Solutio to selected problems i midterm exam i pricipal of statistics PREPARED BY Dr. Nafez M. Barakat Islamic uiversity of Gaza MIDTERM 009-010 1
P(1.99 < x <.0) = p( 1.99.0 0.06 36 < z <.0.0 0.06 = p(z< 0) p (z< -1) = 0.05-0.1587 = 0.3413 36 ) = p(-1 < z < 0) P( X x1 )= 0.99, p(z >z 1 ) = 0.99, p(z < z 1 ) = 1-0.99 = 0.01, z 1 = -1.3 0.06 x1 z.0 1.3 1.9767 36 p(x > 90) = 0.0668, p(z > z 1 ) = 0.0668, p( z < z 1 ) = 1.0-0.0668= 0.933 Z 1 = 1.5, x 90 75 z, 1.5, 10 P(x> 95) = p( z> (95-75)/10) = p(z>.0) = 1- p(z<.0) = 1 0.977=0.08.p(-z 1 < z < z 1 ) = 0.8664, p( 0< z < z 1 ) = 0.433 P(z< z 1 ) 0.5 = 0.433, z 1 = 1.5 3
x z Lower iterval x L = 75-1.5*10 = 60 Upper iterval x u = 75+ 1.5*10 = 90 P (1 ) 0.45(1 0.45) 75 0.05745 P( 0.3 < p < 0.5) = P( (0.3-0.45)/0.5745 < Z < ((0.5-0.45)/0.5745) = P(-.61 < Z < 0.87) = 0.8033 P(Z< Z 1 ) = 0.80, Z 1 = 0.84 (1 ) p Z 0.45 0.84* 0.45(1 0.45) 75 0.498 4
z * e (1.96) * (1000) (50) 1537 5
P( 0.67 P( 0.67 X 0.67 ) X 0.67) P( 0.67 Z 0.67) 0.497 0.5 MIDTERM 18/4/011 SECTION I: MULTIPLE-CHOICE For each questio i this sectio, circle the correct aswer. Each problem is worth 1 poit. 6
1. The process of usig sample statistics to draw coclusios about true populatio parameters is called a) statistical iferece. b) the scietific method. c) samplig. d) descriptive statistics.. A summary measure that is computed to describe a characteristic from oly a sample of the populatio is called a) a parameter. b) a cesus. c) a statistic. d) the scietific method. 3. I a right-skewed distributio a) the media equals the arithmetic mea. b) the media is less tha the arithmetic mea. c) the media is larger tha the arithmetic mea. d) oe of the above. 1. Whe extreme 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.. I its stadardized form, the ormal distributio a) has a mea of 0 ad a stadard deviatio of 1. b) has a mea of 1 ad a variace of 0. c) has a area equal to 0.5. d) caot be used to approximate discrete probability distributios. 1. If a particular batch of data is approximately ormally distributed, we would fid that approximately a) of every 3 observatios would fall betwee 1 stadard deviatio aroud the mea. b) 4 of every 5 observatios would fall betwee 1.8 stadard deviatios aroud the mea. c) 19 of every 0 observatios would fall betwee stadard deviatios aroud the mea. d) All the above.. The Cetral Limit Theorem is importat i statistics because a) for a large, it says the populatio is approximately ormal. b) for ay populatio, it says the samplig distributio of the sample mea is approximately ormal, regardless of the sample size. c) for a large, it says the samplig distributio of the sample mea is approximately ormal, regardless of the shape of the populatio. d) for ay sized sample, it says the samplig distributio of the sample mea is approximately ormal. 7
1. Which of the followig statemets about the samplig distributio of the sample mea is icorrect? a) The samplig distributio of the sample mea is approximately ormal wheever the sample size is sufficietly large ( 30). b) The samplig distributio of the sample mea is geerated by repeatedly takig samples of size ad computig the sample meas. c) The mea of the samplig distributio of the sample mea is equal to. d) The stadard deviatio of the samplig distributio of the sample mea is equal to.. 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 100 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 0.0. 3. A 99% cofidece iterval estimate ca be iterpreted to mea that a) if all possible samples are take ad cofidece iterval estimates are developed, 99% of them would iclude the true populatio mea somewhere withi their iterval. b) we have 99% cofidece that we have selected a sample whose iterval does iclude the populatio mea. c) Both of the above. d) Noe of the above. 8
