Week3 - STA 2023 c B.Presnell & D.Wackerly -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STA 2023 c B.Presnell & D.Wackerly - Lecture 4 42 Thought for the day: Friends may come and go, but enemies accumulate. Assignments Problems : 2.12, 2.14, 2.16, 2.20, 2.24, 2.89, 2.90, 2.92, 2.97–99, 2.102, 2.105, 2.116, 3.3, 3.5–3.8, 3.14, 3.16–3.18, 3.20, PROJECT 1 For Wednesday : Read pages 111–124 Last Time : £ score : value mean standard deviation (p. 65) £ ¤¢ ¡ ¡ Tchebysheff’s Theorem (always works) (p. 56) ¡ Empirical Rule (bell-shaped distns) (p. 57) ¡ Percentiles (p. 64) ¡ Quartiles (p. 69) ¡ Rare Events and Inference - requires the assessment of how likely or probable an occurrence is. STA 2023 c B.Presnell & D.Wackerly - Lecture 4 43 Stem-and-Leaf Displays – P. 28 Given are widths (inches) of the dominant hands (domhwdt) of 37 honors STA 2023 students. From data set you will soon use in the first project. Character Stem-and-Leaf Display Stem-and-leaf of domhwdt Leaf Unit = 0.010 1 1 2 4 5 11 12 (10) 15 13 12 2 2 25 26 27 28 29 30 31 32 33 34 35 36 37 N = 37 0 5 01 0 000000 2 5555555555 07 9 0000000000 55 Descriptive Statistics Variable domhwdt Variable domhwdt N 37 Minimum 2.5000 Mean 3.2446 Median 3.2500 Maximum 3.7500 TrMean 3.2515 Q1 3.0000 StDev 0.2849 Q3 3.5000 SE Mean 0.0468 STA 2023 c B.Presnell & D.Wackerly - Lecture 4 44 ¡ Leaf unit = 0.010 (leafs are in last columns in display) ¡ Stem, middle columns (25, 26, etc.). In the 25, 5 is the “tenths” digit and 2 is the “units” digit. So, entry with stem 28 and leaf 1 is 2.81. ¡ From the top of the first row down to the entry in (), entries in first column are total number of data values last entry in that row. EX. 11 in the row with stem 30 indicates that there are 11 values 3.00. ¡ The stem beginning with the number in () is the stem containing the median. Number in () is the number of leaves on that stem. The rows below that beginning with the number in () are the number of values in that row, on down. Ex. 13 in front of stem 34 indicates that there are 13 ¡ values 3.40. ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 45 ¡ Median is the 19th value from the lowest. ¡ 12 values are in the stems above that labeled (10), seven more in that stem would be the 19th (middle) value. ¡ median is 3.25. ¡ Rotate picture 90 degrees, get something much like a frequency histogram. Locate the point on the stem and leaf diagram corresponding to handwidth 3.37”. ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 Box-Plots – Section 2.8 Box-plots of proportions of undervotes in 2000 Presidential election in various counties in Florida. “O” indicates optical scanner, “P” indicates punch cards. ¡ Top horizontal line (upper hinge) in each box is upper quartile (75th percentile) (approx. 0.02 in P) ¡ Bottom horiz. line (lower hinge) is lower quartile (25th percentile) (approx. 0.01 in P) Middle horiz. line is median (approx. 