MR. STEIN S WORDS OF WISDOM

Transcription

1 MR. STEIN S WORDS OF WISDOM P a g e 1 I am writig this review essay for two tests- the AP Stat exam ad the Applied Stat Fial exam. The topics are more or less the same, so reviewig for the two tests should be a similar process. Either way, you are about to take a comprehesive exam. The AP exam will probably be easier tha our fial, but regardless you have a great umber of ideas that eed to be up frot i your brai. That is the plai truth. While I hope this geerates some degree of healthy ervousess i you, it should ot scare you. I a way, I thik you should be lookig forward to this test. You are well prepared!!! You have studied every topic that will be tested. Your job will be to look at some questios that iitially may seem cofusig to you ad to figure out which of the topics you kow apply to that questio. The Forest ad the Trees I wated to give you what I thik are the importat ideas of this course. These ideas make up the forest. I doig so, I will leave out a lot of the details or what I call the trees. I am ot sayig the details are uimportat. Obviously, without them you caot solve some of the problems. But I thik most of you kow most of the details (although some hard-core studyig over the ext few days will surely help). I wat to make sure that the big ideas are frot ad ceter i your mid as you study for the last few days ad go i to take your exam. Describig Distributios Remember that if you are asked to describe a data set or, more likely, to compare two or more data sets, you must always commet o ceter, spread, ad shape. Ceter ca be expressed with the mea (for symmetrical, well-behaved data) or media (for o-symmetrical or otherwise aughty data). Spread ca be expressed with the stadard deviatio (for symmetrical, well-behaved data) or IQR (for o-symmetrical or otherwise aughty data). Make sure you do ot represet a data set as NORMAL. It ca be approximately ormal, but the ormal distributio is theoretical oly. Do t cofuse Skewed Left with Skewed Right Remember Skewess is Fewess Remember that meas ad stadard deviatios are less resistat- they move towards outliers ad ifluetial poits. Make sure your descriptios are i the cotext of the problem. If the problem tells you a populatio is ormally distributed, get out the ormal table ad draw a sketch. If the problem is about samplig more tha oe idividual you must use the stadard deviatio of the samplig distributio, ot the populatio ( see Cetral Limit Theorem o page 5 ) Kow that a z-score is the umber of stadard deviatios above or below the mea. Be able to calculate z-scores, fid probabilities give a z-score ad fid a z-score give a probability.

2 P a g e 2 Be able to make ad describe a boxplot. Kow what Q1 ad Q3 are. Kow that IQR = Q3-Q1. Kow the formula for determiig outliers. Q1-1.5 IQR or Q IQR Regressio There are some basic terms that you should be prepared to idetify ad iterpret Slope-For every icrease of oe uit i x, there is a certai icrease or decrease i y-hat (i y-hat, ot i y). It is how much our model predicts y will chage with oe icrease i x. Y-itercept- What our model predicts whe x is zero r- Correlatio coefficiet. Idicates the stregth of the liear relatioship. Beware- r ca mea othig if the data is ot liear to begi with. R 2 -the percet of variatio i y that ca be explaied by the regressio of y o x. the sum of the squares of the residuals from the regressios lie are R 2 % less tha the sum of the square of the residuals from Residual- y - y-hat. A positive residual meas the lie is uderestimatig that poit, a egative residual meas it is overestimatig that poit Kow the formula. Uderstad the formula s relatioship to the idea of Regressio to the Mea. For a icrease of oe stadard deviatio i x, we predict a icrease of less tha oe stadard deviatio i y (uless, of course, r = +/-1, but that s ot a real-life possibility) Regressio is useful i determiig the degree to which a respose variable ca be predicted by a explaatory variable. It says othig about whether a explaatory variable is causig a chage i the respose variable. To attempt to determie causatio, you eed to perform a cotrolled experimet. Uderstad the role lurkig or cofoudig variables ca play The appropriate of the liear model should be mostly determied by lookig at the data ad lookig at the residuals. Remember that r ad R-square do ot measure how liear the data is ; i fact, they both assume liearity. Do t forget that you whe you are asked if there is a relatioship betwee two quatitative variables you should do a liear regressio t-test (if they are categorical, you should do a Chi-Square Test of Idepedece), Do t just stop with commetig o r or r-squared. Trasformatios This procedure is basically takig some clearly o-liear data, doig some mathematical trasformatio o the x-data ad/or the y-data to make it liear. We ca the do liear regressio ad fially udo the trasformatio. These trasformatios ca be ay mathematical operatio; two of the more commo oes are Power regressio usig the trasformatio (logx, logy) ad Expoetial regressio uses the trasformatio (x, logy)