SECTION II: TRUE OR FALSE For each questio i this sectio, idicate whether the setece is TRUE or False. Each problem is worth 1 poit. 1. ( F ) The possible resposes to the questio How may times i the past three moths have you visited a city park? are values from a discrete variable.. ( T ) Other thigs beig equal, as the cofidece level for a cofidece iterval icreases, the width of the iterval icreases. 3. ( F ) The t distributio is used to develop a cofidece iterval estimate of the populatio proportio whe the populatio stadard deviatio is ukow. 4. ( F ) The cofidece iterval obtaied will always correctly estimate the populatio parameter. 5. ( F ) I estimatig the populatio mea with the populatio stadard deviatio ukow, if the sample size is 1, there will be 6 degrees of freedom. 6. ( T ) As the sample size icreases, the effect of a extreme value o the sample mea becomes smaller. 7. ( T ) A samplig distributio is a distributio for a statistic. 8. ( F ) The probability that a stadard ormal radom variable, Z, is less tha 5.0 is approximately 0. 9. ( F ) The "middle spread," that is the middle 50% of the ormal distributio, is equal to oe stadard deviatio. 10. ( F ) I a set of umerical data, the value for Q is always halfway betwee Q1 ad Q3. 11. ( F ) The stadard error of the sample mea is affected by the cofidece level 1. ( T )A poit estimator is a fuctio of the radom sample used to make ifereces about the value of a ukow populatio parameter SECTION III: FREE RESPONSE QUESTIONS (i) ( Poits) The assets i billios of dollars of the five largest bod fuds are 19.5, 16.8, 13.7, 1.8, ad 10.9. Compute the stadard deviatio for this populatio of the five largest bod fuds. Descriptive Statistics N Mea Std. Deviatio VAR00001 5 14.7400 3.4095 Valid N (listwise) 5 9
(ii) (4 Poits) You were told that the mea score o a statistics exam is 75 with the scores ormally distributed. I additio, you kow the probability of a score betwee 55 ad 60 is 4.41% ad that the probability of a score greater tha 90 is 6.68%. a. ( Poits) What is the probability of a score greater tha 95? p(x > 90) = 0.0668, p(z > z 1 ) = 0.0668, p( z < z 1 ) = 1.0-0.0668= 0.933 Z 1 = 1.5, x 90 75 z, 1.5, 10 P(x> 95) = p( z> (95-75)/10) = p(z>.0) = 1- p(z<.0) = 1 0.977=0.08 b. ( Poits) The middle 86.64% of the studets will score betwee which two scores?.p(-z 1 < z < z 1 ) = 0.8664, p( 0< z < z 1 ) = 0.433 P(z< z 1 ) 0.5 = 0.433, z 1 = 1.5 x z Lower iterval x L = 75-1.5*10 = 60 Upper iterval x u = 75+ 1.5*10 = 90 (iii) ( Poits) The head of a computer sciece departmet is iterested i estimatig the proportio of studets eterig the departmet who will choose the ew computer egieerig optio. Suppose there is o iformatio about the proportio of studets who might choose the optio. What size sample should the departmet head take if she wats to be 95% cofidet that the estimate is withi 0.10 of the true proportio? e=0.1,.05, z / 1. 96 z 0 * (1 ) 1.96*1.96*0.5(1 0.5) e (0.1) 97 (iv) ( Poits) A hotel chai wats to estimate the average umber of rooms reted daily i each moth. The populatio of rooms reted daily is assumed to be ormally distributed for each moth with a stadard deviatio of 4 rooms. durig February, a sample of 5 days has a sample mea of 37 rooms. Use this iformatio to calculate a 9% cofidece iterval for the populatio mea. 4, 5, x 37, 0.08 4 C I X Z 37 1.75 * 37 8.4 5.8.6 45.5 11
= 16, s=400 from iterval (4739.8, 560.) x 5000 e t s 400 / * 60. t / t /.60, df 15 16 From table T 0.0, cofidece level = (1-0.0)*100%=98% Questios 5-6 refer to the followig iformatio: 5- A 95% cofidece iterval for the mea readig achievemet score for a populatio of third grades is (40, 50). The margi of error of this iterval is a) 95% b) 10 c) 5 d).5 11
e) The aswer caot be determied from the iformatio give 6- The sample mea is a) 0.95 b) 45 c) 4.5 d) 47.5 e) The aswer caot be determied from the iformatio give 1
