Lecture 4: Probability (continued)

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1 Lecture 4: Probability (cotiued) Desity Curves We ve defied probabilities for discrete variables (such as coi tossig). Probabilities for cotiuous or measuremet variables also are evaluated usig relative frequecies. Whe samplig (i.e. SRS), probabilities of evets are evaluated relative to a populatio histogram. Example: I maufactured 1 millio observatios like those i Example 3.27 page 93 of SW. That s certaily eough values to be cosidered a whole populatio. I the made a histogram with 20 classes of all 1 millio values, the a secod with 100 classes. The secod oe looks like a smooth curve, so we ll summarize a set of populatio values more succictly with a smooth curve (the third plot below) Blood glucose Blood glucose Blood glucose Now suppose i this populatio that 42% of all BG levels are i the iterval , ad 2% are greater tha 200. If we radomly select 1 idividual from the populatio, ad defie the radom variable = BG level for selected perso the Pr{ } =.42 ad Pr{ > 200} =.02 The vertical scale o the populatio histograms gets reset so that the area of a bar over a iterval is the populatio proportio i that iterval. With this rescalig, probabilities are obtaied by summig areas of bars. A coceptually similar process,, but oe which is easier to visualize, is to first approximate the rescaled histogram by a smooth curve, called the desity curve from which areas (same as probabilities) are evaluated. The third curve above the is called the desity. Iterpretatio of desity For ay two umbers a ad b, the area uder the curve betwee a ad b = populatio proportio of values betwee a ad b = probability of radomly selectig someoe with a value betwee a ad b Stat 538 Probability Lecture 2, 9/14/2004, page 1 of 16

2 Area = Pr{a b} a Blood glucose b 1 = Total Shaded Area Blood glucose.42 = Pr{ } for BG problem Blood glucose Example: SW example 3.30 p. 95 Stat 538 Probability Lecture 2, 9/14/2004, page 2 of 16

3 Biomial Distributio This is a special discrete probability distributio. We eed to uderstad it to uderstad statistical techiques such as logistic regressio. Idepedet Trials Model for Biomial Experimet A series of trials Each trial results i 1 of 2 outcomes, called Success ad Failure (geeric ames) The probability of success, called p, is the same for each trial The outcomes of the trials are assumed to be idepedet Example: Toss a coi 5 times Outcome for each trial (toss) H (call it success) T (callitfailure) Suppose coi is fair, so p = Pr{H} =.5 Example: SW example 3.42 p. 103 The biomial distributio specifies a formula for evaluatig the probability distributio associated with the umber of successes i a biomial experimet. I the previous problem, = 2 ad, if we defie = No. of Albio Childre, we ca directly evaluate this distributio, but it is much harder to do with trees if is large (too may braches). Defiitios For ay positive iteger x defie x! = 1i2i3 i i( x 1) x What is 2!? 3!? 5!? i. This is called x-factorial. Also defie 0! = 1 The biomial coefficiet is defied as C j have j). C j =! j!( j)! (ote: we will always Table 2 of SW p. 674 gives biomial coefficiets. Notice how fast they get large. The Biomial Distributio Formula For a biomial experimet, the probability of exactly j successes i trials (so -j failures) is j j Pr( j Successes} = C p (1 p), j = 0,1,2,..., Example: 3.45 p. 106 of SW Example: 3.47 p. 108 of SW j Stat 538 Probability Lecture 2, 9/14/2004, page 3 of 16

4 Mea ad Stadard Deviatio of the Biomial If = No. Successes i a biomial experimet with trials ad probability of success = p, the the mea (or expected value) is µ = p ad the stadard deviatio is σ = p(1 p). If we were to repeat the experimet a huge umber of times, each time recordig the umber of successes, we would have a huge collectio of itegers betwee 0 ad. The precedig formulas give the mea ad stadard deviatio we would calculate from that huge collectio. Example: 3.47 p.108 revisited. Here = 6 ad p =.85, so expected o. of Rh+ is 6(.85) = 5.10 ad the stadard deviatio is 6(.85)(.15) =.87. Note that the expected value is ot ecessarily somethig that is observed! The Normal Distributio SW Chapter 4, Sectios 1-4 The ormal or bell-shaped distributio plays a importat role as a approximatio to certai discrete ad cotiuous distributios, but is also useful as a approximate desity curve for certai populatios. Example: SW example 4.1 p Example: SW example 4.2 p Normal Curves The precedig examples illustrate two importat properties of ormal distributio or curves There are may distict ormal curves The ormal curve that approximates a populatio histogram is idexed by the mea ad stadard deviatio of the populatio, labeled µ ad σ, respectively. Note that ormal curves are cetered at µ ad the spread of the distributio is cotrolled by the size of σ (i.e. larger σ implies less probability ear µ). Smaller σ Two ormal curves with same std. dev.,differet meas Two ormal curves with same meas,differet std. dev. Stat 538 Probability Lecture 2, 9/14/2004, page 4 of 16

