Final Examination Solutions 17/6/2010

Transcription

1 The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester Fial Eamiatio Solutios 7/6/00 Name: I: Istructor: r. Samir Safi Mr. Ibrahim Abed INSTRUCTIONS:. Write your ame ad studet I.. You have hours 3. This eam must be your ow work etirely. You caot talk to or share iformatio with ayoe. 4. Show all your work. Partial credit will oly be give where sufficiet uderstadig of the problem has bee demostrated ad work is show. ON'T WRITE ON THIS TABLE Q UESTION # # #3 #4 #5 BONUS TOTAL P OINTS

2 Questio #: [ Poits] For each questio i this sectio, circle the correct aswer. Each problem is worth poits.. The t test for the mea differece betwee related populatios assumes that the a) populatio sizes are equal. b) sample variaces are equal. c) populatio of differeces is approimately ormal or sample sizes are large eough. d) All of the above.. The ull ad alterative hypotheses to determie if the umber of tissues used durig a cold is less tha 60. a) H 0 : µ 60 ad H : µ > 60. b) H 0 : µ 60 ad H : µ < 60. c) H 0 : X 60 ad H : X < 60. d) H 0 : X = 5 ad H : X Whe etreme values are preset i a set of data, which of the followig descriptive summary measures are most appropriate: a. CV ad rage. b. arithmetic mea ad stadard deviatio. c. iterquartile rage ad media. d. variace ad iterquartile rage. 4. The Y-itercept (b 0 ) represets the a) estimated average Y whe X = 0. b) chage i estimated average Y per uit chage i X. c) predicted value of Y. d) variatio aroud the sample regressio lie. 5. The t distributio a) assumes the populatio is ormally distributed. b) approaches the ormal distributio as the sample size icreases. c) has more area i the tails tha does the ormal distributio. d) All of the above. 6. A ecoomist is iterested i studyig the icomes of cosumers i a particular regio. The populatio stadard deviatio is kow to be $,000. A radom sample of 50 idividuals resulted i a average icome of $5,000. What total sample size would the ecoomist eed to use for a 95% cofidece iterval if the width of the iterval should ot be more tha $00? a) = 537 b) = 385 c) = 40 d) = 0

3 7. The width of a cofidece iterval estimate for a proportio will be a) arrower for 99% cofidece tha for 95% cofidece. b) wider for a sample size of 00 tha for a sample size of 50. c) arrower for 90% cofidece tha for 95% cofidece. d) arrower whe the sample proportio is 0.50 tha whe the sample proportio is I testig for differeces betwee the meas of two related populatios, the ull hypothesis is 3. H : 0 µ =. 4. H : 0 0 µ =. 5. H : 0 0 µ <. 6. H : 0 0 µ >. 9. Which of the followig would be a appropriate alterative hypothesis? a) The mea of a populatio is equal to 55. b) The mea of a sample is equal to 55. c) The mea of a populatio is greater tha 55. d) The mea of a sample is greater tha A Type I error is committed whe a) we reject a ull hypothesis that is true. b) we do't reject a ull hypothesis that is true. c) we reject a ull hypothesis that is false. d) we do't reject a ull hypothesis that is false.. If a ecoomist wishes to determie whether there is evidece that average family icome i a commuity eceeds $5,000 a) either a oe-tailed or two-tailed test could be used with equivalet results. b) a oe-tailed test should be utilized. c) a two-tailed test should be utilized. d) Noe of the above.. If the p-value is less tha α i a two-tailed test, a) the ull hypothesis should ot be rejected. b) the ull hypothesis should be rejected. c) a oe-tailed test should be used. d) o coclusio should be reached. 3. It is possible to directly compare the results of a cofidece iterval estimate to the results obtaied by testig a ull hypothesis if a) a two-tailed test for µ is used. b) a oe-tailed test for µ is used. c) Both of the previous statemets are true. d) Noe of the previous statemets is true. 4. We have created a 95% cofidece iterval for µ with the result (0, 5). What decisio will we make if we test H 0 : µ =6 versus H : µ 6 at α = 0.05? a) Reject H 0 i favor of H. b) Accept H 0 i favor of H. c) Fail to reject H 0 i favor of H. d) We caot tell what our decisio will be from the iformatio give. 3

4 5. Suppose a 95% cofidece iterval for µ turs out to be (,000,,00). Give a defiitio of what it meas to be 95% cofidet as a iferece. a) I repeated samplig, the populatio parameter would fall i the give iterval 95% of the time. b) I repeated samplig, 95% of the itervals costructed would cotai the populatio mea. c) 95% of the observatios i the etire populatio fall i the give iterval. d) 95% of the observatios i the sample fall i the give iterval. Questio #: [ Poits] For each questio i this sectio, idicate whether the setece is TRUE or False. Each problem is worth poits.. ( ) Whe we test for differeces betwee the meas of idepedet populatios, we ca oly use a two-tailed test. False. ( ) Repeated measuremets from the same idividuals is a eample of data collected from related populatios. 3. ( ) The statemet of the ull hypothesis always cotais a equality. 4. ( ) The smaller is the p-value, the stroger is the evidece agaist the ull hypothesis. 5. ( ) The t distributio is used to costruct cofidece itervals for the populatio mea whe the populatio stadard deviatio is ukow. 6. ( ) Give a sample mea of. ad a populatio stadard deviatio of 0.7 from a sample of 0 data poits, a 90% cofidece iterval will have a width of.36. False 7. ( ) The sample mea is a poit estimate of the populatio mea. 8. ( ) A poit estimate cosists of a sigle sample statistic that is used to estimate the true populatio parameter. 9. ( ) I a set of umerical data, the value for Q is always halfway betwee Q ad Q3. False 0. ( ) The coefficiet of variatio is a measure of relative variatio.. ( ) The t distributio approaches the stadardized ormal distributio whe the umber of degrees of freedom icreases.. ( ) For a give data set, the cofidece iterval will be wider for 95% cofidece tha for 90% cofidece. 3. ( ) A samplig distributio is a distributio for a statistic. 4. ( ) The type of TV oe ows is a eample of a ordial scaled variable. False 5. The probability that a stadard ormal radom variable, Z, falls betwee.50 ad 0.8 is