7- Usig the same set of data, you compute a 95% cofidece iterval ad a 99% cofidece iterval. Which of the followig statemet is correct? a) The itervals have the same width b) The 99% iterval is wider c) The 95 % iterval is wider d) You caot be determied which iterval is wider uless you kow ad s 9 Suppose you take a simple radom sample from a populatio kow to be ormally distributed, but the value of is ukow. Your sample size is 10. Which formula below should be used to fid the 90% cofidece iterval for the mea? s S a) x 1.645 b) x 1.645 c) x 1.833 d) x 1.833 10 10 10 10 ANSWER C Questios 13-14 refer to the followig iformatio: 13- The height of Palestiia me aged 18 to 4 are approximately ormally distributio with mea 170 cm ad stadard deviatio 6 cm. Half of all youg me are shorter tha a) 164 cm b) 170 cm c) 176 cm d) Ca't tell, because the media height is ot give. 14- Oly about 5% of youg me have heights outside the rage 0.95% C. 170 6 158cm to 18cm a) 164 cm to 176 cm b) 158 cm to 18 cm c) 15 cm to 188 cm d) 146 cm to 194 cm 15- A airplae is oly allowed a gross passeger weight of 6,885 kg. If the weights of passegers travelig by air betwee two cities have a mea of 80 kg ad a stadard deviatio of 18 kg, the approximate probability that the combied weight of 81 passegers will exceed 6,885 kg is: a) 0.3906 b) 0.9938 c) 0.006 d) 0.0000 16. Give a ormally distributed populatio with a mea of 80 ad a variace of l00, we kow that the distributio of sample meas computed from samples of size 5 from that populatio will have a mea of ad a stadard error of. a) 80, l0 b) 80, c) l00, 5 d) 80, l0 Questio #6: [0 Poits] (a) [10 Poits] A article reports that (4.0, 5.6) is a 95% cofidece iterval for the mea legth of stay, i days, of patiets i hospital for a particular operatio. Suppose the sample size is 50, fid the sample mea ad the stadard deviatio. 13
x From the iterval 4.8, e 0.8, 0.05, t / s s e t / * 0.8.009 s.81, 50.009 (b) [10 Poits] How large a sample size is eeded to estimate the mea aual icome of CCC compay to be withi $000 with probability 0.99? Suppose there is o prior iformatio about the stadard deviatio of aual icome of the CCC compay, but we guess that about 68% of their icomes are betwee $10000 ad $40,000 ad that this distributio of icomes is approximately bell shaped At.01, z.58, 15000, e 000 0 / z * e (.58) * (15000) (000) 374.45 375 14
The Islamic Uiversity of Gaza Faculty of Commerce Departmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 130) Sprig Semester 014-8/4/014 Midterm Exam Name: ID: Istructors: Dr. Nafez Barakat Mr. Ibrahim Abed SECTION I: MULTIPLE-CHOICE (Each problem is worth 1 poit) For each questio i this sectio, circle the correct aswer. 4. A summary measure that is computed to describe a characteristic of a etire populatio is called a) a parameter. b) a cesus. c) a statistic. d) the scietific method. 5. Which descriptive summery measures are cosidered to be resistat statistics? a) The arithmetic mea ad stadard deviatio. b) The iterquartile rage ad rage. c) The mode ad variace. d) The media ad iterquartile rage. 6. Which of the followig is a cotiuous quatitative variable? a) The color of a studet s eyes b) The umber of employees of a isurace compay c) The amout of milk produced by a cow i oe 4-hour period d) The umber of gallos of milk sold at the local grocery store yesterday 7. The possible resposes to the questio "How would you rate the quality of your purchase experiece with 1 = excellet, = good, 3 = decet, 4 = poor, 5 = terrible?" are values from a a) discrete umerical radom variable. b) cotiuous umerical radom variable. 15