0.013 in P) 46 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 ¡ Interquartile Range: IQR = upper quartile - lower quartile (measure of variablity) ¡ Top vertical line (whisker) extends up to the largest value that is not more than (1.5)IQR above the upper quartile ¡ Bottom vertical line (whisker) extends down to the smallest value that is not more than (1.5)IQR below the lower quartile ¡ Values beyond the end of the whiskers are marked with a and are “unusually” large (or small) – Outliers. ¡ Why outliers? (mistake recording, measurement from diff. popn., rare event). ¡ 19 counties used punch cards, 31 used optical scanners. Essentially the whole boxplot for P is above that for O. Conclusion? What would be the shape of the histogram for the counties using optical scanners? 47 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 Scatter Diagram: Section 2.9 Plot one variable on horiz axis, another on vert. axis Plotted the number of votes for Buchanan in the ‘00 election versus the number for Perot in the ‘96 election (both candidates from the same “minority party”) Guess which county corresponds to the “unusual” point in the graph. 48 ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 49 Chapter 3: Probability The study of randomness Experiment : the process by which an observation is made. (Def. 3.1, p. 100) EXAMPLES : ¡ Weigh something ¡ Obtain someone’s I.Q. ¡ Toss two coins : a penny and a dime. Intuitively – the PROBABILITY of an outcome is (p. 102) ¡ the long run relative frequency of occurrence ¡ how likely the outcome is. EXAMPLE : What is the probability that both coins turn up “heads”? STA 2023 c B.Presnell & D.Wackerly - Lecture 4 50 A Sample Point is the most basic outcome of an experiment . An event that can occur only one way. (Def. 3.2, p. 101) EXAMPLE : Toss two coins : one penny, one dime ¡ One simple event : ¡ (Head on penny, Tail on dime) ,is the each single SAMPLE POINT often denoted ¥ £¥ ©¨ §¦¤ £ ¡ a subscript, etc.) £ collection of ALL possible sample points. ¢ Def. (Def. 3.3, p. 101) The Sample Space, EXAMPLE : Toss two coins: Simple events: HH HT TH TT or TH : TT £¥ £¥ £¥  § ¨ §¦¤ £ ¨£ HT : £ ¤£ HH : £ : (with ¥ ¥ ¥  ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 51 To each sample point, , we assign a number, the so that: (shaded box, p. 104) ¢ £ 1. the probability of each sample point is between for each £ ¢ ¤ £ ¢ ¡ £ ¡ and . £ £ ¡ , £ probability of ¤ 2. the sum of the probabilities of all of the sample  £ ¢ ¦¥ ¤ all simple events ¤ ¡ points is . EXAMPLE : Coin toss. ¤ ©  ¢ £ ¡  ¢ ¨ £ ¡   ¢ ¨¨ £ ¡  ¢ §¤ £ ¡ Def. (p. 105) An event, , is a collection of sample points. is the sum of the probabilities of the sample points in   (p. 106) The probability of an event . STA 2023 c B.Presnell & D.Wackerly - Lecture 4 52 NOTES : , we must know which sample points ¢  . for any event,  ¤ ¢  .  ¡  ¤ ¡ are in ¡ To find ¢ ¡ £ ¡ ¢ ¡ EXAMPLE : Toss two coins get a match (both coins show the same face). ¡¡ ¡   ¤ ¤  £ ¥ © ¤ ¡ £ TH £ ¢  ¡  : TT ¤ ¤  ¨£ ¥ ¥ ¤  ¥ ©  ¤ © ¢ ¤ © ¢ ¤ ¤ ©  ¡  ¤£ HT : ¢ HH : ¡ : ¡   © ¢  get at least one head ¢ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 53 EXAMPLE : (Similar to several HW problems) Breakdown of the workers in a state according to : ¡ Political Affiliation ¡ Job Type Job Type Political Affiliation Rep. Dem. Ind. White collar 12% 12% 6% Blue collar 23% 43% 4% Select one worker at random from this group. To describe this person, we need to specify: ¡ Political Affiliation ¡ Job Type Republican, White Collar worker.  