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1 Eamiatio Number: Last Name First Sig the Hoor Pledge Below PID # Write Your Sectio Number here: Uiversity of North Carolia Ecoomics 400: Ecoomic Statistics Secod Midterm Eamiatio Prof. B. Turchi Aril 6, 07 Geeral Istructios: Aswer all five (5) questios o this eamiatio, writig your aswers o the eam aer itself. Use the back of the last age of the eam for ay etra work, if ecessary. Sig the Hoor Pledge above. Eress 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 4 ages of the eam. Radom Samle of Bod Yields i Percet Feb 05 to Feb MuiBods CorBAA CorAAA I was recetly aroached by a stock/bod salesma who made claims about three differet bod fuds that he would like to sell me. Oe bod fud cosisted of a collectio of Trile A rated cororate bods (CorAAA). The salesma claims that, over the ast two years, the average yield of these bods was at least 4%. I oly have access to a radom samle of those bod yields as show i the table above. Below, comute the samle mea ad samle stadard deviatio from the CorAAA series ad test the ull hyothesis that the oulatio mea yield is greater tha or equal to 4% agaist the alterative that it is less tha 4%. a) (4 oits) Mea CorAAA bod yield: (Show how you got the aswer) = åcoraaai = i= b) (4 oits) Samle stadard deviatio of CorAAA bod yield: (Show how you got the aswer) mtms7.lw Page of

2 Eamiatio Number: s å ( -) i i=.580 = = = c) (4 oits) Test the ull hyothesis that the mea yield of CorAAA bods is greater tha or equal to 4.0% agaist the alterative hyothesis that it is t. Use a sigificace level of a = 0.05 ad show your work. Ca you reject the ull hyothesis? The test statistic for the ull hyothesis is: t df = -m = = = = =-.53 s s The critical value for the left tail test is for degrees of freedom - ad alha = 0.05: Left-tailed hyothesis test: t-distributio Deg. of freedom t.05 =-.796 Test Statistic = t-value d) (4 oits) Based o this iterval is the salesma's claim credible? Elai. Usig ay tools you have available, estimate the -value at which the ull hyothesis could be rejected. We ca reject the salesma s claim that the yield of CorAAA bods is 4.0 ercet. The test statistic falls outside the do ot reject regio for alha = Lookig at the t-distributio table for df =, we see that the t-value for alha = 0.05 is.0, so give our test statistic of.53, we ca see that the -value has to be slightly larger that 0.05 (It s actually 0.07, but the studets robably would t be able to figure that out.). e) (4 oits) What did you assume about the distributio of the oulatio data before costructig your do-ot-reject iterval? How might you have tested this assumtio? mtms7.lw Page of

3 Eamiatio Number: I assumed that the oulatio data were ormal. This allowed me to use the t-distributio to costruct the cofidece iterval. I could have used qorm, the quatitative ormal commad i Stata.. The bod salesma also claimed that, for ersos who are lookig both for higher yields ad more security i their ortfolios, that Cororate BAA grade bods are suerior to muicial bods because they ehibit lower volatility (i.e., lower variace) ad higher yields tha do muicial bods. Assumig that the CorBAA ad MuiBod data are from ideedet samles: a) (0 oits) Test the hyothesis at the a = 0.05 level that CorBAA ad MuiBod securities ehibit the same variace agaist the alterative that MuiBod securities have a higher variace? Ca you reject the ull hyothesis? Why/Why ot? Show all your work ad elai. F = s / s s / s / = s / = = = A A A A A, sb sb sb sa sb The critical value for this F-test.8536 ad the F-ratio value above is ot large eough to reject the ull hyothesis of equality of variaces. b) (0 oits) Based o your coclusio i art a) erform a hyothesis test o the ull hyothesis that mea yields o CorBAA ad MuiBod securities are the same versus the alterative that they are ot. Assume ideedet samles. Ca you reject the ull hyothesis? Elai ad show the stes you used to comlete your work. We caot reject the hyothesis that the variaces of the two oulatios are equal; therefore, we move ahead with a t-test of the differece betwee the mea yields, assumig equal variaces. That requires first a comutatio of the ooled stadard deviatio of the two series. s ( ) s ( ) s ( ) ( ) = = = = + - Net, because we re usig samle variaces, we have to comute the t-statistic ( ) ( m m) s ( ) + ( ) ( ) t = = = Fially, we have to fid the critical values at a sigificace level of 0.05, ad degrees of freedom + - = 4 - =.!.074 The critical values are ad we ca reject the ull hyothesis at well below the 5% level. NOTE: if studets use the uooled t-test rocedure erfectly correctly deduct oits for ot usig the ooled versio. mtms7.lw Page 3 of

