 Table 22 Frequency Distribution Ages of Best Actresses 20 Age of 3 Actress Frequency 5 tr 21 30 23 Em 31 40 30 4150 12 51 60 2 61 70 2 0 a a a a a s a 71 30 2 9 5 5 n9 3 Ages of 1325 Actresses Table 23 LIDM Relative Frequency Distribution of Best gt 507 ActressAges g 3 Age of Relative El Aclress Frequency L 20 gt 21 30 37 E 3140 39 9 W 41750 16 5160 3 07 61 70 3 s a s a s s a 71 80 3 9 1 939 7 C9 3 9 Ages of 135 Actresses Example Let s create a Histogram and a Relative Frequency Histogram for number of hours worked last semester from the survey taken on the first day of class Critical Thinking Interpreting Histograms One key characteristic of a normal distribution is that it has a bell shape The histogram below illustrates this Sample Value Section 24 Statistical Graphics Key Concept This section presents other graphs beyond histograms commonly used in statistical analysis The main objective is to revealing some important characteristic a data set by using a suitable graph that is effective in Frequency Polygon Uses line segments connected to points directly above class midpoint values Ogive A line graph that depicts cumulative frequencies 80 30 gt U I 60 I 70 of the k 20 E Vauzzs are Li I less than D e e 0 I 505 LL 3 39 IO 4 I g I E 20 I 3 I K I Z55 355 455 555 655 755 A f Best Actresses 0 985 c 205 505 405 505 05 705 805 Ages of Best Actresses Dot Plot Stemplot or StemandLeaf Plot Consists of a graph in which each data value is plotted as a point or dot along a scale of values Dotplot 01 Ages of Actresses Represents data by separating each value into two parts the stem such as the leftmost digit and the leaf such as the rightmost digit i Slcm Icns Leaves unils i 2 12445555666677778888999999 24 32 4 4e 55 5 72 m 3 0011122333334445555555677888899 Actresses 4 011111223569 5 739nu quot 1w quotaiil l 6 o 1 3 7 4 8 o at I I Pareto Chart Pie Chart A bar graph for qualitative data with the bars A graph depicting qualitative data as slices of a pie arranged in order according to frequencies It Am quot r es 1 1 tntemanana Callmg I 6W 7W I Operator SanIces Marke g t 55quot a I 1007 g Crammmg y IZH I g Slammmg Imus wquot lt6 e 6 a w a be e h I A sea A a a y 0a N x o o u at 0 1 2 l c e olt 0 1 WW 00 0Q Rates and SenIces 4473 MATH 2600 Probability and Statistics Chapter 1 Section 11 Overview A common goal of studies and surveys and other data collecting tools is to collect data from a of a group so we can 7 7 something about the 77 7 group In this section we will look at some of the ways to describe data Observations such as measurements genders survey responses that have been collected A collection of methods for planning studies and experiments obtaining data and then organizing 77 7 presenting 7 7 77 interpreting and drawing based on the data 7 7The complete collection of 7 7 elements scores people measurements and so on to be studied the collection is complete in the sense that it includes 7 subjects to be studied Collection of data from 77 7 member of a population Subcollection of members selected from a population Fast Food Is King Arlington TX 7 Fast food restaurants now dominate the quoteating outquot market in the DallasFort Worth MetropleX according to a new study by Decision Analyst Inc a leading national marketing research rm In its survey of 1000 consumers who live in the DallasFort Worth MetropleX conducted during December 1997 and January 1998 595 of all restaurant visits were accounted for by fast food restaurants Sam ple Population Chapter Key Concepts 39239 Sample data must be collected in an 7 such as through a process of selection 0 v If sample data are not collected in an appropriate way the data may be so completely that no amount of statistical torturing can salvage them Serum T 2 nszm Key Cnnczp The subject uf statistics rs largely abuutusmg name make ur that fulluw EXamp Es 1 Parameter 7 out onhe so smes in the u s has a population under1 million Therefore 14 onhe 39 39 n The percentage 39 39 39 based on the emire popumion of all so smes 2 Statistic Deena Inns 5 mu 5 avevagemvgaspvmeswppedmm vseeeurheheemve V931 W a new sAvavsaxd sheer Yheavemge pnue mseneewe assume emppee 22 Demsxmhe pas m v eekS Ed m thundbEVQr puhheher Mme Luneherg suvvev YheavemgeJa hed an Fnday v asthe Wes swaths 173 avevage anMavm 2 2mm Lundhevg Ed Lundherg summedthepvmevedu mmm sump h wade Pncesand demand Pnues mmapmmenmgmrew emrmmccnss nu mmmuesm a ed dmand Maude rheeueme mgh avevage mm M r set an JuWM mmmgm Lundhe g YheLundhevg Survev shased an remanseswam mammary spun seva 31me namnwde dam e numbers represenung taunts urmeasuremmts Example The ufpmfessunal human players msungmshea by sums nunnummc characteristic Ex p The maldfemaleufpmfessmnal athletes am reemwhenunenunber arpasebxemues reenhfrarn