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Introduction to Probability and Statistics 131B Lecture 4

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Introduction to Probability and Statistics 131B Lecture 4
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Mathematics Spherical cap Lecture/Conference Pearson product-moment correlation coefficient Decimal Energy level Total S.A. Mereology Open set Numerical analysis Element (mathematics)
Covering space Standard error Randomization State of matter Multiplication sign Sampling (statistics) Correlation and dependence Mereology Measurement Numerical analysis Grothendieck topology Estimator Mathematics Many-sorted logic Lecture/Conference Modulform Summierbarkeit Position operator
Functional (mathematics) Multiplication sign Expression Sampling (statistics) Variance Thermal expansion Parameter (computer programming) Cartesian coordinate system Flow separation Entire function Derivation (linguistics) Frequency Arithmetic mean Different (Kate Ryan album) Term (mathematics) Order (biology) Hessian matrix Square number
Finitismus Functional (mathematics) Process (computing) Vapor barrier Multiplication sign Prisoner's dilemma Expression Thermal expansion Mereology Grothendieck topology Proof theory Frequency Term (mathematics) Order (biology) Cuboid Summierbarkeit Right angle
Area Sigma-algebra Confidence interval First-order logic Multiplication sign Closed set Sampling (statistics) Thermal expansion Price index Cartesian coordinate system Grothendieck topology Substitute good Numerical analysis Product (business) 4 (number) Sign (mathematics) Lecture/Conference Term (mathematics) Order (biology) Square number Right angle
Estimator Spherical cap Lecture/Conference Real number Maxima and minima Parameter (computer programming) Limit (category theory) Maß <Mathematik>
Group action Link (knot theory) Variety (linguistics) Weight Physical law Sampling (statistics) Maxima and minima Parameter (computer programming) Mortality rate Evolute Cartesian coordinate system Fisher information Positional notation Many-sorted logic Lecture/Conference Oval Right angle Musical ensemble Cycle (graph theory) Gamma function Alpha (investment) Relief
Standard error Sigma-algebra Civil engineering Multiplication sign Closed set Sampling (statistics) Archaeological field survey Algebraic structure Mortality rate Variable (mathematics) Fraction (mathematics) Estimator Root Lecture/Conference Different (Kate Ryan album) Average Resultant
Point (geometry) Coefficient of variation Multiplication sign Variance Water vapor Mortality rate Mereology Grothendieck topology Theory Numerical analysis Arithmetic mean Mathematics Lecture/Conference Term (mathematics) Calculation Right angle Bounded variation Stability theory
Point (geometry) Area Standard deviation Beat (acoustics) Sigma-algebra Multiplication sign Sampling (statistics) Parameter (computer programming) Measurement Event horizon Numerical analysis Entire function Connected space Estimator Lecture/Conference Square number Game theory Set theory
Point (geometry) Standard error Group action Observational study State of matter 1 (number) Similarity (geometry) Water vapor Parameter (computer programming) Element (mathematics) Estimator Different (Kate Ryan album) Term (mathematics) Square number Lie group Area Theory of relativity Weight Sampling (statistics) Variance Grand Unified Theory Measurement Numerical analysis Arithmetic mean Combinatory logic Right angle Figurate number
Standard error Randomization Functional (mathematics) State of matter Multiplication sign Insertion loss Rule of inference Event horizon Grothendieck topology Estimator Lecture/Conference Term (mathematics) Average Physical law Selectivity (electronic) Circle Nichtlineares Gleichungssystem Set theory Social class Random variable Area Standard deviation Multiplication Constraint (mathematics) Autocovariance Weight Gradient Expression Sampling (statistics) Stress (mechanics) Independence (probability theory) Maxima and minima Mortality rate Variable (mathematics) Flow separation Subgroup Numerical analysis Arithmetic mean Order (biology) Right angle Summierbarkeit Pressure
Art assassin a ratios it way
of 2 values assigned to the elements of a part our population would be a running example in the book will be as reason b the number of beds in the hospital I VXI and these should be the number of discharges in January 1968 and hospitalized with an elevated level so each with XI will beat XIY I represents 2 distinct measurements of of a new individual life and some populist again the young population from Iris are you axis BOY they say where x squared you go floor say square sick ex-wife and so far and role and by the way of another way to express is as new supply over the use of acts as the reason being you divide top and bottom by 1 over capital and not change it and then becomes wine is not that as and I guess is but call the population correlation coefficient so at the a goalless decimate ah I guess they relate to 2 things we don't want as we are all I forgot there's also acts tall why our it is the population total tossed backed up with but why that's hot why real enough that fills up that over but such a it's like individual summer population capital X 1 Kabir elects to cap of 11 and capital Y 1 capital light to capitalize 11 will be there I can't believe it over the
weekend as reading a regular last week I was reading newspaper