5 y µ ( ) The fuctioal form for a ormal desity curve is ( ) = σ f y e σ 2π We will see presetly how to evaluate probabilities for ormal distributios. The followig diagram does show some of the most importat areas (approximately). µ - 3σ µ - 2σ µ - σ µ µ + 2σ µ + σ µ + 3σ 68% 95% 99.7% Stadard Normal Distributio The stadard ormal curve correspods to a ormal distributio with mea 0 ad stadard deviatio 1. The letter Z (for Z-score) is used to idetify a stadard ormal variable. All ormal distributios ca be related to the stadard ormal through the so-called z-score trasformatio, which allows oe to calculate probabilities for arbitrary ormal distributios as areas uder the stadard ormal curve z Stat 538 Probability Lecture 2, 9/14/2004, page 5 of 16

6 Areas uder the stadard ormal curve SW Table 3 p , but more coveietly o the iside frot cover, gives lower-tailed areas uder the Z-curve of the form Area uder curve for all values z z I the table z rages from to 3. 49, although it is positive above. Example: If z = 0.47, the Area uder curve =.6808 This picture also tells you the area to the right of How? It also tells you the area to the right of How? z = 0.47 It also tells you the area to the left of How? We exploit symmetry extesively to use these tables, i.e. lower tailed areas ca be used to compute upper tailed areas. We also use the fact that the total area uder the curve is 1. Example: Area uder curve right of ca be computed two ways. It is the same as the area to the left of +1.47, or It also is the 1-area left of -1.47, or , which also is The first way probably is most direct; the secod ca be easiest sometimes z = Stat 538 Probability Lecture 2, 9/14/2004, page 6 of 16

7 We fid cetral areas by subtractig: = z = 1.23 z = -1 z = 1.23 z = -1 Desired area = =.7320 Commets 1. Always draw a picture 2. Recogize symmetry about 0 (area to left of 0 = area to right of zero =.5) 3. Areas are probabilities. If Z is a stadard ormal variable the we computed i the three examples (a) Pr{Z.47} =.6808 (c) Pr{-1 Z 1.23} =.7320 (b) Pr{Z -1.47} =.9292 Example: compute Pr{Z 2.12}. Draw a picture! Usig the stadard ormal table we ca show, for a Z, that the followig areas are true. These imply the more geeral result o page 5. We will retur to this shortly. Stat 538 Probability Lecture 2, 9/14/2004, page 7 of 16

8 Z 68% 95% 99.7% Areas for Geeral Normal Curves Suppose for a certai populatio of herrig, the legths of idividual fish follow a ormal distributio with mea 54 mm ad stadard deviatio 4.5 mm (populatio mea ad std. dev.). What percet of fish has legth less tha or equal to 60 mm? Let = legth of radomly selected fish. The the percet of iterest is just Pr{ 60}. The picture is Normal Curve µ = 54 ad σ = Z To get the area of iterest we use the Z-score trasform µ Z = to create a stadard ormal σ variable from. We the covert the limits of the area from the -scale to the Z-scale i the 60 µ same maer. That is for y = 60, z = = = We the compute the σ 4.5 correspodig area uder the Z-curve usig Table 3, Stat 538 Probability Lecture 2, 9/14/2004, page 8 of 16