3 P a g e 3 Experimet ad Survey Desig Please kow the differece betwee a experimet ad a observatioal study. Experimets require a treatmet. Kow what a Simple Radom Sample (SRS ) is: A sample i which each group of size has a equal chace of beig chose. Kow some other radom samplig techiques: systematic, stratified, multistage. Uderstad the differece betwee variatio ad bias. Do you remember the aalogy of the dart board? If you aim towards the ceter but do t hit it every time- that s variatio!! If your aim is off ad all your shots are goig right œ that s bias. Thik about the differet types of bias (udercoverage, o-respose etc.) that we have discussed. Study their ames ad, more importatly what they mea. Uderstad that if asked to carry put a experimet you must iclude (i detail) radomizatio, cotrol, ad replicatio. Be sure that you leave othig the grader s imagiatio. Make sure to discuss how you will aalyze the results. If you are ot asked to metio a specific hypothesis test, the at least metio which results will be compared. Be very clear o the cocept of blockig. We block to cotrol for the variatio caused by a variable other tha the oe we are studyig. We separate our sample ito two or more blocks ad the essetially carry out multiple, idetical experimets. We the compare results withi blocks. Probability Kow the multiplicatio ad additio rules P(A ad B) = P(A) * P(B A) P (A or B) = P(A) + P(B) P(A ad B) Idepedece meas that the outcome of oe evet will ot affect the outcome of the others or P(B A) =P(B) Disjoit (or mutually exclusive) meas that the two evets caot happe simultaeously, or P(B A) =0 Be able to substitute to get special cases of the two geeral rules. For example, if A ad B are idepedet the P(A ad B) = P(A) * P(B), or if A ad B are mutually exclusive the P(A ad B) = 0 Do t forget how to calculate coditioal probabilities. First calculate the space, the ask yourself ( self? ), of these, how may meet some coditio. Never forget that tree diagrams ca really help with some complicated probability problems, particularly coditioal probabilities

4 P a g e 4 If you are asked to calculate a probability through simulatio, you will probably have to use a radom umber table. Make sure that you clearly state your model: which digits represet which outcome, how may digits you will take i each trial, how you will kow whe to stop each trial, ad how may trials you will perform. Probability Distributios of Radom Variables Kow that a probability distributio cosists of all possible outcomes ad each outcome s probability. The mea of a probability distributio (the expected value) is the sum of each outcome times its probability. µ x,= The stadard deviatio of a probability distributio is the square root of the variace Rules for Combiig Meas ad Stadard Deviatios I geeral meas behave exactly as we would expect, while stadard deviatios ca be a little trickier. Let ad Y be radom variables ad let a ad b be costats: if ad Y are idepedet Pay close attetio to the last formula, kow as the Pythagorea Theorem of Statistics (it is the basis for the two sample Z ad T tests). If ad Y are ot idepedet the stadard deviatio of ad Y will go i the directio of the correlatio; it will be higher if ad Y are positively correlated ad it will be lower if ad Y are egatively correlated.

5 P a g e 5 Biomial ad Geometric Radom variables Biomial: ( ) Geometric: Make sure you ca idetify the biomial ad geometric settigs. They both have either success or failure, idepedece, ad the probability of success is always equal. They differ i that biomial deals with how may successes i a fixed umber of trials whereas geometric deals with how may trials util you get a success Be sure you ca use the pdf ad cdf features o your calculator. Pdf calculates the probability of exactly some, cdf calculates some or less. To calculate some or more, you eed to use 1 - cdf. Samplig Distributios Kow the differece betwee a parameter ad a statistic. A parameter is a umerical descriptio of a populatio; a statistic is a umerical descriptio of a sample Uderstad that a samplig distributio is the distributio of statistics from all possible samples from a give populatio. I iferece, we are usig the mea ad stadard deviatio (or Stadard Error if we are approximatig). You eed to kow the followig formulas for the meas ad stadard deviatios of the samplig distributios of sample meas ad sample proportios Remember, if you do t kow p or sigma, you will estimate them with p-hat or s ad calculate the stadard error istead of the actual stadard deviatio

