Examination Number: (a) (5 points) Compute the sample mean of these data. x = Practice Midterm 2_Spring2017.lwp Page 1 of KM

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1 Last Name First Sig the Hoor Pledge Below PID # Write Your Sectio Number here: Uiversity of North Carolia Ecoomics 4: Ecoomic Statistics Practice Secod Midterm Examiatio Prof. B. Turchi Srig 7 Geeral Istructios: Aswer all five (5) questios o this examiatio, writig your aswers o the exam aer itself. Use the back of the ages for ay extra work, if ecessary. Sig the Hoor Pledge above. Exress all aswers to a recisio of at least 3 decimal oits. Show your work to be eligible for artial credit. Be sure to ote that tables ad formulas are o the last ages of the exam.. (3 oits total) Last week a bomb exloded i a crowded Baghdad market lace killig ad woudig a umber of civilias. The Iraqi govermet blamed America cruise missiles for the attack; however, the U.S. forces sokesma i Qatar raised doubts, sayig that the U.S. does ot target civilia areas ad that the accuracy of the missiles is so good that the Iraqi claim is doubtful. The market was about 5 meters beyod the most likely target, a ower lat ad the missiles were fired from a distace of km. (see icture below) from the target. 5 meters KM Marketlace Target (ower lat) Missile Laucher Suose that the U. S. govermet sulies you with the followig test data o the accuracy of cruise missiles fired from a distace of km: These data show the umber of meters short or log of the target resultig from the test firig of 6 cruise missiles i tests i the Nevada desert. Assume that a missile must actually hit the target to damage it. Aswer the followig questios. (a) (5 oits) Comute the samle mea of these data. 6 å xi i.8 = = = x = Practice Midterm _Srig7.lw Page of

2 (b) (5 oits) Comute the samle stadard deviatio of these data 6 å ( ) x -x i i= s = = = Þ s= (c) ( oits) Comute the 95 ercet cofidece iterval for the samle mea. Is ero icluded i the cofidece iterval?_yes_ [ ] x ± ta/sx = x ± t.5sx = 6.38 ± = 6.38± Þ ,7.578 (d) ( oits) Here is the ormal robability lot for these data. Do the data look ormally distributed? Yes Is it likely that they come from a ormally distributed oulatio of targetig errors? Yes dist Iverse Normal (e) ( oits) If the errors are, i fact, ormally distributed with a mea =, what is the robability (based o this samle) that a missile could have reached the market or beyod? (You ca assume that the samle stadard deviatio, s, equals the oulatio stadard deviatio, r. Now we re dealig with the robability distributio of the radom variable x: the distace i meters log or short. The first thig we do is comute the -value for a distace of +5 meters: Practice Midterm _Srig7.lw Page of

3 = 5 meters 5 - = = Note that this assumes that the uderlyig oulatio mea =. I do this because the cofidece iterval cotais ero ad ay reasoable hyothesis test could ot reject the ull hyothesis that l =. I m sure that may studets will use the samle mea istead of ero. I do ot cosider this as correct; however, they should oly receive a small ealty for comutig a -value usig l = I ll show that calculatio below. The, to comute the robability of beig at least 5 meters log, we have:..... =.3849 = =-. Þ.7. = =.3863 Þ =.37 5 m..37 ( x ) \ Pr ³ 5 =.37.7 Alterate calculatio: Alterate usig samle mea: = = =.3766 Pr ³ 5 = =.384 ( x ) 5 m Gradig Stadard: Oe oit should be deducted if the studet did the etire calculatio correctly but used 6.38 istead of Practice Midterm _Srig7.lw Page 3 of

4 . (8 oits total) Husbads ad wives ofte disagree about how much housework the husbad does aroud the house i a give -week eriod. Below you will fid a table cotaiig data from ideedet samles of husbads ad wives. Each samle cotais resodets. Samle is husbads who reort o their housework i a tyical -week eriod. Samle is wives (ot ecessarily married to the husbads i samle ) who reort o the housework erformed by their husbads i the same two week eriod. At the a =. level ca you reject the ull hyothesis that, o average husbads ad wives reort the same umber of hours versus the alterative that husbads reort more hours? (a) erform this hyothesis test assumig equal oulatio variaces; the (b) erform the test assumig uequal oulatio variaces. Do your results differ deedig uo which assumtio you make about the resective oulatio variaces? For arts (a) ad (b) be sure to show your calculatios of test statistic, critical values ad degrees of freedom. Show all your work icludig all relevat iformatio for erformig the sigificace test. Husbads Wives This roblem asks the studet to test the hyothesis that samle meas from two oulatios are the same vs. the alterative that the husbads estimates of housework are greater. Because the husbads ad wives are ot ecessarily married to each other, the samles are ideedet ad we ca use the t-test for ideedet samles: However, sice we are usure as to whether the uderlyig stadard deviatios from the two oulatios are the same, art (a) asks for the calculatio usig ooled variaces (assumed equal) ad art (b) asks for the same test assumig uequal variaces. (a) The roer formulas are: Practice Midterm _Srig7.lw Page 4 of

5 t = s ( x-x) -( m-m) s ( ) + ( ) = df = + -. ( - ) + ( -) s s + - t=.849 df = s =.49 critical t(.)=.3 Caot reject Ho (b) For the uequal variace t-test, the roer formulas are: t =.849 delta (df) = ~9 critical t(.) =.38 caot reject Ho ( - ) -( m-m) ( s ) + ( s ) t= x x D= é( s ) ( s ) ù ë + û ( s ) ( s ) rouded dow. While the results are slightly differet, we caot reject the ull hyothesis i either case. Practice Midterm _Srig7.lw Page 5 of