and 1 of these newspapers my home state they want to use this word this form of their but they wrote T HERE ran a front-page fog will be there to X and Y measurements X and Y values respectively and Abelard eagled the capital Alexei are over cover why bar will be the site for the lock e says ingots often acts New often next values are known that have come and I'm sorry it did against is why borrower Y and the ratio estimates the can be used to give investment of YBa new life but but but I don't think so so so here we take C of the known population means servers Apple in the example the number of beds in the hospitals you get that information sites are numbered discharges changes every every month random so you might know something I never beds hospital ahead but you wanna have some estimate of only people are being discharged from the hospital given what use this because there's some sort of a positive relationship between number of beds and number of discharges more beds a hospital has more documents was beds will have more discharged so sums are positive correlation between the 2 you might under sample somehow on the number of discharges and that would mean you're underestimated the was white might be an underestimate for the true number of discharges and that would mean you probably underestimating new acts with the ex parte words you take select and little in hospitals and you find that the number of people that mean number of discharges assists this might be underestimate if it's an under a slate of new wife than the should be an underestimate of new acts and that means racially new acts of Rex bar should be bigger than 1 that serve corrects this this plan that would correct be estimate of our y bar conversely if you serve or unlucky in selected in hospitals and why bar was overestimating you buying new wine then ex parte Bobby overestimating acts because a possibly correlated in some sense and that would mean it a bigger that ratios smaller Vermont and so you could have multiplied number smaller 1 times a rest McClatchy crept back to better estimates of this just minutes for them to further means the Y values has some advantages in some situations not only that we are our standard error for the assessment can be better than the standard bearer for just a simple estimate using white parts this has some nice features assessment for for me Alastair Sim
Bergantin now last period we looked at as meaning they are at Guinea expressions for at least 2 things and in terms of the young population parameters they can be expressed as follows where you going to but the core Cook half OK so you are organizing a fine population correction here and where they staff from um have or 1 arms from a 1st order Taylor expansion other function GOX y eagle eye over X around could appoint new severity so Musa and then squaring tool of problems from a 2nd order Taylor expansion of the same function a so brought still builders summary briefly strokes gifted computer driven with extra is minus why erect squared BY durative is 1 or axis but we put we evaluate these gurus at New XBY to Sasabe time are the the of the sea and actually mines fly Red Square to evaluate here I get my y all acts where these 1st derivative Y of G violated New actually life Java that respect lies 1 racks that way here I get 1 or new X G X X evaluated at Newark but he said that in acts Austria's my direct squared the next to y your acts due to wire racks to make it through New Y overview execute the mixed derivatives that you XPY I differentiate 1st respect to Wiergate 1 acts different respect exit minus 1 over X squared off as being minus 1 or new X squared and finally the 2nd rivers and why the 0 and sold but Jeannine our X wide by these 2 characters should be nearer new acts and knew why member what's a variance the X its Sigma Sigma x squared over and an entire population Gretchen so it's a very very small agrees abide by the sample size this emphasized they made
a small sum is to be very close to you at some should be very close to new life so Taylor expansion around New XVY should be valid for a while with very high probability will be better protect probably near New XMU wife sites like this works noaa GXX prepared here limit value-added out I guess I'm very out what Taylor expansion looks like and will subsidize next door but for it's always a 2nd order expansion Mr. B R 1st firms are new prison new why or new acts was review X Box E R is wife or new acts as 1st term years are there may yet we just write it as G sub x time sex are minus BOX I don't care what this is because my take expected banish work that it's OK so plus plus what this is to y over X Q dragged somewhat my my half I get new Y over you X 2 times X bar Myers Musa backs square 0 and then this 1 was minus 1 over X squared so here I get minus 1 over abusive X squared X bar minus new X Y our players new what do they OK so if we just take what is this here this is capital R this function as the burial of that are Susan expansion capita lock up Lars Krochmal equaled send sex bars very close to the new selected probably why bars because news while would I probably start this so we wanted to make it 4 Onyx expressions the expected by or are we just take expected by right side tho Abbas take expected either right-hand side let them die or are there it is equal to approximately little plus now taking 2nd I here 0 because expected by this 0 recognize that it's period supply over new X Q times sub media next bars winners that's a very next far and get minus 1 over news of Axis Anderson be stigma next far wide by now we helped finite population expressions for these things we've got before I actuator prove this so we prove this but if you follow the proof of that offices and substitute a Y was Signetics part X borrowed be this thing you just modified approval this is the same OK this goes in here this goes in here and say job rearrange things any a hats were these terms and further barriers of ah we