4 Eamiatio Number: 3. (a) (3 oits) Write the formula for the samle roortio draw from a samle of size. The samle roortio draw from a samle of size is: (b) ˆ = where is umber of "successes." (7 oits) For a samle of size, rove that the samle roortio is a ubiased estimator of the oulatio roortio. E[ / ] = [ ] ( ) E = =. (c) (7 oits) For the samle roortio to be a cosistet estimator of the oulatio roortio, it must also be true that the variace of the samle roortio teds to zero as samle size goes to ifiity. Prove that the variace of the samle roortio goes to zero as samle size goes to ifiity. [ ] [ ] V / = V = ( ) q q q = Þ lim æ è ç ö = 0 ø (d) (3 oits) A biased estimator of a oulatio arameter ca ever be cosistet. True or False? Why? This statemet is false. Cosistecy is the roerty of a estimator as samle size aroaches ifiity. For a estimator to be cosistet, its eected value must aroach the oulatio value ad its stadard error must aroach zero i the limit. A cosistet estimator s small samle roerty may be either biased or ubiased as log as it aroaches the oulatio arameter i the limit. 4. As a ivestigative reorter for the Daily Tar Heel, you are istructed by your editor to survey the studet oulatio i order to determie whether me ad wome agree that seual assault is a roblem o the UNC camus. Each resodet is asked if he/she believes seual assault is a roblem o camus. Code a yes = ad a o = 0. a) (5 oits) First, you eed to geerate two ideedet samles of me ad wome. You wat your margi of error to be o more tha 0. for a 95% mtms7.lw Page 4 of

5 Eamiatio Number: cofidece iterval. How large must each samle be? (Roud dow to the earest whole umber) The aroriate samle size is comuted usig the formula: za /.96 = = = 96.04» 96 4D 4 0. b) (5 oits) Your editor, citig time ad budget costraits, allows you to iterview 4 wome ad 4 me studets. At the same cofidece level, what is your ew margi of error? D = za/ / 4 =.96 =.96 = = c) (0 oits) 0 wome out of 4 thought seual assault was a roblem at UNC, while 8 me out of 4 thought there was a roblem. Test the ull hyothesis at the a = 0.05 level that there is o differece i their oiios versus the alterative hyothesis that there is a differece. Ca you reject the ull hyothesis? Show your work ad elai your aswer ˆ = = = z ˆ - ˆ = = = = =- ˆ ( ˆ ) ±.96 are the critical values for a = 0.05 Therefore, we caot reject the ull hyothesis that me ad wome have equal values. (It turs out, that if we d bee able to samle 96 me ad wome each, we could easily have rejected the ull hyothesis. mtms7.lw Page 5 of

6 Eamiatio Number: 5. A olie retailer has data relatig the amout of time customers sed lookig at its catalog to the dollar amout they sed. Usig a radom samle of 5 customers from the revious quarter, the retailer wats to develo a quatitative estimate of the dollars set er miute customers sed viewig the olie catalog. Use the regressio outut followig to aswer the followig questios:. reg urchases_ time_miutes Source SS df MS Number of obs = F(, 49) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = urchases_ Coef. Std. Err. t P> t [95% Cof. Iterval] time_miutes _cos reg urchases_ time_miutes Source SS df MS Number of obs = F(, 49) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = urchases_ Coef. Std. Err. t P> t [95% Cof. Iterval] time_miutes _cos Filled i regressio results above. a) ( oits) Write the Stata commad that would give the results above reg urchases_ time_miutes b) ( oits) Comute the t-statistic for the sloe coefficiet. Fill i the aroriate bo i the results table above. Show your work below. Ca you reject the ull hyothesis of o effect at the alha = 0.05 level? mtms7.lw Page 6 of