6 3. (8 oits total) Provide defiitios for: a) Simle radom samlig: Simle radom samlig: A samlig rocedure for which each ossible samle of a give sie is equally likely to be the oe obtaied b) Samlig error: Samlig Error: Samlig error is the error resultig from usig a samle to estimate a oulatio characteristic. c) The differece betwee robability distributios ad samlig distributios: A robability distributio of a radom variable gives, for each value of the radom variable, the robability that a give value of the radom variable will be obtaied. The Samlig Distributio of a Samle Statistic shows, for a samle statistic comuted from a give samle sie, the robability distributio of all ossible values of the samle statistic that ca be comuted from samles of that sie. So, for a variable x-bar ad a give samle sie, the distributio of the variable that is, the distributio of all ossible samle meas -- is called the samlig distributio of the samle mea. d) The Power of a Hyothesis Test: The robability of ot makig a tye II error (i.e., it is the robability of rejectig a false ull hyothesis) e) Ubiasedess of a estimator: Ubiasedess: A estimator is said to be ubiased if the exected value of the estimator is equal to the arameter beig estimated, or [ Q] E $ = Q. f) Efficiecy of a estimator Efficiecy: smallest variace. The most efficiet estimator amog a grou of ubiased estimators is the oe with the g) -value of a hyothesis test: Defiitio: The P-value of a hyothesis test equals the smallest sigificace level at which the ull hyothesis ca be rejected, that is, the smallest level for which the observed samled data results i rejectio of Ho. (Problem #4 o the ext age) Practice Midterm _Srig7.lw Page 6 of

7 4. ( oits total) Suose the Carolia Poll wats to gauge studet oiio regardig the resigatio of Matt Doherty as UNC s me s basketball coach. a) To obtai a estimate of the roortio favorig the resigatio, how may studets would have to be samled if the maximum error is to be.5 at a level? a =.5 a) We ca use the formula with D=.5: 4D = a / a /.96 = = = Þ384 : 385 4D 4.5 ( ) b) Suose the samle is take with 5 radomly selected resodets. If 46 resodets said that they are oosed to the resigatio at what level ca we reject the ull hyothesis that a majority favor the resigatio i favor of the alterative that a majority are agaist the resigatio? H : =.5 where = roortio agaist resigatio H a : >.5 ˆ = = = = Þ a = - = ( - ).5(.5) Therefore, we ca reject the ull hyothesis that a majority are i favor of the resigatio at the a =.394 level. Do ot reject Ho Reject Ho Pr =.394 =.656 Practice Midterm _Srig7.lw Page 7 of

8 5. ( oits) The followig Stata outut shows the regressio model: AvgWaterUse = a + b $ AvgRaiFall + e. as a basis for your aswers. Aswer the followig questios usig these results. regress house rai Source SS df MS Number of obs = F(, 35) =.57 Model Prob > F =.7 Residual R-squared = Adj R-squared =.4 Total Root MSE = house Coef. Std. Err. t P> t [95% Cof. Iterval] rai _cos (a) Draw a causal diagram ad exlai what this regressio is attemtig to discover. (a) The causal diagram would look like: Rai Huse This regressio is attemtig to determie if there is a causal relatioshi betwee average raifall ad water use by households. (b) What roortio of the total variatio i water use does raifall exlai? (b) The R =.5, which says that oly about.% of the variatio i water use is accouted for by the regressio. (c) Does raifall have a sigificat imact o the level of water use? Why/Why Not? (c) No, raifall aears ot to have a sigificat effect o water use. The coefficiet o "rai" is small (-.3933) ad it has a small t-statistic, which allows us to reject the ull hyothesis at oly the.% level. Practice Midterm _Srig7.lw Page 8 of

9 t = s = ( x-x) -( m-m) s ( ) + ( ) df = + -. ( - ) + ( -) s s + - ( - ) -( m-m) ( s ) + ( s ) t= x x D= é( s ) ( s ) ù ë + û ( s ) ( s ) rouded dow. ˆ x = + x + æ x - m Prç- ta < < t è s/ ö = -a ø / a/ = = ˆ ˆ - ˆ - ˆ - + ( ˆ ) ( - ) s = D a/. 4D = a/ Practice Midterm _Srig7.lw Page 9 of

10 N å i= s = ( x -m) i N ( x) C C = = r N -r x -x N C r N -r ( x)( -x) N ( ) æ r ö Mea: m = ç è N ø æ r öæ N - r öæ N - ö Variace: s = ç ç ç è N øè N øè N - ø Stadard deviatio: s = s ( ) ( ) f x~ - x-m / s = e - < x < s ì ï f( x) =í ïî ( ) P A B i ( b-a) = å all ( i) ( i) PBA ( ) PA ( ) PBA PA k, a x b, otherwise k ( b-a) m = ( b+ a) ad s = c s ( - ) - = s k - x ìle l, l >, x ³ f() x = í î, otherwise m = ad s = l l P( x a) -la ³ = e, a ³ ad l > F x ( lt) for x,,,,, l, ì t e - l, = K ï > Px ( ) = í x! ïî, otherwise. s / s A A ( A-; B- ) = sb / sb [ ] l = the mea umber of evets i a give segmet of time ( t = ) ( t ) t = the legth of a articular subsegmet E x = m = lt = the exected umber of evets i oe subsegmet legth t x Practice Midterm _Srig7.lw Page of

11 Critical values of c a a c.995 c.99 c.975 c.95 c df c. c.5 c.5 c. c.5 df Practice Midterm _Srig7.lw Page of

12 F-Distributio Table: Uer 5% Probability (or 5% Area) uder F-distributio Curve f(f) F= df=d=5 D= F F-Table for alha =.5 / df= if df= if Practice Midterm _Srig7.lw Page of