a) 100 b) 40 c) 10 d) 0 c) categorical radom variable. d) parameter 8.. I samplig from a large populatio with = 0, the stadard error of the mea is foud to be. The size of the sample used is: 9. The smaller the spread of scores aroud the arithmetic mea, a) the smaller the iterquartile rage. b) the smaller the stadard deviatio. c) the smaller the coefficiet of variatio. d) All the above 10. For sample size 16, the samplig distributio of the sample mea will be approximately ormally distributed e) regardless of the shape of the populatio. f) if the shape of the populatio is symmetrical. g) if the sample stadard deviatio is kow. h) if the sample is ormally distributed 13. I left-skewed distributios, which of the followig is the correct statemet? a) The distace from Q1 to Q is smaller tha the distace from Q to Q3. b) The distace from the smallest observatio to Q1 is larger tha the distace from Q3 to the largest observatio. c) The distace from the smallest observatio to Q is smaller tha the distace from Q to the largest observatio. d) The distace from Q1 to Q3 is twice the distace from the Q1 to Q. 1.Accordig to the Chebyshev rule, at least what percetage of the observatios i ay data set are cotaied withi a distace of 3 stadard deviatios aroud the mea? a) 67% b) 75% c) 88.89% d) 99.7% 13.For some positive value of Z, the probability that a stadard ormal variable is betwee 0 ad Z is 0.3340. The value of Z is a) 0.07 b) 0.37 c) 0.97 d) 1.06 14. A cofidece iterval was used to estimate the proportio of statistics studets that are female. A radom sample of 7 statistics studets geerated the followig 90% cofidece iterval: (0.438, 0.64). Usig the iformatio above, what total size sample would be ecessary if we wated to estimate the true proportio to withi ±0.08 usig 95% cofidece? a) 105 b) 150 c) 40 d) 597 16
z p 1p e =( 1.96/0.08) * 0.54* (1-0.54) = 150 15.Whe extreme 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 rags Sectio II: TRUE Or FALSE (Each problem is worth 1 poit). For each questio i this sectio, idicate whether the setece is TRUE or False.. ( F ) A statistic is usually used to provide a estimate for a usually uobserved parameter. 3. ( T ) As a geeral rule, a observatio is cosidered a extreme value if its absolute value of Z score is greater tha3. 4. ( F ) The aswer to the questio How do you rate the quality of your busiess statistics course is a example of a ordial scaled variable. 5. ( T ) The coefficiet of variatio is a measure of relative variatio. 6. ( F ) The t distributio is used to develop a cofidece iterval estimate of the populatio proportio whe the populatio stadard deviatio is ukow. 7. ( T ) The width of a cofidece iterval equals twice the samplig error 8. ( T ) A poit estimate cosists of a sigle sample statistic that is used to estimate the true populatio parameter 8. ( F ) The amout of bleach a machie pours ito bottles has a mea of 36 oz. with a stadard deviatio of 0.15 oz. Suppose we take a radom sample of 36 bottles filled by this machie. The samplig distributio of the sample mea has a stadard error of 0.15. 9. ( F ) A uiversity dea is iterested i determiig the proportio of studets who receive some sort of fiacial aid. Rather tha examie the records for all studets, the dea radomly selects 00 studets ad fids that 118 of them are receivig fiacial aid. Use a 90% cofidece iterval to estimate the true proportio of studets who receive fiacial aid. The aswer is 0.59 0.068165 P=118/00 = 0.59, P(1 P) P Z 0.59 1.645* 0.59*(1 0.59) 00 0.59 0.057 10. ( F ) As a aid to the establishmet of persoel requiremets, the director of a hospital wishes to estimate the mea umber of people who are admitted to the 17
