THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management

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1 THE UNIVERSITY OF TEAS AT AUSTIN Departmet of Iformatio, Risk, ad Operatios Maagemet BA 386T Tom Shively ESTIMATION AND SAMPLING DISTRIBUTIONS The purpose of these otes is to summarize the cocepts regardig estimatio ad samplig distributios that we discussed i class. ESTIMATING THE POPULATION MEAN (µ) Cosider a example dealig with the salaries of last sprig s MBA graduates. The radom variable will represet MBA salary. We will assume the populatio of last sprig s MBA salaries are ormally distributed with ukow mea µ ad a kow variace of = (0,000). The distributio is draw below. = 0,000 µ : MBA Salary The iterpretatio of µ is the followig: µ represets the mea salary of all MBA graduates from last sprig. We could determie the exact value of µ (i.e. the exact mea salary of all MBAs i the populatio) by obtaiig the salary of every MBA graduate, addig them up, ad dividig the total by the umber of MBA graduates. I practice, it is too expesive ad time cosumig to obtai every MBA salary, which meas i practice, that we caot determie µ exactly. The importat poit to uderstad from this discussio is that the populatio mea µ is a umber, but we do ot kow what the umber is without gettig the salary of every MBA graduate i the coutry. Note that mea ad average mea the same thig. They ca be used iterchageably. For example, sample mea ad sample average mea the same thig, as do populatio mea ad populatio average.

2 Aside I am assumig the variace of the populatio of MBA salaries ( ) is kow. I particular, I am assumig that = (0,000). I practice, if we do t kow the mea (µ) of the populatio, we will ot kow the variace ( ) of the populatio either. I am assumig we kow the variace to remove oe level of complexity from the problem of estimatig the populatio mea µ ad determiig the quality of the estimate. I practice, we will have to estimate the populatio variace. This will be discussed below i further detail. Ed Aside Next Aside It is much easier to iterpret the stadard deviatio of the populatio tha it is to iterpret the variace of the populatio. The reaso for this is that we kow 68% of the probability falls withi oe stadard deviatio of the mea, ad that 95% of the probability falls withi two stadard deviatios of the mea. The probability calculatio for oe stadard deviatio is give below. µ - µ µ+ : MBA Salary pr(µ - < < µ + ) = ( µ ) µ µ ( µ + ) µ pr < < = pr(-.0 < Z <.0) = pr(z <.0) - pr(z < -.0) = = Also, the uits for the stadard deviatio i this example are dollars, while the uits for the variace are dollars squared. It is cosiderably easier to thik of a measure of dispersio i dollars tha i dollars squared.

3 I the MBA salary example, 68% of the MBA graduates make withi = $0,000 of the populatio mea salary. A equivalet way to thik about this probability is that if we pick a MBA graduate at radom from the populatio, there is a 68% chace that we will get a perso that makes withi $0,000 of the average salary of all MBA graduates. Similarly, 95% of the MBA graduates make withi = $0,000 of the populatio mea salary. A equivalet way to thik about this probability is that if we pick a MBA graduate at radom from the populatio, there is a 95% chace that we will get a perso that makes withi $0,000 of the average salary of all MBA graduates. Ed Aside To estimate the populatio mea, we collect a sample of MBA salaries from the populatio of last sprig s MBA graduates ad use the mea of the salaries i the sample as a estimate of the mea of the salaries i the etire populatio. The steps i the logic uderlyig this idea are the followig: () The sample is represetative of the populatio. If the sample is relatively large, it will represet the etire spectrum of MBA salaries i the populatio. For example, a few people i the sample will make low salaries (because a small percetage of the MBA populatio makes low salaries), a large portio of the people i the sample will make salaries right aroud the populatio average (because most of the people i the MBA populatio make a salary close to the populatio mea salary), ad a few people i the sample will make high salaries (because a small percetage of the MBA populatio makes high salaries). () The sample mea ( ) is represetative of the populatio mea (µ) This follows from step (). If the sample is represetative of the etire spectrum of MBA salaries, the the sample mea ( ) must be represetative of the populatio mea (µ). Aother way to say this is that the sample mea ( ) is a good proxy for the uobservable populatio mea (µ). (3) The sample mea ( ) is a good estimator of the populatio mea (µ) This follows from step (). Step (3) just formalizes the idea i step (), i.e. if is a good proxy for µ, the we say provides a good estimator for µ. 3