Uzi so this is easing the secular Taylor expansion than they expected value to the variants of ah auto where his use
of first-order expansion all light that approximately equal to the IGC backs Amissville user after the 1st order that means stop here and uses a slice write-down GX dissidents and of wine or is he selects here's supply I guess the minus sign here minus 0 why or abusive acts where is best use of X time acts are buying new axis and then plus what she supplied the time wife Barbara Nardi get the variants of capital are you subtract will are square take expected Vice subtract the La from both sides bank square this of a body blow to this things where which would be what you why squared or new acts to the 4th time exit bar by the use of X squared plus those things squared member product terms B minus 2 y all over you X Q X bar minors why are biased view I think so I subtracted locks suicides and squared books Chu now IT expected value below sites again to be closed to new acts why are so are cap because of the right hand side so please expected value should be about the same and what's expected by this that's stigma Bob squared plus 1 over the use of X squared they will squared biased 2 new why overview of setbacks and this be Sigma X white or and now as a substitute in these 3 things of this this and this here here it here and rearrange fact out stuff you'll get E spot so that explains released compromises that Taylor expansion and that basic ideas French and should be good With a high probability when these sample taken is large enough because X concentrating near new acts surrendered rebel doesn't vary too much from new X and Y concentrating near new life so that means side should be close to each other means a Taylor expansion should be very close to being correct that Taser truncated him to stop any Russian had far right is just populace right to put some real a formed a and say really love I so we're I confidence
intervals for who are the OK so let's think about how we got a cop reasonable for a new and user backs in these cases of simple random sampling of population about where we do reduced the is he use back next are view over screw race bars nearly normal when the sample size is large enough and not too close population size but the problem using this to get top was there's a stigma X squared in here that we don't know yet various index Pa all ever of this was it is but but a but here she was at acquisition that 1 was that for was a risky irate less square was for say depends on the sample square depends on sample in these numbers colonel so something you contribute once it got that we use that broke ranks of so it to something get here it turns out the thing you use areas S our squared the G take so to compare
curies of ah has sacked Popeye population trucks Internet and so we keep that and then we have to do what
these are the real population parameters and we don't know what they are slated to replace them by the things we asked them that's what we put cap great estimates were Weisberg OK so they how would you give it counted as for capital are well you know this is a will prove this but don't bring use we was set a limit their a for true so it have ah plus or minus Z E of Elf over to our S a bar give end points 0 0 approximate 100 1 myself a cent CI 4 they promoter the La so
probably 1 minus Alpha the law it isn't the and Vicki rubber the notations you allow for 2 it was formally precedes the same way as Kinnock copper sales from use of acts about then again the year various are has to be estimated by things you get from the sample you can't put in population parameters there said Maxim that's why or are so it wouldn't be a sample estimates they are so that's pretty much for Stillwater of Let's go onto the 2nd technique that uses ratios nets isolating Piazza from you why using racial the surmise
neckwear no link webpage page how radio look to the webpage higher hobby all right is it to get the homework and better next rate Monday saw Roby posted this afternoon army you looked at EU videos there online what about us it's sort of a from a record of my mistakes where I also put up the link to web page Wikipedia Web Web entry on Gossett His image that you might that I've put 1 up for another statistician Fischer was very interesting guy they also Fisher gossip communicated with Fischer aren't saying this because I just don't want tactical stuff I could tell that you do overwhelms psychiatry will break so Fisher also was very interested and genetics and evolution and you might be able to act astounding added Neville evolutionary argument for why these ratio males of females in the population would be 50 50 years there was not incestuous if there wasn't a incestuous reproduction and that turns out there is a very interesting insect particularly might in which the male to female ratios exactly 1 10 and this but particular insect has has incestuous reproduction and use a life cycle of this might as this is a very example show why the shares are likely to distract the lights life cycle begins with the female pregnant is born pregnant actually taxes itself to leave she has 10 infants in 1 male 90 females and Ismail reprove died fertilizers and eggs all of his sisters and in variety dies before is born before Exocet never to who will mobilize Mesecar Riady axis of gloom and dies middle words and and so the females are born pregnant may test cells to relief and it's itself so in the face of incestuous reproduction male-female ratio of obesity