rnnnypasebxe valuesthat rs anther a number ax a caunmble39 canespandta same eannnuaue scale um savers a rang m s afwlues gape mm ne hr amle number re Lhznumbexufpasa evalu 15 Lnua n12z Jumps Example The ufeggthatahznlays Example The mauntafmxlk matacawpzaduceseg 2 343115 gallanspex day Levels ofMeasurement level of measurement characterized by data that consist of names labels or categories only and the data 7 be arranged in an ordering scheme such as low to high Example Survey responses 7 77 undecided 2 level of measurement involves data that can be arranged in some order but differences between data values either be 39 39 39 or are Example Course grades 77 77 77 D or F 3 7 7 level ofmeasurement like the ordinal level with the additional property that the difference between any two data values is 7 7 however there is no 7 7 zero starting point where none of the quantity is present Example Temperatures 20 385 45 98 4 77 7 level of measurement the interval level with the additional property that there is also a natural zero starting point where zero indicates that 777 of the quantity is present for values at this level differences and ratios are meaning Example Prices of college textbooks 77 77777 Section 1 3 Critical Thinking Key Concepts 39239 Success in the introductory statistics course typically requires more 7 77777 than mathematical expertise 39239 This section is designed to illustrate how common sense is used when we think critically about data and statistics Misuses of Statistics Example or selfselected sample one in which the respondents themselves decide whether to be included In this case valid conclusions can be made only about the speci c group of people who agree to participate Note the following poll on a news webpage that makes it clear that this is not a good use of statistics Quick Vote Is Leon Panetta a good choice to head the CIA Yes 321er No 565512 Toial Votes 325m This is not a scienti c poll 2 Conclusions should not be based on samples that are far too small E 39 The Children s Defense Fund published Children Out of School in America in which it was reported that among secondary school students suspended in one region 67 Were suspended at least 3 times The media reported this but failed to mention that the sample size Was 3 students In reality in that Whole region 3 students Were suspended and 2 of them Were suspended multiple times 3 To correctly mm mm a quot7 mm interpret a graph you must analyze the numerical information given in the graph so as not to be misled by the gmph s shape Q 2 sumo sim 1 5Wquot w my rm W cm 2 7 mmm cap 200 Daily on Consumption millions a barrelsl 39 4 Part b is E g designed to exaggerate the difference 3 j 70 by increasing each dimension in S E 5 proportion to the actual amounts of oil EE 5 consumption 0 USA Japan USA Japan a b 5 Misleading or unclear percentages are sometimes used For example if you take 100 ofa quantity you take it all 110 ofan effort does not make sense Example In referring to lost baggage Continental Airlines ran ads claiming that this Was an area Where We ve already improved 100 in the last six months This Would mean no baggage is noW being lost which is not true 6 The following 2 questions Were asked in a survey The subjects Were asked if they agree With the statement 1 Too little money is being spent on Wel e sai they agree 2 Too little money is being spent on assistance to the poor said they agree Read in Text about these Other Misuses of Statistics er of Questions 8 als 9 Correlation amp Causality 10 Self Interest Study 11 Precise Numbers 12 Partial Pictures 13 Deliberate Distortions Section 1 4 Design of Experiments Key Concept If sample data are not collected in an appropriate way the data may be so completely useless that no amount of statistical tutoring can salvage them study observing and measuring speci c characteristics without attempting to modify the subjects being studied 0 v Cross sectional study data are observed measured and collected at 7 7 point in time 39239 Retrospective or case control study data are collected from the 777 by going back in time 39239 Prospective or longitudinal or cohort study data are collected in the future from groups called cohorts sharing common factors apply some treatment and then observe its effects on the subjects subjects in experiments are called experimental units 0 v Confounding occurs in an experiment when the experimenter is not able to distinguish between the effects of different factors 39239 Blinding subject does not know he or she is receiving a treatment or placebo Read Example from text 777777 39239 Blocks groups of subjects with similar characteristics Read Example from text 7 7 o v Completely Randomized Experimental Design subjects are put into blocks through a