9 Area =.9082 Symbolically, 1.33 Z Pr{Z 1.33} = Pr{ 60} = proportio of fish with legth 60 mm =.9082 = 90.82%. What percetage of fish is more tha 45 mm log? Follow steps draw a picture of desired area 3. area = = Z For y = 45: 2 - Trasform to z-score scale z = = = i. e. Pr{Z -2} = Pr{ 45} = proportio of fish 45 mm =.9772 = 97.72%. Let us work out, together, the followig problem: What proportio of fish is betwee 56 ad 62 mm log? SD rule The Z-score tells us how may stadard deviatio (SD) uits a value is from the mea. For ( µ + 2 σ ) µ 2σ example, y = µ + 2σ is a value 2 stadard deviatios above µ, so z = = = 2. σ σ ( µ σ ) µ σ Similarly y = µ σ is a value 1 stadard deviatio below µ, so z = = = 1. The σ σ SD rule says that for a ormal populatio, Stat 538 Probability Lecture 2, 9/14/2004, page 9 of 16

10 68% of values are withi 1 std dev of µ (i.e. i ( µ σ, µ + σ )) 95% 2 (i.e. i ( µ 2 σ, µ + 2σ )) 99.7% 3 (i.e. i ( µ 3 σ, µ + 3σ )) This follows from a direct trasformatio of iterva l edpoits to z-scores. µ - 3σ -3 µ - 2σ -2 µ - σ µ µ + σ µ + 2σ µ + 3σ Z 68% 95% 99.7% I essece, this result implies 1. That the distace from the ceter is best measured i terms of multiple of stadard deviatios (i.e. Z-scores). 2. The stadard deviatio determies how extreme or uusual a value is. Example: Fish legths. Here µ = 54 mm ad σ = 4.5 mm 68 % of fish have legths betwee ad , or 49.5 ad % 54 2(4.5) ad (4.5), or 45 ad % 54 3(4.5) ad (4.5), or 40.5 ad 67.5 Example: SW exercise 4.3 (a) (c) p Stat 538 Probability Lecture 2, 9/14/2004, page 10 of 16

11 Percetiles of Normal Distributios These are iverse problems where you are give a area ad eed to fid a value that will produce it (istead of the direct p roblem of fidig a area like we have bee doig). The p th percetile for a probability distributio is the value such that there is probability p of beig less tha or equal to that value, ad probability 1-p of beig greater. The picture below idetifies the placemet of the p th percetile for a ormal distributio. Area = p Area = 1 - p p th percetile (y*) To compute this percetile, we first compute the p th percetile for a stadard ormal Area = p z Z This ca be obtaied by a iverse process from what we re used to. Give the stadard ormal percetile we solve for the y-value that gave rise to it: y * µ z= y* = µ + zσ σ Examples (fish legths, revisited): What is the 40 th percetile of the fish legth distributio? What legth must a fish be so that oly 5% have loger legths? Example: SW ex 4.16 p. 133 Stat 538 Probability Lecture 2, 9/14/2004, page 11 of 16

12 Samplig Distributios Read SW Chapter 5, Sectios 1-4 Suppose that I wish to kow what the mea is for a specific populatio (target or study populatio. The followig diagram summarizes the samplig from the populatio. SRS of Radom sample Populatio of iterest Mea µ (ukow) The pop. is what we re iterested i but caot see completely. i 1 i = =,, 1. Calculate as best guess f or µ. This is our data. We re iterested i the populatio, but this small sapshot is all we have to work with. Importat poits The sample mea estimates µ, but there is error associated with the estimate (we would eed to access the whole populatio to exactly calculate µ). Differet samples give differet s, so the size of the error depeds upo the sample. Of course i practice we do t kow if we got a good sample or a bad oe (i.e. we do t kow if is close to µ or ot). T he value of caot be p redicted prior to the sample beig selected, so before collectig data w e thik of as beig a radom variable. O a coceptual level we ca evisio listig all possible samples of size ad the that results. The collectio of sample meas that is obtaied ca be collected ad plotted as a histogram, leadig to what is commoly called the samplig distributio of the sample mea. The samplig distributio of is, i essece, a probability distributio of that specifies the probability of observig specific rages of values for whe we sample the populatio. The samplig distributio is real but is mostly a coceptual costruct. We ca list the samplig distributio oly if we kow the populatio values. If we kew the populatio distributio we would ot eed to sample it! Noetheless, the idea of the samplig distributio is crucial to uderstadig the accuracy of. Stat 538 Probability Lecture 2, 9/14/2004, page 12 of 16

