Test of Statistics - Prof. M. Romanazzi

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1 1 Uiversità di Veezia - Corso di Laurea Ecoomics & Maagemet Test of Statistics - Prof. M. Romaazzi 19 Jauary, 2011 Full Name Matricola Total (omial) score: 30/30 (2 scores for each questio). Pass score: 18/30. Portable computer, computer programs (specifically, R program): allowed; textbook or class otes: ot allowed. Detailed solutios to questios must be give o the draft sheet (foglio di brutta copia); fial aswers/results must be copied o the exam sheet, ear to the small squares. Mark below your choice about oral discussio (required whe the total score is betwee 18 ad 21). Default: YES. Oral Discussio Optio NO YES Questio Score Questio Score Questio Score Total score

2 2 1. A ad B are two radom evets satisfyig P (A) = 0.5, P (B C ) = 0.4 ad P (A B) = 0.9. What is the probability of A B? Are A ad B stochastically idepedet? P (A B) = No, they are depedet because Yes, they are idepedet because 2. I a class of 20 studets there are 8 males. Suppose to draw at radom ad without replacemet a sample of 3 studets. What is the probability of observig at least oe female i the sample? 3. The probability of a studet to pass Statistics is kow to deped o whether the studet has already passed Mathematics. Deote with S the evet: the studet passed Statistics ad with M the evet: the studet passed Mathematics. From past data P (M) = 0.6, P (S M) = 0.8 ad P (S M C ) = 0.3. Use these data to evaluate P (S) ad P (M S). P (S) = P (M S) = 4. Compute expectatio ad media of the radom variable X with pdf { 2x, 0 x 1, f X (x) = 0, elsewhere. Is there ay differece? Explai thoroughly. Expectatio: Commets: Media: 5. The legth R of the radius of a radom circle has a uiform distributio R(0, 1). What is the expectatio of the legth of the circumferece C = 2πR ad of the area of the circle A = πr 2? E(C) = E(A) = 6. Mr Smith usually goes to work by trai ad bus. Trai A takes him to C*** where he stops ad catches bus B to the work place. Let T A ad T B deote the times (miutes, waitig times are take ito accout) spet by A ad B, respectively, ad assume they are idepedet ormal radom variables, T A N(µ A = 20, σ A = 5), T B N(µ B = 10, σ B = 3). Let T be the total time spet by Mr Smith to go to work. What is the distributio of T? What is the probability that T > 35 miutes? Distributio of T : P (T > 35) = 7. I the game of darts, a player hits the target with probability 0.6 i each throw. Cosider a sequece of 10 idepedet throws by this player. Compute the probability of a) o hit ad b) exactly 5 hits. Probability of o hit: Probability of exactly 5 hits: 8. You are i a queue at the post office ad 15 customers must be served before you by just oe clerk. Service times (miutes) X 1,..., X 15 are assumed to be pairwise idepedet with the same expoetial distributio Exp(λ = 1/4). What is your expected waitig time? What is approximately the probability that your waitig time is less tha 45 miutes? Expected waitig time: Probability of a waitig time less tha 45 miutes:

3 3 9. The plot below shows the probability desity fuctios of a stadard ormal distributio, a Studet t distributio with 3 degrees of freedom ad a Studet t distributio with 6 degrees of freedom. Which distributio has the highest stadard deviatio? Explai briefly. PDF f(x) A B C Value of x A: B: C: Commets: 10. The stem-ad-leaf display shows the total fertility rate (TFR, childre per woma) of a sample of = 20 italia provices i Compute quartiles, IQR ad feces ad report the results i the table below = is read i=1 x i = i=1 x2 i = Sample Statistic Q 1 Q 2 Q 3 IQR Left Fece Right Fece Value 11. Cosider agai the fertility data. Draw the boxplot i the space below ad describe the distributio features: locatio, shape, outliers.

4 4 Distributio features: 12. Cosider agai the fertility data ad deote with µ the mea of TFR i the populatio of all italia provices i Derive the cofidece iterval for µ(cofidece level: 0.95). 13. The demography of firms is a ew field of ecoomic research dealig with the life cycle of firms. The mai idicators are birth ad death rates, that is, the umber of ew firms ad the umber of closed firms i the referece time uit. The figure below shows the the birth (X, %) ad death (Y, %) rates of a sample of = 21 italia provices i 2008 ad the table provides some data summaries. Compute the sample meas, the stadard deviatios ad the liear correlatio. i=1 x i i=1 y i i=1 x2 i i=1 y2 i i=1 x iy i x y s X s Y r X,Y Demography of Firms, Italy 2008 Death Rate (%) Birth Rate (%) 14. Cosider agai the birth ad death rates of firms. Cosider the liear predictio model y = α 0 +α 1 x ad derive the least squares estimates of the parameters. What is the estimated goodess of fit of

5 5 the model? Estimates of the parameters: Goodess of fit: 15. Cosider agai the birth ad death rates of firms. Like huma populatios, the growth rate of a populatio of firms is defied to be the differece betwee the birth ad death rates. Deote with G = X Y this ew variable ad let µ G be the correspodig mea i the populatio of all italia provices i Cosider the statistical hypothesis H 0 : µ G = 0, H 0 : µ G < 0. Suggest a test statistic, describe its distributio uder H 0 ad compute the p-value. observed sample support H 0? What is your iterpretatio? Test statistic: Distributio uder H 0 : p-value: Commets: Does the

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be