4 DETERMINING THE QUALITY OF AS AN ESTIMATOR FOR µ A atural questio to ask is how good the estimate of µ is that we get usig. For example, suppose we collect a sample of size = 00. It is possible (although ulikely) that we get 80 people i the sample that make salaries well above the populatio mea µ, ad oly 0 people i the sample that make salaries below the populatio mea µ. If this happes, the the sample mea will be far above the populatio mea µ, ad we will get a bad estimate of µ. The appropriate way to phrase the questio cocerig the quality of as a estimator for µ is the followig. First, we must defie what a good estimate is. I will say ay estimate withi $900 of the populatio mea µ is a good estimate. The defiitio of a accurate estimate (i.e. sayig we wat a estimate withi $900 of µ) is a subject matter questio, ot a statistical questio. This meas you must cosider the cotext of the problem to determie the degree of accuracy that is required to have a good estimate. (The $900 figure I chose is admittedly a bit arbitrary. A more atural figure would be $000 but I chose $900 to make it easy to differetiate from = 000, which is used below.) The questio we wat to aswer is: What is the probability that we get a sample of MBA salaries from the populatio of last sprig s MBA graduates that gives a that is withi $900 of µ? To aswer this questio we must cosider the samplig distributio of. First, ote that is a radom variable. The radom experimet used to obtai is the process of collectig a radom sample, ad the outcome of the radom experimet (i.e. the sample mea ) is a umerical value. Therefore, is a radom variable ad has a distributio associated with it. This distributio represets the ucertaity regardig the value of that we obtai due to the ucertaity regardig the radom sample we will obtai. The distributio of is N(µ, = ), where is the sample size, = (0,000) is the variace of the populatio, ad = Var( ) = is the variace of the sample mea. The distributio is draw below. 4

5 = µ Ituitio uderlyig the distributio of The distributio of has a mea of µ. The reaso is that there is a 50/50 chace we will obtai a sample of MBA salaries from the MBA populatio that is weighted towards good studets (i.e. there are more good studets i the sample whose salaries are above the populatio mea µ tha poor studets whose salaries are below the populatio mea µ), ad therefore there is a 50/50 chace the sample average is above the populatio mea µ. This is represeted i the samplig distributio because half the area i the distributio for is above µ, i.e. half the time we get a sample that gives a greater tha µ. A similar argumet ca be used to explai why half the area i the distributio for is below µ. Aside There is a theorem i statistics called the Cetral Limit Theorem. It says that if ~ N(µ, ), the ~ N(µ, = ). This theorem backs up the ituitio we developed i class. It is a formal statemet of the ituitio (which is what all theorems are). Ed Aside Suppose we collect a sample of size = 00. Give the samplig distributio for is N(µ, = = (, ) =,000,000 = (,000) ), we ca compute the probability 00 that we get a sample of MBA salaries that give a withi $900 of µ (i.e. we ca compute the probability that we get what we defie to be a good estimate of µ). 5

6 = 0,000 = = µ-900 µ µ+900 ( µ 900) µ µ ( µ + 900) µ pr(µ < < µ + 900) = pr < < = pr(-0.9 < Z < 0.9) = pr(z < 0.9) - pr(z < -0.9) = = () Suppose we collect the followig sample of = 00 salaries from the MBA populatio: The otatio we will use is the followig: i represets the salary of the i-th perso i the sample. Thus, = $56496 is the salary of the first perso i the sample, = $6446 is the salary of the secod perso i the sample,, 00 = $7786 is the salary of the last perso i the sample. The sample mea is 00 = i = i =

7 Thus, = $645 is our estimate of the populatio mea µ. We do t kow whether = $645 is close to µ or ot because we do t kow µ. If we kew µ we would ot have to bother estimatig it. However, we ca make the followig statemet based o the probability calculatio i equatio (). Of all the possible samples of size = 00 we could collect from the populatio of MBA graduates, 63.% of them give a withi $900 of the populatio mea µ. 36.8% of the possible sample we could collect give a more tha $900 from the populatio mea µ. You do t kow which kid of sample you get (i.e. you do t kow whether you get oe of the 63.% that give a withi $900 of µ or oe of the samples that gives a more tha $900 from µ). However, i my opiio, a 36.8% chace of failure is too high (i.e. a 36.8% chace of gettig a bad estimate is too high). To icrease the probability of gettig a good estimate we eed to icrease the sample size. If we icrease the sample size we are collectig more iformatio about the populatio mea µ so we should get a better estimate. This will be reflected i a smaller probability of collectig a sample that gives a more tha $900 from µ. Ituitively, with a larger sample (say = 400) we are less likely to get a strage sample. A large sample is more likely to be very represetative of the populatio. For example, we are highly ulikely to get all = 400 people i the sample from good schools. Give the samplig distributio for is N(µ, = = (, ) = 50,000 = 400 (500) ) whe = 400, we ca ow compute the probability that we get a sample of 400 MBA salaries that give a withi $900 of µ (i.e. we ca compute the probability that we get what we defie to be a good estimate of µ whe = 400). = 0,000 = = µ-900 µ µ+900 ( µ 900) µ µ ( µ + 900) µ pr(µ < < µ + 900) = pr < < = pr(-.8 < Z <.8) 7