13 Meta - Experimet Histogram of possible s Pop.... Desity Curve Differet samples, differet s The followig properties of the samplig distributio of ca be show mathematically: If we have a SRS from a populatio with mea µ ad stadard deviatio σ, the 1. The average or mea i the samplig distributio of is the populatio mea µ. I symbols, µ = µ (the mea of the mea is the mea?) 2. The stadard deviatio of the samplig distributio of is the populatio stadard deviatio σ divided by the square root of the sample size. I symbols, σ =. 3. Shape of samplig distributio a) If the populatio distributio is ormal, the the samplig distributio of is ormal, regardless of. b) Cetral Limit Theorem: If is large, the samplig distributio of is ormal, eve if the populatio distributio is ot ormal. Importat poits 1. The typical error i as a estimate of µ is The accuracy of icreases with sample size (i.e. smaller variability i samplig distributio). Stat 538 Probability Lecture 2, 9/14/2004, page 13 of 16

14 3. For a give sample size, will be less accurate for populatios with larger stadard deviatio σ. 4. Kowig that the shape of the samplig distributio is ormal, or approximately ormal, allows oe to evalua te probabilities for. Thes e ideas are fairly subtle but importat. Let us go through a umber of examples i the text. Examples: SW example 5.8 p p p.168 SW example 5.9 p p. 163 SW example 5.10 p p.167 Dichotomous observatios Dichotomous = biary (successes or failures) Suppose we select a SRS of size from a populatio ad cout the umber of successes or idividuals i the sample with a give characteristic. We kow this has a Biomial distributio. As a alterative way to summarize the data, we might cosider the sample proportio ˆp, where ˆp = # successes /. If the populatio proportio of successes is p, we ca show that the samplig distributio of ˆp has 1. Mea = p p(1 p) 2. Stadard deviatio = 3. Shape that is approximately ormal for large These results ca be used to approximate biomial probabilities usig a ormal distributio. Example: SW example 5.16 p I do ot wat to overemphasize probability calculatios based o samplig distributio. The importat place for samplig distributios is where they directly lead to procedures for statistical iferece. Stat 538 Probability Lecture 2, 9/14/2004, page 14 of 16

15 Cofidece Itervals SW Chapter 6 Course otes p This chapter itroduces a stadard statistical tool, the cofidece iterval, which will be used to estimate ukow populatio meas µ ad proportios p. Stadard Errors of the Mea Give a SRS of size from a populatio with ukow mea µ ad stadard deviatio σ, our i = 1 i bes t guess for µ is the sample mea =. The accuracy of as a estimate of µ is σ dictated by the size of the stadard deviatio of the samplig distributio of, σ =. I practice this quatity is ukow (we do t kow σ), but ca be estimated by usig the sample stadard deviatio S as a estimate of σ. This leads to the so-called stadard error of, s SE = as a estimate of σ. Example: SW ex 6.3 p 181 SW ex 6.5 p The Studet s t-distributio If the populatio stadard deviatio σ were kow, the a Z-score trasformatio of the µ µ samplig distributio of, i.e. Z = = could be used to geerate cofidece σ σ itervals for µ, ad this idea is illustrated by SW o p Because σ is ukow i practice, we eed a alterative approach ad that is to stadardize by usig SE istead of σ, µ µ computig a t-score istead o f a z-score, t = =. If the populatio we sampled from SE s is ormal, the Z is stadard ormal ad t has the so-called t-distributio. The desity curve for a t-distributio looks like a bell-shaped curve, but has more probability i the tails tha the Z-distributio. Furthermore, there are may t-distributio, but the family or collectio of t-distributios is idexed by oe parameter called the degrees of freedom (df). For our problem the df = -1. As df icreases the t-desity curve more closely resembles a stadard ormal curve. For df = (ifiity) the t-curve is the stadard ormal curve. See Figure 6.7 p. 187 of SW. Stat 538 Probability Lecture 2, 9/14/2004, page 15 of 16

16 For our applicatios we will eed upper tailed critical values from the t-distributio of the form Area = α 0 t α t Table 4 (iside back cover) gives upper tail critical values. Example: for df = 10 fid t.05, t.025, t.01 Repeat for df = 3 Questio: For a give upper tail area, how do critical values chage as df icreases? What do you get whe df =? Stat 538 Probability Lecture 2, 9/14/2004, page 16 of 16

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