process of random selection Read Example from text 777777 39239 Rigorously Controlled Design subjects are very carefully chosen 39239 Replication repetition of an experiment when there are enough subjects to recognize the differences from different treatments or to verify previous results v Sample Size use a sample size that is large enough to see the true nature of any effects and obtain that sample using an appropriate method such as one based on randomness Random Sample Gives the best most useful results and is preferred over any other method Other methods of sampling that are used but are inferior to a random sample 1 Systematic Sampling Select some starting point and then select every k th element in the population Would this be a random sample 2 Convenience Sampling Use results that are easy to get 3 Stratified Sampling subdivide the population into at least two different subgroups that share the same characteristics then draw a sample from each subgroup or stratum 4 Cluster Sampling divide the population into sections or clusters randomly select some of those clusters choose all members from selected clusters MATH 2600 Probability and Statistics Chapter 4 notes Section 42 Fundamentals Key Concept This section introduces the basic concept of the Different methods for nding probability values will be presented The most important objective of this section is to learn how to probability values Event 7 Simple Event 7 Sample space 7 31 32 33 34 35 35 515253 5455 55 Notation denotes a probability denote speci c events denotes the probability of event A occurring POSSible Values for Probabilities v 7 cmquot Probability Limits 139 probability of an impossible event is 7 my My dog Woof Woofwill start to y V The probability of an event that is certain to occur is Someone in this class is not a fan ofmatli 7 SD 50 Chnnze For any event A the probability of A is between 0 and l inclusive That is 0 S PA S l 7 myde 0 7 impossvbk Basic Rules for Computing Probability Rule 1 Conduct or observe a procedure and count the number of times eventA actually occurs Based on these actual results PA is estimated as follows PA Rule 2 Requires b Assume that a given procedure has 7 different simple events and that each of those simple events has an 7 7 chance of occurring If eventA can occur in s of these n ways then PA Rule 3 PA the probability of eventA is 7 fiby using 7 7 of the relevant Example 1 for Rule 1 Can you wiggle your ears a Since one can wiggle their ears or not is the probability that one can wiggle their ears equal to 05 b Actually determine if you can you wiggle your ears Then determine what the actual of students in class than can wiggle their ears Example 2 for Rule 1 Do couples get engaged or not If they are engaged how long did they date before becoming engaged A poll of 1000 couples conducted by Bruskin and Goldring Research gave the following information Length of Dating Number of Time Before Couples Engagement Never engaged 200 Less than 1 year 240 1 to 2 years 210 More than 2 350 Example 3 for Rule 1 A person can have one of four blood types A B AB or 0 If people are randomly selected is the probability they have blood type A equal to 1 Why Example 1 for Rule 2 Let event A rolling a 4 on a fair die PA Let event A rolling an even number on a fair die PA Let event A rolling at least a 2 on a fair die PA 7777 Let event A rolling a sum of 4 on a pair of fair dice PA Example 2 for Rule 2 Let event A tossing a H when ipping a coin PA iiiiiii Let event A tossing 2 heads HH when ipping 2 coins or getting 2 heads in a row when tossing 1 coin twice PA Let event A tossing 3 coins and getting 1 head and 2 tails PA77 Example 3 for Rule 2 Bob is asked to construct a probability model for couples with three children He determines that couples with three children can have all boys all girls or a mix of girls and boys Because Bob considers three outcomes he reasons the probability of each outcome is 13 What is wrong with Bob s reasoning Find the real probabilities of the three outcomes Example 4 for Rule 2 Two balls are drawn out of a bag without replacement containing two red balls and four green balls a List all simple events b Determine the probability of drawing a red and a green ball Law of Large Numbers As a procedure is repeated again and again the relative frequency probability from Rule 1 of an event tends to approach the actual probability Give example De nition The consists of all outcomes in which the eventA Rounding OffProbabilities When expressing the value of a probability either give the i 7 or decimal or round off nal decimal results to 39 T digits n 39 When the probability is not a simple fraction such as 23 or 59 eXpress it as a decimal so that the number can be better understood Section 43 