STA 2023 c B.Presnell & D.Wackerly - Lecture 4 54 Sample space consists of all political affiliation, job type combinations: ¡ ¡ ¤ ¤ ¥£  ©  ¡ £¡ ¢ £  ¡ £¡ ¤ ¢ ¤ ¢¡ £ ¥  © ¤ ¢ ¢¡ ¤    ¤ ¢ ¢  ¤ ¢ ¥ £ ¤   ¢ ¡ ¡ ¢ ¡ ¡ £¡ ¢ ¢¡ ¢ ¥ ¥  ¤ ¥£  ¢ ¥ ¢  ¡  ¢ £ ¤  £   ¢ ¡ £ person is a white collar worker ¢ ¤   ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 55 ¡ person is a Republican ¤  ¡ ¤ ¥ ¡ ¤ ¢ ¢ ¢ ¡ ¥  ¥  £ ¡   ¤ ¢ ¡  £ ¢ ¤  LETS MAKE A DEAL!! Three Doors ¡ One good prize. ¡ Two “duds” Select a door “at random”. Win the good prize. ? ¡ ¢  ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 56 Simple Events ¡  Good door ¤ Dud #1. ¨ Dud #2. ¡ ¡ Sample space : ¨ ¤ ¥  ¢ ¨ ¤ ¢¡  ¢  ¢¡ ¤ ¢  ¢ ¡ After you choose a door, Monte Hall shows you a “dud”. (He can always do this!!) If you “STAY” with your initial choice, ¤   ¢ ¡  ¢ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 4 57 If you “SWITCH” initially and switch – you lose! initially ¤ – If you selected ¨ Monte will show you – If you selected If you switch – you win! initially ¤ Monte will show you ¨ – If you selected If you switch – you win! – If you “switch” £ ¡ ¥ ¡  ¢ ¨ ¥¤ ¡  ¢ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 58 Thought for the day: Drive-in banks were established so most of the cars today could see their real owners. Assignments For tomorrow: Exercises : 3.21–23, 3.27, 3.32, 3.33, 3.35, 3.37–3.39, 3.41, 3.43, 3.46 For Monday : Read pages 127–135 LAST TIME : ¡ Experiment (p. 100), ¡ Probability (p. 102), ¡ Sample Point (p. 101), ¡ Sample Space, ¡ Properties of Probabilities of Sample Points (p. 96), ¡ Events (p. 105), ¡ Find the probability of an event (p. 106) , (p. 101), ¢ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 are two events in ¤ and , ¢ Def. If 59  ¡ , is all points in or or both ¤  ¤ their UNION,  (Def. 3.5, p. 111). ¢ ¤  ¤  Entire shaded area is In the Job Type–Political affiliation example ¡ ¡ ¥ ¡ ¥  white collar worker .   ¡ ¡ ¤ Republican ¥  ¡ :  ¥ ¥  ¡ ¤ ¥ £ ¥ £ ¤  ¤ ¥  £   ¤   ¢ ¤ ¥£  ¢ ¤ ¢ ¤  ¢  ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 60 ¡ their INTERSECTION, ¤  and simultaneously ¤ sample points in both , is the set of all  (Def. 3.6, p. 111). ¢ ¤  ¤  In the Job Type–Political affiliation example ¥ ¡ ¥    ¡ ¡ ¤ ¥   ¤  ¡ ¤  £ . : ¡ Republican ¡ white collar worker . ¤  ¢ ¡ ¤    ¤ ¢  ¡  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 have no sample  contains no points), then ¤  are mutually exclusive (or disjoint ) events. ¤ and  points in common, ( and ¤ Def. (Def. 3.8, p. 118) If 61 See figure 3.11, p. 118. ¢ ¤  EXAMPLE : Toss two dice sum of up faces is 11 ¤  up face on first die is 3  are mutually exclusive, then (p. 118) ¢ ¡ ¢ ¢  ¡  ¤ ¢  ¡ and ¤  Note. If are mutually exclusive. ¤ and ¤ Then £ ¤  ¢  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 62 ,  Def. (Def. 3.7, p.115) The complement of an event , is the set of all sample points not in .   