7 Eamiatio Number: t ˆ b -b ˆ b s s = = = = bˆ bˆ 7.70 Yes, you ca easily reject the ull hyothesis at the 5 ercet level c) ( oits) Comute the t-statistic for the itercet coefficiet. Fill i the aroriate bo i the results above. Show your work. Ca you reject the ull hyothesis of zero itercet at the alha = 0.05 level? t aˆ-a aˆ s s = = = = aˆ aˆ Yes, you ca easily reject the ull hyothesis at the 5 ercet level d) ( oits) Write the degrees of freedom for the F-statistic i the bo above. (,49) e) ( oits) Comute the F-statistic, show your work, ad eter the comuted F-statistic i the bo above. F (,49) ( - ) ( - ) ( ) ( ) ESS / K ESS / = = = = RSS / K RSS / f) ( oits) Does the -value associated with this F-statistic all you to reject the ull hyothesis of o effect at the alha = 0.05 level? Why? Elai your aswer, usig suortive umbers. Yes, the F-statistic is Lookig at the F-distributio table o the eam, oe sees that for df = (,49) or as close as oe ca aroimate it, the required F-statistic is about So the F-statistic far eceeds the critical value g) ( oits) Comute the R-square, show your work, ad eter the comuted R-square i the aroriate bo above. What roortio of total variatio i the value of olie sales is elaied by the regressio? ESS R = = = TSS The regressio elais ercet of the total variatio i eeditures. h) ( oits) Comute the cofidece iterval for the sloe coefficiet. Show your work ad eter the lower ad uer limits i the aroriate bo above. mtms7.lw Page 7 of

8 Eamiatio Number: ˆ s b ± t = ˆ b ± t gs = ±.96g = 0.68,.06 e a a bˆ TSS ( ) The studets robably will use t df=º which is OK. Check for 3 sigificat digits. i) ( oits) How much does dollar eediture chage (ositively or egatively) for each additioal miute set lookig at the catalog? Elai your aswer. The sloe coefficiet shows the margial imact of a oe miute chage i browse time: d ( urchases _($)) ( _ mi utes) d time = ˆ b = So, each additioal miute set browsig yields, o average a icrease of $0.843 i eeditures. j) ( oits) Fill i the blak sace i the equatio below ad elai its meaig: æ ö ç ˆ i Eébù b ç ë û = + å Ee [ i] = b i= ç i è å i= ø I the bo should be b, which meas that the estimate b is a ubiased estimator, sice E[e i ] = 0. mtms7.lw Page 8 of

9 t = s = ( -) -( m-m) s ( ) + ( ) df = + -. ( - ) + ( -) s s + - ( - ) -( m-m) ( s ) + ( s ) t= D= é( s ) ( s ) ù ë + û ( s ) ( s ) rouded dow. ˆ = + + æ - m Prç- ta < < t è s/ ö = -a ø / a/ z = z = ˆ ˆ - ˆ - ˆ - + ( ˆ ) 0 ( - ) 0 0 z s = D a/. z 4D = a/ Page 9 of

10 N å i= s = ( -m) i N ( ) C C = = r N -r - N C r N -r ( )( -) 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 ~ - -m / s = e - < < s ì ï f( ) =í ïî ( ) P A B i ( b-a) = å all ( i) ( i) PBA ( ) PA ( ) PBA PA k, a b 0, otherwise k ( b-a) m = ( b+ a) ad s = c s0 ( - ) - = s k - ìle l, l > 0, ³ 0 f() = í î 0, otherwise m = ad s = l l P( a) -la ³ = e, a ³ 0ad l > 0 F ( lt) for 0,,,,, l 0, ì t e - l, = K ï > P ( ) = í! ïî 0, 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 = m = lt = the eected umber of evets i oe subsegmet legth t Page 0 of

11 Critical values of c a a c c 0.99 c c 0.95 c df 0 c 0.0 c 0.05 c 0.05 c 0.00 c df Page of

12 f(f) F-Distributio Table: Uer 5% Probability (or 5% Area) uder F-distributio Curve 0.05 F= df=d=5 D=0 F F-Table for alha = 0.05 / df= if df= if Page of