Addition Rule Key Concept The main objective of this section is to present the as a device for finding probabilities that can be expressed as the probability that either eventA occurs 7 eventB occurs as the single outcome of the procedure Compound Event any event combining 2 or more simple events Notation for the addition rule P in a single trial eventA occurs event B occurs they both occur General Rule for a Compound Event When finding the probability that eventA occurs or event B occurs find the total number of waysA can occur and the number of ways B can occur but Formal Addition Rule where PA and B denotes the probability thatA andB both occur at the same time as an outcome in a trial or procedure Intuitive Addition RuleTo nd PA or B nd the sum of the number of ways eventA can occur and the number of ways event B can occur adding in such a way that every outcome is counted only once PA or B is equal to that sum divided by the total number of outcomes in the sample space event PA PCB 7 Find PA U B intuitively Addition rule PA or B PA U B GO 0000 IIIDDOOOO Let s de ne some events A randomly selecting an object that is square PA B randomly selecting an object that is solid black PB C randomly selecting an object that is an open circle PC Find PA U B by the Addition rule Events A and B are disjoint or time That is disjoint events do not overlap if they cannot occur at the same Total Area 7 Total Area l PM PB PM PB Fifi and B Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events Complementary Events PA and PA are disjoint It is impossible for an event and its to occur at the time Find PA U B GO 0000 IIIDDOOOO A randomly selecting an object that is square B randomly selecting an object that is solid black PB Find PA U C CO 0000 IIIDDOOOO Since A 0 C has no samBle Boints we could modify the Addition rule PA C randomly selecting an object that is an open circle PC A randomly selecting an object that is square B randomly selecting an object that is solid black C randomly selecting an object that is an open circle Are events A and B mutually exclusive Are events A and C mutually exclusive Summary of the Addition Rule Additive Rule for events that are n0t mutually exclusive Additive Rule for events that m mutually exclusive Section 4 4 quot39 quot quot Rule Basics Key Concept If the outcome of the rst eventA somehow affects the probability of the second event B it is important to adjust the probability of B to re ect the occurrence of eventA The rule for nding PA B is called the 7 Notation PA and B Pevent 7 occurs in a trial and event occurs in a 7 i trial Tree Diagrams A tree diagram is a picture of the possible outcomes of a procedure shown as line segments emanating from one startng point These diagrams are helpful if the number of possibilities is not too large Example 1 1 True or False The probability that a randomly picked student is wearing a hat is 05 2 The probability of getting a sum of 3 when rolling 2 dice is a 1 12 b 136 c 16 1 236 e Huh Reviewlast section If you randomly guess at both questions what is the probability that you get the 1st correct or the second correct This section If you randomly guess at both questions what is the probability that you get the 1st correct and the second correct Notice that Pthat you get the 1st correct AND the second Key Point Conditional Probability The probability for the second event B should take into account the fact that the rst eventA has already occurred Notation for Conditional Probability represents the probability of event B occurring after it is assumed that eventA has already occurred read B LA as B A Example 2 Two balls are drawn out of a bag without replacement containing two red balls and four green balls a List all the simple events b Determine the probability of drawing a 1 red and a green ball 2 2 red balls 3 2 green balls Independent Events TWo events A and B are Seveml events are similarly independent if the occurrence of any does not affect the probabilities of occurrence of the others IfA and are not independent they are said to be Formal Multiplication Rule PA and B Note that if A and B are events is really the same as Intuitive Multiplication Rule Applying the Mumplica lm RUIB ding the probability that eventA occurs in one trial and event B occurs in the next trial multiply the probability of eventA by the probability of event B but be sure that the probability of event B takes into account the previous occurrence of eventA dB PM an Multipllcation my 39an A and 1 ye ltndependenv PM and 13 pm pm I we l and llPt l ll Are the events Mutually Exclusive Are the events Independent Show why 2 Find the probability of selecting one passenger who was a Man and a boy Use the two evens de ned above Mutually exclusive Independent Show why