6 P a g e 6 CENTRAL LIMIT THEOREM The CLT says, i essece, that if the sample size is large eough, the samplig distributio will be approximately ormal ad will i fact be ormal as the sample size approaches ifiity. This theorem is why we ca calculate p-values ad cofidece itervals usig a ormal probability table. Cofidece Itervals Be sure you completely uderstad the formula for a cofidece iterval Cofidece iterval = estimate ± critical value * stadard error Make sure you ca calculate a critical value ( Z star or T star) for ay give cofidece level. Completely uderstad the iterpretatio of a cofidece iterval. We are 95% (or whatever) cofidet that the true mea (or whatever) lies betwee # ad #. If we take samples may, may times, 95% (or whatever) of our itervals will capture the true mea (or whatever). Be clear i your mid that is differet tha sayig that the true mea has a 95% chace of beig i the iterval or that this iterval has a 95% chace of capturig the mea. The parameter does t chage, the itervals do. Make sure you remember that assumptios must be checked o cofidece itervals. Uderstad that there are certai trade-offs with cofidece itervals. The larger the cofidece level, the wider the iterval. The larger, the arrower the iterval. Make sure you ca calculate a miimum sample size give a cofidece level ad margi of error Hypothesis Tests Keep i mid the logic of a hypothesis test. We are assumig the truthfuless of the ull hypothesis ad, if so, calculatig the likelihood of this sample occurrig. If the likelihood is low, we will reject the ull hypothesis; if it is ot low, we will fail to reject the ull hypothesis Whe decidig o which test go through the followig decisio process Is the data categorical or quatitative? Categorical - oe sample two categories - Oe proportio Z-Test - Oe sample more tha two categories Chi Square Goodess of Fit - Two samples (or treatmets) two categories 2 proportio z test

7 P a g e 7 - Two (or more) samples ( or treatmets) more tha two categories Chi Square test of Homogeeity - Oe sample, two variables, more tha two categories - chi Square test of Idepedece Quatitative - If you kow the populatio stadard deviatio you ca use Z-tests for meas, but this extremely ulikely, so you almost certaily should be usig a T-test - Oe Sample T-test - Differece betwee two Idepedet Samples 2 sample T-test - Differece betwee two liked samples - Matched Pair T-test - Relatioship betwee two samples- Liear Regressio t-test Make sure each of your hypothesis tests cotais all of the followig steps I. Check (ot just state) assumptios II. Null ad alterative hypotheses III. Correct formulas with correct umbers filled i appropriately IV. p-value ad decisio V. Iterpretatio i the cotext of the problem. Make sure this iterpretatio explais the coditioal probability that you have calculated. There are some defiitios that are very importat for you to uderstad. p- value- the probability of gettig results equal or more extreme as the sample assumig the ull hypothesis is true p-value - the probability of falsely rejectig the ull hypothesis p-value - the probability of makig a Type I error Type I error - Rejectig the ull hypothesis whe it is true Type II error- Failig to reject the ull hypothesis whe it is false Alpha - the probability of a Type I error Beta - The probability of a type II error Power -The complemet of beta ( 1 - beta) Power- the probability of rejectig the ull hypothesis whe it is false. Keep i mid that there are some trade-offs here. The lower the alpha, or the sigificace level, the higher the beta. We ca lower beta (ad raise power) without adjustig alpha by icreasig the sample size. This requires more work o the experimeter s part, so that also is a trade-off. Be able to calculate beta for a give alterative value of the parameter (ot a AP topic) Here are all of the details of the various procedures ad a example of a Miitab output