that went in the case will be 50 50 Fischer gave a very interesting argue for why did you know but this might be curious you to read about it in Stephen Jay Gould how would like to be that might 1 addressing life you have it so it actually Fisher and many others very interesting ideas 1 thing done everywhere have been the sentry possible human channel in India idea but information and acted CD called Fisher information alright these will resume that's to the 2nd technique is a guest from your wife Irish restaurants a solo here so it back so again there's a served self correcting effect here if y bars under sampling than ex parte in certain cases will be also and this would be bigger than 1 would increase white vestments why bars over sampling and in certain instances wood probably true that expires as well and so this would be lesson 1 correct back like so he the various survived bar owner will be approximately what is
Newark's times appraisal are ah Senor squared times rates are what
exactly equal to new Exorcist squared times are an and that's then multiply
that by squared and does not the Oxford denominator and submitted 1 around on a and 1 might
slowly minus 1 or as 1 that sars words in a single squared minus To our Sigma ex-wife Frank they're everywhere but at the authority is expected value of arrest mayor from the thing rest make isn't it if it's differed 0 then this is an unbiased estimator while his wife Teresa why art estimated knew why this is on call for an average of week away from your wife and well we have an estimate for expected value of R sub their E. if we get rid of ideas by radio sir broke up if we modified to nearby new Exorcist bonds are not quite to hear my new acts of this in this fight me you when you modify despite new get right he still get down new X Our minus BUY and there may be a bit equal to love this time as new X-ray and that would be also are that's are what Weiser bar are minors BY is not as a saving lives and right this is a why borrower expired and the led Emulex that's life of twice a lock for the expected value of this difference here would be missing over here I should say this is expected not expect so I just copy that thing over here Her saw financed gala focused at sunrise says there but the sample size is big and these things act to this doesn't tragedy 2 big then not bitter repeat not to buy say so let's take a break here mold with thought as missing after the British results single isolated I topic that I heard there's a civilized pertinent Kestner lot so there would be no here a misprint Sokol backed a very 1st whether braver status the incorrectly wrote down Weiss out and that means it's not a these estimates that over the square root the variously estimated skull the standard error anyone make the standard error smaller words I have a very speedy small that means it is less swell variability solar estimator will be likely close to me it's estimated so let's compare they stand there for why far and wide well over there the Centers a scorer the various throw on whether we get center square in various why a goal or they fired a population correct it'll be the same for both soloist with your that earlier than we get 1 or an are squared sigma X squared plus single squared minus 2 are Sigma ex-wife and raise OR surveys of wire is signal twice ignoring again by population structure because it's the same for both ahead are that means this means we should let rewrite the 1st thing you wanna compare them as single Y squared overland plus are squared segments where minors 2 are the same ex-wife over nothing at this some here I pray over amusive
acts all I it was right there for us all ran I guess today still when is it isn't are smaller the variance why bar well as its 80 s when this term is negative right about all the course of a fight but the Baxter I am missing something here resist role here but if I write call but this is correct but let me right where wrote rowers I think I adhered E. here Signetics or say Mexico I substance that here as equal that observed say next Michael cigarettes Wrexham I think and somebody at the extreme sites here this His lesson that becomes are is less than true role stabilize over acts by the by both sides by Sigma expert but received little part the water the both sides your by said spreading christening calling so leave it like that but my point of theory raters row is bigger than 1 half are sick Rex over Sigma why I E injury rate that rewrite Sebert right on the official New wire Ramiro acts ours me acts by that it can rewrite it this way the new acts under the cigarettes knew Iowa under the singer wiped fear fractional their places by now and so there's no writers has a name but where would Cesar backs why so bro was bigger one-half a CX over this implies are as smaller stand there then why are better estimated that lifeline user called coefficients of variation these sorrow this somewhat the variable X varies as its means if the variants of the Nexus can seek nexus 10 but newest 10 million I would say as much variation compared to the value of new acts or maybe put another way you're stock went up by a dollar share would that make you will have a happier very happy well what the stock was worth a million dollars a share you would notice that but it was worth and the you know it's a lot right so this is like the stockinette somewhat changing your more intrusiveness racial Benin are much changes euro don't care bye Ira finesse like it's a hobby of a denominational bill would have down before Bill Gates would bend over pick it up with at what stage would be worth his time we pick up a dollar at the because 1 dollars compared to our cost billions you might hurt his back so so these are reserved for interesting in some sense that trick OK so let's look at an example of this is a kind surprising that you have this indirect ways as they knew why it's better than the drug the fighting there where means using light bar are using the back way y so they're actually