denoted ¢   ¡  ¡  ¡ Thus, ¡ Since are mutually exclusive  and ¢ . ¢ ¢ ¡ ¡  ¢  (p. 115) ¤  ¢  ¡  ¢ ¢ (why?), we have  ¤ ¡   ¡ ¡ ¢ ¢ ¢ ¢  so ¢  ¡ £ ¤  ¢  ¡ and ¢  ¡ £ ¤  ¢  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 63 Ex. Odds are 2:1 that when Henry and Thomas play racquetball, Henry will win. H and T play 3 matches, and the winner of each match recorded. There are eight sample points: ¤£ ¨£ 4/27 £ 4/27 £ 2/27 £ 4/27 HTH HTT ? £ THT THH ¡ £ TTH ¢ £ £ ¢ ¡ £ ¡ a. Find £ 8/27 HHT TTT ¢¥ HHH ¡ Sample Point 2/27 1/27 . ¡ H wins at least two matches ¡  H wins an odd # of matches . Find ¤ . ¢ ¤ ¡  ¢  ¡ c. Find and  b. Let . STA 2023 c B.Presnell & D.Wackerly - Lecture 5 64 ,  ¤ a. Since the sum of the probs. of all sample points ¡ £ £ ¡ ¢ £ £  £ £  ¤ ¢ £ ¡ ¡ b. Let ¡ H wins at least 2 matches   ¡ HHH, HHT, HTH, THH £ ¢ ¡ ¢ ¨ ¢ ¡ ¢ £  ¡ £ ¢ ¢ © ¨£ ¡ ¢ §¤ ¢ © © £ ¡ ¡ £ ¡ £ ¢ £ ¡ ¢ ¢ £  ¡¢ ¤ £ £  ¢ £ ¢  ¡  £ c. ¡ H wins an odd number of matches ¤  ¡ HHH, HTT, THT, TTH £ ¡ £ £ © ¤ ¡ £ £ ¢ £ ¡ ¢ £ ¢ £ ¡ ¢ ¢ £ ¡£ ¤  £ ¤  ¢ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 65 ¡ H wins at least 1 match  HHH, HHT, HTH, HTT,  ¡ THH, THT, TTH © £ ¡ ¢ ¡ £ ¢ £ £ ¡ ¢ ¢ £ ¡ £ ¢ £ £ ¡ ¢ ¡£ ¤ © ¢ £ ¢ £ © ¡ £  ¢ £ ¡ ¢ £ ¡ £ £ £  ¤ Last one much easier using idea ofcomplement : ¡ TTT ¡ ¢ £ ¤ £ H wins no matches  ¤  ¡¢ £ £ ¤  ¢ ¡  ” ¡ £ “not  ¡  ¤ ¢ ¡ H wins at least 2 and H wins an odd # ¤   ¡ HHH (NOT mut. excl.) ¡ £ ¤  £  ¤ ¢  ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 66 EXAMPLE : Five equally qualified job applicants, 2 Women and 3 Men. Randomly select 2 to hire. ¡ ¡ hire at least 1 woman  ¡ ¡ hire two of the same sex ¢ ¤ ¥ ¢ ¡ mutually exclusive? and  ¤ ¢  and ¨ ¥ ¡ ¨  ¡ ¡ ¤ ¥¤ Men : Women : ¡ and ¤ Are ¤ Find . STA 2023 c B.Presnell & D.Wackerly - Lecture 5 67 Additive Rule of Prob. (p. 117) ¡ ¤ ¢  £ ¢ ¤ ¢ ¡ ¢ ¢  ¡ ¡ ¤  ¢  ¤  ¤  are mutually exclusive £  ¢  ¡ ¢ ¤ ¡ ¢ ¢  ¡  ¤  ¢  ¡ OLD STUFF!! since ¤ ¤ and  If ¤ Entire shaded area is © ¢ ¥    ¢ ¤ £ ¢ ¤ ¢ ¤  ¢  ¤ ¥  ¤  £  ¥   £ ¤  ¡  £ ¢ ¥ ¡  ¥   ¢ ¤ ¥  £ £ ¡ £ ¢ ¤ £ ¢ ¡ ¤ ¥ ¥   £ ¢ ¡ ¡  ¡ ¤ ¡   ¡ ¡  £ ¢ ¡ £¡ ¤  ¤ ¥£  ¢ © ¤ ¥ ¡ £¡   ¢ ¢¡ ¤  ¢   ¢¡ £ ¢ ¤ ¥ £  ¤ ¡ ¡ ¡ ¥  Republican white collar worker ¡ ¢ ¡ £ EXAMPLE : Political Affiliation/Job Type STA 2023 c B.Presnell & D.Wackerly - Lecture 5 68 STA 2023 c B.Presnell & D.Wackerly - Lecture 5 69 Conditional Probability be two events, where  . The conditional probability of ¡ ¤ ¢ ¤ ¢ ¡  ¡ ¤ £ ¤ ¢ is  ¢  ¡ ¡ ¤ ¢ ¤  is also in  ¤ (What proportion of ?)  and ¤ Def. (p. 122) Let given STA 2023 c B.Presnell & D.Wackerly - Lecture 5 70 EXAMPLE : Toss a balanced die. ¡ ¤ . © ¥ ¤   ¤ ¥ ¥  ¡ ¡ ¥   ¤  ¡ ¤ ¢ ¡ ¤ ¢ ¡   ¡  ¡ ¢  ¡ ¡ ¡     ¡ ¥ ¤ £  © ¥ ¨ ¡ ¤  ¢  ¨  £ ¥ ¡ ¤ ¡ ¤  ¡ ¡   © ¡ . Thus, ¡ ¥ ¡ £ ¢¥ ¡ ¢ £ get a number greater than 3 ¡  ¥ ¥ get an even number ¡ ¥ ¤  © Balanced ¢ ¤ ¢ ¡  ¥ ¤  ¡ ¤  ¡ ¢ ¡ ¤   ¡ ¤  ¢  ¡ ¡ ¡ STA 2023 c B.Presnell & D.Wackerly - Lecture 5 71 EXAMPLE : Pol. party – job type. ¡ ¡ person is a white collar worker .   ¡ ¡ person is a Republican . ¤  If I know that a person is a Republican, what is the ¡ probability that (s)he is a white collar worker? ¢  RW DW IW RB DB IB ¤ ©  ¢ ¡ £¡ ¤ £  ¢ ¢¡ £  ¥ © ¤ ¢ ¢¡ ¤   ¢   ¤ ¡ ¥   ¢ ¢ ¤ ¡ ¡ £ ¤   ¡ ¥ ¢ ¥ ¤ ©  ¤ £ ¥ ¡ £  ¢ ¡ ¤  ¢  ¡ probability that (s)he is a blue collar worker? ¤ ¥£  ¡ £¡ If a person is NOT a Democrat, what is the ¡ ...
View Full Document

Ask a homework question - tutors are online