8 = pr(z <.8) - pr(z < -.8) = = 0.98 () We ca ow make the followig statemet based o the probability calculatio i equatio (). Of all the possible samples of size = 400 we could collect from the populatio of MBA graduates, 9.8% of them give a withi $900 of the populatio mea µ. 7.% give a more tha $900 from the populatio mea µ. Therefore, we ca be highly cofidet that the we obtai from a sample of size = 400 will be withi $900 of µ. ESTIMATING THE POPULATION VARIANCE The logic for estimatig the populatio variace is the same as the logic used to estimate the populatio mea µ. The three steps are the followig: () The sample is represetative of the populatio. () The sample dispersio is represetative of the populatio dispersio ( ) The questio is how to measure the dispersio of the sample. A atural way to do this is to use the average distace squared that each poit i the sample is from the ceter of the sample. The distace from the i-th poit ( i ) to the ceter of the sample ( ) is ( i - ). Thus, the average distace squared is s = ( i ) Rather tha divide by, we divide by -. Ituitio tells use to divide by but some mathematics (which we will ot discuss ad you are ot resposible for) tells us to divide by -. Therefore, we will use s = ( i ). to represet the dispersio of the sample. (Techical poit: We use average distace squared istead of average distace because if we added up all the distaces, they would always add to zero, which meas the average distace would also be zero. This would 8

9 clearly be a bad measure of dispersio. To avoid the problem of positive ad egative values cacelig out, we use distaces squared.) (3) The sample variace ( s ) is a good estimator of the populatio variace ( ) The reaso is that s is a measure of the dispersio of the sample, ad the dispersio of the sample is represetative of the dispersio of the populatio. Thus, s is represetative of the populatio dispersio, ad is therefore a good estimator of. Example Cosider the MBA salary example. The salaries i the sample we collected are give o page 6. The followig dotplot of the salaries (which is ot from Excel output) gives a feel for the dispersio of the sample... :.: :. : :...::. :..:.: :.... ::. :::.::::::.::::: :::::::::...: :: Salary For this sample, = 645 ad s = ( i ) = 00 ( i ) 00 = = (973). Thus, s = (973) is our estimate of the populatio variace. The sample stadard deviatio (called s ) is s =

10 EPLANATION OF THE THREE TYPES OF VARIANCES The three types of variaces are: () : is the populatio variace. is the populatio stadard deviatio. The populatio stadard deviatio ad populatio variace provide a measure of the dispersio of the populatio. 68% of the probability falls withi of the populatio mea µ, while 95% of the probability falls withi of the populatio mea µ. For example, i the MBA salary example, = $0,000 so 68% of the MBA populatio makes withi $0,000 of the average salary of all MBAs i the populatio, while 95% of the MBA populatio makes withi $0,000 of the average salary of all MBAs i the populatio. () s = ( i ) : s is the sample variace. s is the sample stadard deviatio. The sample stadard deviatio ad sample variace provide a measure of the dispersio of the sample. Because the sample is represetative of the populatio, the sample variace s is represetative of the populatio variace, ad therefore s is a estimator for. (3) = : is the variace of the sample mea. It provides a measure of the ucertaity regardig the value of the sample mea we obtai from our radom sample. is the stadard deviatio of the sample mea. Of all the possible samples we could collect from the populatio, 68% of the them will give a withi of the populatio mea. For example, i the MBA salary example (with = 00 ad = = (, ) =,000,000 = (,000), so 00 = 000), 68% of the possible samples we could collect from the populatio will give a withi = $000 of µ. This provides a measure of the quality of as a estimator of µ. To summarize, is a measure of the dispersio of the populatio ad s is a measure of the dispersio of the sample. is a measure of the quality of the sample mea as a estimator of the populatio mea µ. The populatio stadard deviatio = $0,000 tells us the probability that the salary of a sigle radomly chose graduate from the populatio beig withi $0,000 of the populatio average salary (µ) is 68%. 0

11 ( 0000) The stadard deviatio of the sample mea = = = 000 tells us 00 the probability that we collect a sample of size = 00 from the populatio that gives a sample mea withi = $,000 of µ is 68%. The sample stadard deviatio s is a estimate of.

12 NOTATION () ~ N(µ, ): is ormally distributed with populatio mea µ ad populatio variace. () µ: Populatio mea. µ represets the ceter of the populatio. It is the value of that we expect o average. For example, i the MBA salary example, µ is the average salary of all MBAs i the populatio. (3) : Populatio variace. provides a measure of the dispersio of the populatio. See the descriptio o the previous page. (4) : Populatio stadard deviatio. also provides a measure of the dispersio of the populatio. It is easier to iterpret tha the populatio variace because the uits are appropriate, e.g. dollars (ot dollars squared) i the MBA salary example. (5) i : i-th value i the radom sample. For example, i the MBA salary example, represets the salary of the first perso i the sample, represets the salary of the secod perso i the sample, etc. (6) = i : Sample mea (or equivaletly, sample average). represets the ceter of the sample. It is a estimator for the populatio mea µ. (7) ~ N(µ, =. (8) = : previous page. = ): is ormally distributed with mea µ ad variace is the variace of the sample mea. See the descriptio o the (9) s = ( i ) : s is the sample variace. It represets the dispersio of the sample ad is a estimate of. See the descriptio o the previous page. (0) s = s = i ( ) : s is the sample stadard deviatio. s also provides a measure of the dispersio of the sample. It is easier to iterpret tha the sample variace because the uits are appropriate, e.g. dollars (ot dollars squared) i the MBA salary example.

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