emergecy room durig a 4-hour period. The director radomly selects 64 differet 4-hour periods ad determies the umber of admissios for each. For this sample, X 19. 8 ad s = 5. The 95% cofidece iterval for the populatio mea is 19.8 4.96 SECTION III: FREE RESPONSE QUESTIONS Questio #3: (5 Poits) The ower of a fish market determied that the average weight for a catfish is 3. pouds. He also kew that the probability of a radomly selected catfish that would weigh more tha 3.8 pouds is 0% ad the probability that a radomly selected catfish that would weigh less tha.8 pouds is 30%. a) Fid the probability that a radomly selected catfish will weigh less tha 3.6 pouds? 3.8 3. 0.6 p ( w 3.8) 0.0, p ( Z ) 0.0, P( Z ) 0. 80 0.6 0.84, 0.714 3.6 3. P ( w 3.6) P( Z ) P( Z 0.560) 0.714 0.713 b) The middle 40% of the catfish will weigh betwee which two umbers? P ( Z Z1) 0.70, Z 1 = 0.55 18
X U Z 3. 0.55*0.714 3.574 X L Z 3. 0.55*0.714.85 Questio #4: (5 Poits) A study at a college i the west coast reveals that, historically, 45% of their studets are miority studets. If a radom sample of size 75 is selected e) Fid the probability that the sample proportio of miority studets lies betwee 30% ad 50%? 0.45, 75 0.3 0.45 0.5 0.45 P(0.3 P 0.5) P( Z ) 0.45(1 0.45) 0.45(1 0.45) 75 75 P(.611 Z 0.87) P( Z 0.87) P( Z.611) 0.8078 0.0197 0.8058 f) 95% of the samples proportios of miority studets will be greater tha 19
what value? P(Z>Z1) = 0.95, P(Z< Z 1 ) = 0.05 Z 1 = -1.645 p Z (1 ) 0.45 1.645* 0.45(1 0.45) 75 0.355 The Islamic Uiversity of Gaza Faculty of Commerce Departmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 130) Sprig Semester 014 Fial Examiatio Date : 31/5/014 Name: ID: Time: Two hour s Istructor's: Mr. Ibrahim Abed ad D. Nafez Barakat DON'T WRITE ON THIS TABLE QUESTION #1 # #3 #4 #5 #6 #7 TOTAL POINTS Questio #1: [15 Poits] For each questio i this sectio, circle the correct aswer. Each problem is worth 1 poit. 1
1. A summary measure that is computed to describe a characteristic from oly a sample of the populatio is called a) a parameter b) a cesus. c) a statistic d) the scietific method. Which of the followig statemets about the media is ot true? a) It is more affected by extreme values tha the arithmetic mea b) It is a measure of cetral tedecy i bell-shaped "ormal" c) It is equal to Q. d) It is equal to the mode distributios 3. Accordig to the Chebyshev rule, at least what percetage of the observatios i ay data set are cotaied withi a distace of 3 stadard deviatios aroud the mea? a) 67% b) 75% c) 88.89% d) 99.7% 4. For some positive value of Z, the probability that a stadard ormal variable is betwee 0 ad Z is 0.3770. The value of Z is a) 0.18 b) 0.81 c) 1.47 d) 1.16 5. If we kow that the legth of time it takes a college studet to fid a parkig spot i the library parkig lot follows a ormal distributio with a mea of 3.5 miutes ad a stadard deviatio of 1 miute, fid the poit i the distributio i which 75.8% of the college studets exceed whe tryig to fid a parkig spot i the library parkig lot. a) 4. miutes b) 3. miutes c) 3.4 miutes d).8 miutes 6. The stadard error of the mea a) is ever larger tha b) decreases as the the stadard deviatio sample size icreases of the populatio c) measures the variability of the mea from sample to sample. d) All of the above 7. The Dea of Studets mailed a survey to a total of 400 studets. The sample icluded 100 studets radomly selected from each of the freshma, sophomore, juior, ad seior classes o campus last term. What samplig method was used? a) Simple radom sample b) Systematic sample c) Stratified sample d) Cluster sample 8. I the costructio of cofidece itervals, if all other quatities are uchaged, a icrease i the sample size will lead to a iterval. a) arrower b) wider c) less sigificat d) Biased 9. A uiversity dea is iterested i determiig the proportio of studets who receive some sort of fiacial aid. Rather tha examie the records for all studets, the dea radomly selects 00 studets ad fids that 118 of them are receivig fiacial aid. If the dea wated to estimate the proportio of all studets receivig fiacial aid to withi 3% with 99% reliability.how may studets would eed to be sampled P=118/00 = 0.59 Z p(1 p).575*.575*0.59*(1 0.59) e 0.03* 0.03 178.164653 a) 1844 b) 1784 c) 1503 d) 1435 10. Which of the followig is most likely a parameter as opposed to a statistic? 1