that these calculations and the in the tax on the write down what members are try right on the the 1st numbers the city sizing this we don't need
a man us so so I is a hospital to measurements for that brief hospital for fun winners in our beds in hospital the measuring the number of discharges in January I think 68 a house on the true 0 population parameters here for the example of the book are new X equal to 7 4 . 8 new equal to 8 1 4 . 6 sigma x equal to 2 1 3 . 2 signal wife equal to 5 8 9 . 7 are equal to 2 . 9 6 row equal to . 9 1 item c-axis see why were released and see exit beat 2 . 1 2 1 3 . 2 or 2 7 4 . 8 I think this buy and that turns out to be . 7 8 see is this divided by absurd this but it by this 5 9 . 7 divided by 1 4 . 6 and that's the point 7 to sewer 1 FCX oversee were choosing quandary here it is some 1 one-half or 1 . 0 8 4 . 5 4 and you could see growers here and role is bigger than I have 6 firms you what this example of why bar would have a smaller standard than me and that why bar for your discharges and so why borrowed recoverable Dr. can computes a things so From what I hear our areas it you we said white bar of connection if we think that's 1 for an equal and 9 x-ray Anagallis 64 self bigger sample size 64 remember that population size for this example of 393 signal Bart are or the wire it is 1 0 vs. 1 over square event for 8 times where best which becomes 206 quite want an entire population corrections which is 1 minus 63 over 392 0 turns out to be 30 where z true various areas 29 . 9 pct in various for the estimators very close to the 2 rents wears a computer various for why bar that's a signal y over rude and squared of one-liners and minus 1 Apple was 1 puts wife their put 9 64 near here and 393 here and this becomes 66 . 3 so the various years twice is more than twice that so is a big game for as standard area is twice the standard for the big game and accuracy by using Y Ba ba OK so there will be some exercises based on ratios there too gap the alliance well here all of this is not a signal why that's as the best the wife former s as from the sample and this is the true what this is from a set of here over well well maybe these are the squares With this but maybe it is authorities square so most things squared sorry that OK at they while you're right the Ira what
back off and it is being done they delay on her see him I think the high I their actual station OK so this is our race that don't but then so for free what that
would be main point is this difference a much better this has been sold out get rid us Standard area is much smaller than half the standard thanks for quite a while now on
that's it for ratios so again there were 2 things we did with ratios 1 is got estimate for the ratio using a couple both ratios using settlement terms and the other 1 was To assonate knew why using this ratio us why are here and in the case
as this is bigger than that standard error for the assessment was better than the standard bearer for using white bar the next topic in Chapter 7 as called stratified sampling and in this type of sampling publishers barges in the new Subpart lifts area capital L soccer relations with travel 1 traveling to travel and they'll members respectively where some of these numbers she is capital and it is a R all of them but as the weight of the L so popular so water some examples maybe you're starting on employment in the United States some estimate of the number of unemployed days females aged 30 to 40 in U.S. well you could just take a random sample from the whole population and down it that way or you might divide the U.S. into either could be 15 Capra 50 in a new check the unemployment where between 30 and 40 in each state and and you might make some combination of your sample tried as to make a lot of fun figures the group for AL could be the number of regions maybe you divide the U.S. into regions like the West Coast of the States west of the Rocky Mountains maybe Great Plains states Southeast Midwest Northeast and then in that case Capel might be something like 5 study unemployment figures in those regions and combined them somewhere the Bogor but look at that population I that um parameters and measurements so over art effort let us see our if I like I think under his right eye Friday like ice which sees I shared by Xu reversing case on the 2nd number induced indicates that group number so it all these groups 1 and this is the 1st individual Group 1 2nd individual Group 1 and 1 individual and group 1 died here I should do a similar thing group This is a beauty 1st individual the measure the 1st individual Rubell most 2nd visual Rubell measurability blasts individual Bell said a school we take a population parameters of new X 1 I think scores of next goddess column you want don't need to use UCX new ones would be won over capelan won some I people wanted capelan 1 X lie 1 else means me new else group would be won over the number of elements in that group and then this just as some of the numbers you see in their group etc ever similarly Sigma 1 squared be won over and 1 I equal 1 To capelan 1 X I 1 my this new ones where did sigma sub blue elsewhere would be 1 over or capital and sung by people want to capital and Luelle X I will new elsewhere etc. down through signal elsewhere people who won over couple Al Capone capital L want to capital X capital by new capital Square case they form the mean and variance in each group in separately the tall meaning at all we're
