Week3-2up - mean standard deviation value(p 65 ment of how...

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