a) The average score of the first five studets completig a assigmet b) The proportio of females registered to vote i a couty c) The average height of people radomly selected from a database. d) The proportio of trucks stopped yesterday that were cited for bad brakes 11. I aright _ skewed distributio a) The arithmetic mea equals the media. b) The arithmetic mea is less tha the media. c) The arithmetic mea is larger tha the media. d) Noe of the above 1.The width of a cofidece iterval estimate for a proportio will be A) arrower whe the sample proportio is 0.10 tha whe the sample proportio is 0.45. B) wider for 90% cofidece tha for 95% cofidece. C) arrowest whe the sample proportio is 0.5. D) arrower for a sample size of 50 tha for a sample size of 100. E) wider whe the sample proportio is 0.95 tha whe the sample proportio is 0.55. Questio #: [15 Poits] For each questio i this sectio, idicate whether the setece is true or false. Each problem is worth 1 poit. 1- ( T ) The possible resposes to the questio How log have you bee livig at your curret residece? are values from a cotiuous variable - ( F ) The aswer to the questio What is your favorite color? is a example of a ordial scaled variable 3- ( T ) I a sample of size 40, the sample mea is 15. I this case, the sum of all observatios i the sample is X i 600. 4- ( T ) The coefficiet of variatio measures variability i a data set relative to the size of the arithmetic mea 5- ( F ) The probability that a stadard ormal radom variable, Z, is betwee 1.00 ad 3.00 is 0.574. 6- ( T ) As the sample size icreases, the effect of a extreme value o the sample mea becomes smaller 7- ( F ) The stadard error of the samplig distributio of a sample proportio is p 1 p
8- ( T ) Other thigs beig equal, as the cofidece level for a cofidece iterval icreases, the width of the iterval icreases 9- ( T ) A poit estimate cosists of a sigle sample statistic that is used to estimate the true populatio parameter 10- ( F ) For a give level of sigificace, if the sample size is icreased, the probability of committig a Type I error will icrease 11- ( T ) For a give sample size, the probability of committig a Type II error will icrease whe the probability of committig a Type I error is reduced Bous Questio #7: [5 Poits] Suppose that the icomes i a populatio have mea $000 ad stadard deviatio $4000. A sample of size 40 is selected a) What is the probability that the sample mea will be withi $000 of the populatio mea? µ=000, 4000 =40 p( X 000) P( 3.16 Z 0.999 0.0008 0.9984 000 P( Z ) P( P( Z 3.16) 4000 / 40 3.16) P( Z 3.165) P( Z 3.16) b) Suppose that we wat the probability of X beig withi $000 of the populatio mea to be 0.95. How large a sample do we eed? α=0.05 Z α/ = 1.96 e = 000 Zα/ e σ 1.96*1.96* 4000*4000 16 000*000 A Itroductio to Statistics Course (ECOE 130) Sprig Semester 01 Fial Examiatio Date : 8/5/01 3-Asummary measure that is computed to describe a characteristic of a etire populatio is called 3