in this way the sums all the possible population values and abides by the population size tell this thing inside here is light find of the sale this term here term is what you see here a thing in the circle here so this would be equal to capitalize times Mueller His following that equation there further terminal circle and then I could rewrite this some L. Eagle One to Kaplan by pulling the capital and inside and recalling calling Capel l over traveling the weight of the news of El awaits class for the whole average is as some of the weighted averages of the individual populist weighted by these populations are soft pop so for example of but the U.S. into state by where California which Llewelly 1 1st state are the best thing would be about 1 AM what these 1 8 it fell 21 8 . 8 per people cent for California but the weight of 5 nautical order would be almost 0 because I think there are more people in jeter lighting Hunter Beach Irvine more people there and got it so it makes sense to wait loss populations that are bigger with more weight suffer populations their smaller because they quite figures in Northborough won't have much influence on the packages but the whole country should tho on right here are big see things him some 1 of also ah sorry yeah kept banks that edits knew that we want to rest there were want if possible stratified population but in a way that minimizes the standard error of the estimate so start 5 population and up selected sample Olivia at stratified oppression Ed select a sample it away that realizes Saturday to actually there should be more careful but WA life possible given the stratification to select a random sample of simple random sample in a way that minimizes standard so what we do sample each strata independently all can for a war selecting little events Abell individuals from me capital and Isabel individuals and Estrada not be resulting render rebels by but Texas 1 L X to L queued up to x and so do l l see Rio blue and Sibelle observations by this random selection and assume sets of grammar rules on how but but take sale sa I means is set around the Robles as a hundred of us that is independent of the of independent of the set it doesn't mean that the random Robles needs that are these
dependent these are from these are simple random sample from 1st subgroup or strapped as a simple random sample from a 2nd stress so each In each set the runner rules are weakly dependent of the sample size is big enough member the covariance would be Miners guessing whispered over the sample size minus 1 so population size minus 1 population size would be capital capelin numbers but from 1 set of next independents in other words we sample year what we did we don't that doesn't affect how we sample the 2nd starter does affect sample of efforts so from starters travel samples were day so what's our estimator for a new program and see what kind I guess of a drum roll please expire asks for this article ex parte where x 4 L N. C sorry xxx it would be an estimator for new Al His unbiased estimator for new oil Texas Isa separate population who else population would take a random sample that it's me is the use of what's the unbiased estimator from news about this sample me sample so the expected value of X L it is a new wealth I'd expected L his new book ID expected I hear a tape expected by the sunlight exchange expected values the WLZ a constant that's a number so I take expected by this what's expected that new album expected by excess bar would be some weld W L and the some LWL the lists new so it's very simple this gives an unbiased estimator so you just take sample averages in each substratum stances saying some populations Estrada sample averages each strata wake them according to the proportion of individuals and so population that strap connectors you an unbiased estimator for that told me broke it by using the young independence from Sept the that we can get it Guy express further various of X R S and they E but we Setzer independent so where Barbara random variables X bar 1 of the rare this is formed from Betty Robles in the 1st set this is among 4 men rebels in the 2nd set but is far from Ranna rebels and offset go sets are independent these also be independent we hear what we have these are numbers fixed numbers times these random variables to render rebels in the summer does that ran rebel in the some are independent from each other 1 California's 1 is independent of the case 1 when from a said we're O'Sullivan the and rebels were taking the variants of that what's a somewhat of what's rates of some of grammar rebels at the summit of the various Nouri resort area with multiple of X al sites here and I thought a W squared a way out of a subway sample me and using simple random sampling we know what that is From earlier in this chapter where signal where I think I wrote Thunder 9 batteries as a baby buried somewhere else so population the thing is a farce define it means correction by a population pressure OK so see barracks of our S OK so next time slot answer this to be the little they'll here and little and elder next time we'll see how we can pick blew a mail to make this well severe take it taking a course were a couple what grant most buyers the as a functional capital L variables and want to the Lemont food health and it gradient that constraint a constraint will be that there are some of the and bells will be some number bill and a lot of demise is subject to a series of