:;%24'-< =/-24'#)2-/(*"%(-7Draw a single individual from a certain population!"#$%&'((() !"#$%&"'() *)+),* *+,-'(.'/ -./% 0#1./% 23 $+$45671.451 89'9 88: 89;< ='':>4519=?<;?(8::'=(;@P(“female”) = 9116 / 18174 = 0.5002P(“employed” and “female”) = 4313 / 18174 = 0.2367P(“employed” | “female”) = 4313 / 9116 = 0.4731P(“employed”) = 8240 / 18174 = 0.4534>4519=?<;?(8::'=(;@A+5B.#C$+$456 @?8( =:: @=:@ '@'<8:;%24'-< !"#74-"&()*%/0*-035%(&#/Draw a single household from a certain populationA = household is “educated”B = household is “prosperous”Known profile of the population:P(A) = 0.254, P(B) = 0.134, P(Aand B) = 0.080P(A) = 0.254, P(B) = 0.134, P(Aand B) = 0.080Are A and B independent?P(B | A) = P(A and B) / P(A) = 0.080 / 0.254 = 0.315 ≠ 0.134 = P(B)Not independent.:;%24'-< C-0&5%'*75"--/&/9Tree diagram for breast cancer screening of women in twentiesCancerIncidenceCancerDiagnosisAB0.8“Sensitivity”False Womenin 20sAcBcBBc0.00040.9996S0.20.10.9“Specificity”False negativeFalse positive:;%24'-< C-0&5%'*75"--/&/9*D5#/(&/3-0 EGiven a cancer diagnosis, what is the probability of a cancer incidence?ABBcB0.00040.9996S0.80.20.1AcBc0.99960.9:;%24'-< F-8-5(&A-*4%"(7Tree diagram for parts from two different suppliersSupplierDefectiveABBc0.650.020.98PartAcBcBBc0.35S0.980.050.95ABBcB0.650.35S0.020.980.05:;%24'-< F-8-5(&A-*4%"(7*D5#/(&/3-0EGiven a defective part, what is the probability it came from Supplier A?AcBc0.350.95:;%24'-< .#'#"*$'&/0/-778% of white male population is colorblindSample n = 140 white males, p = P(“color blind”) = 0.08What are µXand σX?µ= n p = (140)(0.08) = 11.2µX= n p = (140)(0.08) = 11.2σX= √{n p (1 – p)} = √{(140)(0.08)(0.92)} = 3.21:;%24'-< .#'#"*$'&/0/-778% of white male population is colorblindSample n = 140 white males, p = P(“color blind”) = 0.08What is P(X ≤ 5)?=binomdist(5, 140, 0.08, 1) = 0.0284 =binomdist(5, 140, 0.08, 1) = 0.0284 Note: np = (140)(0.08) = 11.2 ≥ 10n(1 – p) = (140)(0.92) = 128.8 ≥ 10:;%24'-< 6/037("&%'*%55&0-/(7Manufacturing center averages 7 accidents per monthX = # accidents this month ! Poisson with µ = 7What is P(X =6)?=poisson(6, 7, 0) = 0.1490 What is P(X ≤ 6)?=poisson(6, 7, 1) = 0.4497:;%24'-< 6/037("&%'*%55&0-/(7*D5#/(&/3-0EManufacturing center averages 7 accidents per monthX = # accidents this year ! Poisson with µ = (12)(7) = 84What is P(X ≤ 75)?=poisson(75, 84, 1) = 0.1774=poisson(75, 84, 1) = 0.1774What is P(75 ≤ X ≤ 90)?P(X ≤ 90) – P(X ≤ 74) = = { =poisson(90, 84, 1) } – { =poisson(74, 84, 1) } = 0.7637 – 0.1495 = 0.61426/0-4-/0-/(*7327*#8*!#&77#/*"KAK7If X and Y are independent Poisson random variables, then Z = X + Y is a Poisson random variable with mean µZ= µX+ µY.Example: Counts of a car’s “paint sags”X = # sags on sq. yard of car’s roof, µX= 0.7Y = # sags on sq. yard of car’s side panel, µY= 1.4Z = X + Y = # sags in both areas combined, µZ= µX+ µY= 0.7 + 1.4 = 2.1General approach to inferenceProbability calculations help distinguish patterns seen in data between those that are due to chance and those that reflect a real feature of the phenomenon under studyExample:Weights of brown eggs is N(65, 5) gramsExample:Weights of brown eggs is N(65, 5) gramsSelect (by SRS) a dozen white eggs. SupposeHow do white eggs compare to brown eggs?Treat as known(more later)Note: The setup has population distribution N(μ, σ= 5), hence is N(μ, )Target of inferenceσ/√n = 5/√12Target of inference(mean white egg weight)HypothesesExample: Weights of brown eggs is N(65, 5) gramsHow do white eggs compare to brown eggs?Possibilities: Suspect that white eggs weigh less, on average H0: μ≥65 versusHa: μ< 650μ65esusaμ65 Suspect that white eggs weigh more, on averageH0:μ≤65versusH:μ>65H0: μ≤65 versusHa: μ> 65 Suspect different weights, on averageH:μ=65versusH:μ≠65H0: μ= 65 versusHa: μ≠65Test statistic The test statistic is typically an unbiased estimate of the parameter relevant to H0versus Hap0a for hypotheses about  The observed distance of the test statistic from H0indicates evidence against H0 The direction away from H0is determined by Ha.y0yaExample: How do white eggs compare to brown eggs?H0:  65 versus Ha:  < 65Evidence against H0when is smallHa: Do white eggs weigh less?P-values A P-value measures how surprising the patterns in the data would be if H0was true0 A smaller P-value indicates a more surprising patternCalculate as the probability of observing data“at leastCalculate as the probability of observing data at least as extreme” as was observed if H0was trueExample:How do white eggs compare to brown eggs?Example:How do white eggs compare to brown eggs?H0: 65 versus Ha:  < 65F12 ith5bFrom n = 12 eggs, with = 5 grams, observeSignificance level The significance level is the decisive level, , at which the P-value is small enough to “reject H0”gj0 The data are statistically significant when the P-value equals or falls belowvalue equals or falls below  Typical value is 0.05Example:How do white eggs compare to brown eggs?Example:How do white eggs compare to brown eggs?H0: 65 versus Ha:  < 65Pl02987005AtHP-value = 0.2987 0.05 Accept H0Conclude that white eggs do not weigh less, on averageExample: White egg weightsThe mean of a sample of n = 12 white eggs is N()gramsN(, ) grams.Observea = 3/1.44 = 2.08Consider a margin of error of grams State 96.23% confidence that  lies within the boundsExample: White egg weights (continued)The mean of a sample of n = 12 white eggs is N()gramsN(, ) grams.ObserveFor 95% confidence, deduce P(Z  -1.96) = 0.025 = (1 – C)/2,  P(-1.96 Z1.96) = 0.95 = C z* = 1.96.St t 95% fid th tli ithi th b dState 95% confidence thatlies within the boundsExample: Loan-to-deposit ratioThe st. dev. of the loan-to-deposit (LTDR) ratio among a certain population of banks is known to be = 12.3.pp In a sample of n = 110 such banks, the mean is acorresponding 95% CI has boundsm = 2.3acorresponding 95% CI has bounds3 In a separate sample of n = 25 such banks, the mean ism=48 a corresponding 95% CI has boundsm= 4.8Example: Loan-to-deposit ratio (continued)In planning a new sample of banks, what sample size is needed for a margin of error no more than m = 3, with g,95%