a) a parameter b) a cesus c) a statistics d) The scietific method 4- Accordig to the chebyshev rule, at least 93.75% of all observatios i ay data set are cotaied withi a distace of how may stadard deviatios aroud the mea 1 1 1 100% (1 ) 100% 93.75% k 4 a) 1 b) c) 3 d) 4 5- The mea age of five people i a room is 30 years. Oe of the people whose age is 50 years leaves the room. the mea age of the remaiig four people i the room is a) 40 b) 30 c) 5 d) Not able to be determied from the iformatio give 6- For some value of Z, the probability that a stadard variable is below Z is 0.791.The value of z is a) 0.81 b) -0.31 c) 0.31 d) 1.96 7- Samplig distributios describe the distributio of a) parameters b) Statistics c) Both a ad b d) Neither a or b 9- The symbol for the power of a statistical test is a) b) 1- c) d) 1-10- A type II error is committed whe a) we reject H 0 that is true. b) we reject H 0 that is false. c) we do t reject H 0 that is true. d) we do t reject H 0 that is false 11- A uiversity dea is iterested i determiig the proportio of studets who receive some sort of fiacial aid. Rather tha examie the records for all studets, the dea radomly selects 00 studets ad fids that 118 of them are receivig fiacial aid. If the dea wated to estimate the proportio of all studets receivig fiacial aid to withi 3% with 99% reliability.how may studets would eed to be sampled sampled? P=118/00 = 0.59 Z p(1 p).575*.575*0.59*(1 0.59) e 0.03* 0.03 178.164653 a) 1844 b) 1784 c) 1503 d) 1435 1) I a survey of 300 T.V. viewers, 0% said they watch etwork ews programs. Fid stadard error for the sample proportio. A) 0.0071 B) 0.0865 C) 0.071 D) 0.014 E) 0.0649 4
For each questio i this sectio, idicate whether the setece is true or false. Each problem is worth 1 poit. 1- ( F ) A statistic is usually uobservable while a parameter is usually observable - ( F ) As the sample size icreases, the stadard error of the mea icreases. S X S 3- ( T ) Other thigs beig equal, as the cofidece level for a cofidece iterval icreases, the width of the iterval icreases 4- ( T ) The t distributio is used to develop a cofidece iterval estimate of the populatio mea whe the populatio stadard deviatio is ukow 7- ( T ) The mea of the samplig distributio of a sample mea is the populatio mea 8- ( F ) The quality ("terrible", "poor", "fair", " acceptable", "very good", ad " excellet" ) of a day care ceter is a example of a umerical variable 9- ( T ) The coefficiet of variatio measures variability i a data set relative to the size of the arithmetic mea 10- ( T ) A box plot is a graphical represetatio of a 5 _ umber summary 11- ( F ) The probability that a stadard ormal radom variable Z is below 1.96 is 0.4750 13- ( T ) A 95% cofidece iterval for will be wider tha a 96% cofidece iterval for 15- ( T ) The a mout of water cosumed by a perso per week is a example of a cotiuous variable A Itroductio to Statistics Course (ECOE 130) Sprig Semester 015-8/4/015 Midterm Exam Name: ID: Istructors: Dr. Nafez Barakat Mr. Ibrahim Abed SECTION I: MULTIPLE-CHOICE (Each problem is worth 1 poit) For each questio i this sectio, circle the correct aswer. 9. The collectio ad summarizatio of the socioecoomic ad physical characteristics of the employees of a particular firm is a example of a) iferetial statistics. b) descriptive statistics. c) a parameter. d) a statistic. 10. Researchers are cocered that the weight of the average America school child is icreasig implyig, amog other thigs, that childre s clothig should be maufactured ad marketed i larger sizes. If X is the weight of school childre sampled i a atiowide study, the X is a example of a) a categorical radom variable. b) a discrete radom variable. 5
c) a cotiuous radom variable. d) a parameter. 3.Which of the followig is the easiest to compute? a) The arithmetic mea. b) The media. c) The mode. d) The stadard deviatio. 4.Accordig to the Chebyshev rule, at least 93.75% of all observatios i ay data set are cotaied withi a distace of how may stadard deviatios aroud the mea? a) 1 b) c) 3 1 1 d) 4 1 100% (1 ) 100% 93.75% k 4 5. For some positive value of X, the probability that a stadard ormal variable is betwee 0 ad +1.5X is 0.433. The value of X is e) 0.10 f) 0.50 g) 1.00 h) 1.50 6.. I its stadardized form, the ormal distributio a) has a mea of 0 ad a variace of 1. b) has a mea of 1 ad a variace of 0. c) has a area equal to 0.5. d) caot be used to approximate discrete probability distributios. 