8 P a g e 8 T-test SRS, id Pop is ormal Or is large Or check data 1-Proportio Z-test SRS, id Pop size>10 p>10 (1-q)>10 2-Sample T-test 2 id. SRS s or radom assigmet of treatmets Pops are ormal Or is large Or check data Matched Pair T-test 2 liked SRS s Pop is ormal Or is large Or check data 2-Proportio Z -test 2 id SRS s or radom assigmet of treatmets pop s>10 s Ho: μ= # Ha: depeds o questio Ho: p = decimal Ha: depeds o questio Ho: 1 2 or 12 0 Ha: depeds o questio Ho: diff 0 Ha: depeds o questio Ho: p1 p2 or p1 p2 0 Ha: depeds o questio Picture Put Ho # i the middle Shade from Estimate: s T= Cofidece. It t * s Put Ho dec. i the middle Shade from Estimate: ˆp p(1 p) p p Z= Cofidece It * (1 ) Z Put zero i the middle Shade from estimate: 2 2 s s 1 2 ( 12) 0 t 1 2 Cofidece It 2 2 * s s ( ) t Put zero i the middle: Shade from estimate: diff sdiff diff Test Statistic diff 0 t Cofidece It diff t s * diff Put zero i the middle Shade from estimate: Formulas (1 ) (1 ) Test Statistic ( ˆ 1 p2) 0 Z 1 2 Cofidece Iterval ( ) Z (1 ) (1 ) * 1 2 -To fid t-star, use -1 for the df -If you kow sigma, you ca use a z-test For CI, assumptios ad SE have p-hat istead of p. For df. Either use exact value from calculator or use the smaller of the two df s Notes; ALL work should be doe o the differeces. Do ot use orig data For assumptios ad SE formula :I hyp test, used pooled ˆp ; for cof-its use ˆp ad ˆp 2 1

9 2 GOF 2 Test of Idepedece SRS SRS All expected All expected cells cells greater greater tha 5 ( is tha 5 ( is big big eough) eough) Test of Homogeeity SRS All expected cells greater tha 5 ( is big eough) 2 Liear Regressio T test P a g e 9 SRS Li. model appropriate (look at scatterplot) St. dev aroud the lie is costat (looka t residual plot) Residuals ormal(uless is big draw resids : Ho: Data fits expected distributio Ha: it doest Ho: 2 vars are id Ho: 2 vars are related Ho: proportios are equal Ha: they re ot Ho: Ha: depeds o questio Picture Picture 0 to the left Shade to the right of the 2 statistic 0 to the left Shade to the right of the 2 statistic 0 to the left Shade to the right of the 2 statistic 0 i the middle Shade from Estimate: b 2 = 2 = 2 = Cofidece. It NONE Cofidece It NONE Cofidece It NONE Cofidece Iterval b ± t* - df = # of categories -1 df= (R-1)(C-1) df= (R-1)(C-1) df = 2 You will almost certaily do this from computer output. Be familiar with output (see followig page)

10 P a g e 10 Regressio Aalysis The regressio equatio is weight = xxxx + xxxx legth Predictor Coefficiet St Error T P Costat Legth S = R-Sq = 12.2% R-Sq(adj) = 1.6% Aalysis of Variace Source DF SS MS F P Regressio Residual Error Total Please Note: This is oe of may computer packages that you may see. But they should be similar eough for you to fid the iformatio you eed

11 P a g e 11 Coclusios: Here are some more geeral poiters for you: O the AP, do problem #1 first, it should be easy ad straight forward. The move to questio #6; it is loger ad worth twice the poits. It also may cotai parts that are ot exactly what you have doe before. You should be able to do these, but they might require a little more thought. The proceed to fiish problems 2-5. Do ot assume that you kow what the test is askig you. Read the etire questio (all parts) ad the aswer choices before you aswer. Pay attetio to key words (ormally distributed, predictio, idepedece, ull hypothesis, etc.) O multiple choice, use process of elimiatio. Focus o the differece betwee the aswer choices. Do t be scared off by log ad wordy multiple choice questios. Usually several of the aswer choices are obviously icorrect. Focus o key words to decide betwee the remaiig choices. Do ot skip multiple choice questios. There is o additioal pealty for gettig them wrog. O part II aswer exactly what is beig asked. Give solid statistical reasoig for your aswers. Use hypothesis tests, cofidece itervals, regressio etc. Whe usig formulas, write dow the formula, show how the umbers are plugged i, ad the use the calculator to come up with the fial aswer. If you are ruig out of time, skip the calculatios ad iclude assumptios, Ho ad Ha ad coclusios,. This will get you most of the poits. Try ot to leave ay part II questios blak. If you do t uderstad what to do, try to get at least oe poit. If you eed a aswer from a previous part which you could t do, make up a aswer ad cotiue with the subsequet parts Keep tellig yourself (self): I am well prepared. I have take a rigorous college-level statistics course. I kow what I am doig. I just have to coect some piece of kowledge i my head with the questio i frot of me. GOOD LUCK!