5 Questio #3: [6 Poits] To test the effectiveess of a busiess school preparatio course, 8 studets took a geeral busiess test before ad after the course. eote X : Before Course (), X : After Course ():X, ifferece =X -X. Suppose = 50, S = a. ( Poits) Sate the ull ad alterative hypotheses to determie the effectiveess of a busiess school preparatio course. b. (6 Poits) Usig the sample iformatio provided, calculate the value of the test statistic. c. (4 Poits) Compute the P- value. d. ( Poits) State your decisio at level of sigificace α =.05 e. ( Poits) State your coclusio Questio #4: [4 Poits] The dea of a college is iterested i the proportio of graduates from his college who have a job offer o graduatio day. He is particularly iterested i seeig if there is a differece i this proportio for accoutig ad ecoomics majors. I a radom sample of 00 of each type of major at graduatio, he foud that 65 accoutig majors ad 5 ecoomics majors had job offers. If the accoutig majors are desigated as Group ad the ecoomics majors are desigated as Group, perform the appropriate hypothesis test usig a level of sigificace of a. ( Poits) Sate the hypotheses the dea should use a) H 0 : π π = 0 versus H : π π 0 b) H 0 : π π 0 versus H : π π = 0 c) H 0 : π π 0 versus H : π π > 0 d) H 0 : π π 0 versus H : π π < 0 b. (6 Poits) Usig the sample iformatio provided, calculate the value of the test statistic. Z=.866 c. (4 Poits) costruct a 95% cofidece iterval estimate of the differece i proportio betwee accoutig majors ad ecoomic majors who have a job offer o graduatio day. Iterpret. 0.0 to 0.7 d. (4 Poits) Compute the P- value d. ( Poits) State your decisio based o your result i Part (c) e. ( Poits) State your decisio based o your result i Part (d) f. ( Poits) Are the two decisios i (d) ad (e) cosistet? Why or why ot? g. ( Poits) State your coclusio 5

6 Questio #5: [5 Poits] The maagers of a brokerage firm are iterested i fidig out if the umber of ew cliets a broker brigs ito the firm affects the sales geerated by the broker. They sample brokers ad determie the umber of ew cliets (X) they have erolled i the last year ad their sales amouts i thousads of dollars (Y). X = 30, Y = 549, X Y = 4868, i i i i i= i= i= i i X Y i= i= X = 853, Y = 668,S = 89.7,S = 4.0 a) (6 Poits) Compute the values of the estimated itercept ad slope. Eplai Slope =., Y-itercept = 7.7 b) (5 Poits) Compute the value of the coefficiet of correlatio. Iterpret c) ( Poits) Compute the value of the coefficiet of determiatio. Iterpret 0.785: 78.5% of the total variatio i sales geerated ca be eplaied by the umber of ew cliets brought i. d) ( Poits) Compute the predictio for the amout of sales (i $,000s) for a perso who brigs 5 ew cliets ito the firm إضافي Bous: (a) (3 Poits) For a ormally distributed variable, what is the probability betwee µ. 67σ ad µ +. 67σ equals.50? (b) (3 Poits) A fast food chai sells hamburger that they claim has sodium cotet of 650 milligrams. A simple radom sample of 35 hamburgers was aalyzed for sodium cotet. A 99% cofidece iterval for the populatio mea sodium cotet, µ, of such hamburgers is (65, 67). If we were to use the precedig data to test the hypotheses H o : µ =650 versus H a : µ 650. At a % sigificace level, would we reject the ull hypothesis? Eplai. 6

7 Formulas: IQR = Q3 Q i = S y iy i y yˆ = a + b, b = r, i = r = S a = y b i y i y i = i = ± z σ µ z = σ t µ =, df = t = s S, i = Xi Xi, df = µ σ = µ = µ, σ = σ i = ( ) ± t sp + t = sp + df = + df = + ˆp p σ = S + z =, p p p ˆp = ad ˆp S = i i = ( ) s ± t, df = ˆp± z p( ˆ p) ˆ µ µ µ π ( π ) z = ( ) = pˆ = pˆ pˆ pˆ( pˆ) µ = π, σ = S p p = p ( ) + ( ) S S + ˆ ˆ ˆ ˆ ˆp = ( pˆ pˆ ) ± z + z σ = m z = pˆ p m p( p) p( p) ( ˆ) 7

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