7. At a computer maufacturig compay, the actual size of computer chips is ormally distributed with a mea of 1 cetimeter ad a stadard deviatio of 0.1 cetimeter. A radom sample of 1 computer chips is take. What is the stadard error for the sample mea? a. 0.09 b. 0.050 c. 0.091 d. 0.10 8. If you were costructig a 99% cofidece iterval of the populatio mea based o a sample of =5 where the stadard deviatio of the sample s = 0.05, the critical value of t will be a..7969 b..7874 c..49 d..4851 9. I the costructio of cofidece itervals, if all other quatities are uchaged, a icrease i the sample size will lead to a iterval. a. arrower b. wider c. less sigificat 6
d. biased 10. For sample size 16, the samplig distributio of the mea will be approximately ormally distributed a. regardless of the shape of the populatio. b. if the shape of the populatio is symmetrical. c. if the sample stadard deviatio is kow. d. if the sample is ormally distributed. Sectio II: TRUE Or FALSE (Each problem is worth 1 poit).for each questio i this sectio, idicate whether the setece is TRUE or False. 1.( F ) A statistic is usually uobservable while a parameter is usually observable.. ( F ) Marital status is a example of a ordial scaled variable. 3. ( F ) The lie draw withi the box of the boxplot always represets the arithmetic mea. 4. ( T ) The Z score of a observatio measures how may stadard deviatios is the value from the mea 5. ( T ) The probability that a stadard ormal radom variable, Z, is betwee 1.50 ad.10 is the same as the probability Z is betwee.10 ad 1.50. 6. ( T ) If the populatio distributio is skewed, i most cases the samplig distributio of the mea ca be approximated by the ormal distributio if the samples cotai at least 30 observatios 7.( F ) I formig a 90% cofidece iterval for a populatio mea from a sample size of, the umber of degrees of freedom from the t distributio equals 8.( T ) Other thigs beig equal, as the cofidece level for a cofidece iterval icreases, the width of the iterval icreases. 9. ( T ) The sample mea is a poit estimate of the populatio mea. 10. ( F ) The cofidece iterval obtaied will always correctly estimate the populatio parameter SECTION III: FREE RESPONSE QUESTIONS Questio #3: (5 Poits) A uiversity dea is iterested i determiig the proportio of studets who receive some sort of fiacial aid. Rather tha examie the records for all studets, the dea radomly selects 00 studets ad fids that 118 of them are receivig fiacial aid. a) Use a 90% cofidece iterval to estimate the true proportio of studets who receive fiacial aid? 7
90%C.I PZ α/ p(1 p) 0.59 1.645 0.54 0.46 00 ( 0.4806876, 0.59797314) b) A ecoomist is iterested i studyig the icomes of cosumers i a particular regio. The populatio stadard deviatio is kow to be $1,000. A radom sample of 50 idividuals resulted i a average icome of $15,000. 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 $100? Z σ 1.96*1.96*1000*1000 α/ 1536.64 1367 e 50*50 Questio #4: (5 Poits) The amout of time required for a oil ad filter chage o a automobile is ormally distributed with a mea of 45 miutes ad a stadard deviatio of 10 miutes. A radom sample of 16 cars is selected. 1. Fid the probability that the sample mea is betwee 45 ad 5 miutes? 45 45 5 45 p (45 x 5) p( z ) 10 / 16 10 / 16 p ( 0 z.8) p( z.8) p( z 0) 0.9974 0.5 0.4974. 95% of all sample meas will fall betwee what two values? p ( z z z1) 0.95 1 p ( 0 z z1) 0.475 p z z ) 0.975 z 1. 96 x u x l z 45 1.9610 64.6 z 45 1.9610 5.4 ( 1 1 You may use the followig formulae: x 1 1 i i1 x, S x x, x x Z x i1 i, xi i1 N N X i 1 N N i1 1 x, x, p, p, 8
9 s x t, x z p(1 p) p z, Z e, z p 1 p e, z m
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