STAT 211-200EXAM 1 - FORM ASUMMER03
The U.S. Census Bureau produces estimates of total resident population for each state on an annual basis. The following table outlines it.
StateBirthsDeaths(birth-death) international internaltotal
migration migrationmigration
______
Alabama6226745798164695849 -43421507
Alaska10429314272871514 13002814
Arizona84876422534262339843 66511106354
Arkansas375532830492495339 8246163
California529610234012295598326917 -108595218322
Colorado66380291753720526118 1160837726
Connecticut42854308331202116074 -140914665
Delaware10958731536432565 46607225
District of Columbia8263577324904517 -10059-5542
Florida21012217058039542122484 176531299015
Georgia138575660517252444606 3602380629
Hawaii189427620113226513 3176830
Idaho2029810051102473462 664010102
Illinois1866191080847853574769 -73821948
Indiana85356565752878112520 -86723848
Iowa370802858784937085 -10431-3346
Kansas3865225528131249200 -8656544
Kentucky5431840632136866434 423710671
Louisiana6856142624259374665 -18004-13339
Maine13503128316721001 84299430
Maryland76420451143130624621 1626340884
Massachusetts81408584622294632244 -280744170
Michigan134882887534612925678 -26728-1050
Minnesota62768376882508016270 -63489922
Mississippi4464028621160192576 -6648-4072
Missouri75731560991963210413 524615659
Montana1094785232424448 12201668
Nebraska246711559290795483 -5287196
Nevada31496159501554616352 4300559357
New Hampshire144061008243242298 917011468
New Jersey115529753664016362813 -2375939054
New Mexico2719214482127106516 487711393
New York25900616133497672146392 -170828 -24436
North Carolina120942720194892337524 2809865622
North Dakota766761761491821 -4790-3969
Ohio1591921115504764216358 -31303-14945
Oklahoma4910934066150438835 6649499
Oregon45174310471412716449 1770434153
Pennsylvania1441471297371441022501 -322319278
Rhode Island12197987723203935 40307965
South Carolina5786637347205198231 1668324914
South Dakota1056771863381928 -1594-666
Tennessee79576575442203211643 1457926222
Texas349206143899205307161325 41174202499
Utah47163130073415611689 -83773312
Vermont62315389842962 18612823
Virginia101282568244445832551 1858151132
Washington79610458543375630173 1171441887
West Virginia1831420068-1754718 20822800
Wisconsin69052482382081410592 435814950
Wyoming603541751860520 25593079
Descriptive statistics using MINITAB software
Variable n Mean Median TrMean StDev SE Mean
Births 51 79366 54318 64220 93998 13162
Deaths 51 47958 34066 41493 48872 6843
total migration 51 28418 9430 17383 59526 8335
Variable Minimum Maximum Q1 Q3
Births 6035 529610 18942 85356
Deaths 3142 234012 12831 57544
total migration -24436 299015 1507 34153
Comparative Boxplot using the MINITAB software.
Answer the following 8 questions using the information above.
- Which of the following can be concluded about the skewness of the datasets?
(a)All three boxplots show negatively skewed data
(b)Some boxplots show negatively skewed and the others positively skewed data
(c)Some boxplots show skewed and the others symmetric data
(d)All three boxplots show positively skewed data
(e)All three boxplots show symmetric data
- Which of the data sets have the highest variability?
(a)Births
(b)Deaths
(c)Total migration
- Which of the following is the IQR for the Deaths data set?
(a)32646
(b)44713
(c)48872
(d)66414
(e)230870
- What is the percentage of the states with more deaths than the births?
(a)0
(b)1/51
(c)2/51
(d)3/51
(e)4/51
- Which of the data sets have the lowest median?
(a)Births
(b)Deaths
(c)Total migration
- Which of the following is the numeric measure (coefficient) of the skewness for the births data?
(a)-0.7994
(b)-0.2665
(c)0.2665
(d)0.7994
(e)0.8444
- Which of the following is the largest 25% of the births data?
(a)12831
(b)18942
(c)54318
(d)57544
(e)85356
- Which of the following is the smallest 25% of the births data?
(a)12831
(b)18942
(c)54318
(d)57544
(e)85356
- A sample of 50 widgets is taken one at a time directly from production. Each item is classified as defective or satisfactory. How many outcomes are possible in the sample space?
(a)2
(b)50
(c)250
(d)52
- A sample of 50 widgets is taken one at a time directly from production. Each item is classified as defective or satisfactory. How many involve exactly 2 defectives?
(a)50
(b)100
(c)1225
(d)2450
Enterprises has a central computer connected to district offices in telecommunications network. At any moment there is a 20% chance that all teleports are busy and incoming messages must be places on hold. Any particular office sends messages infrequently, so that the teleports at successive inquiries have independent status. Suppose three inquiries are received. Answer the following 3 questions using this information.
- What is the probability that the all three encounter a busy computer system?
(a)0.008
(b)0.032
(c)0.488
(d)0.512
(e)0.992
- What is the probability that at least one of the three encounter a busy computer system?
(a)0.008
(b)0.032
(c)0.488
(d)0.512
(e)0.992
- What is the probability that the first message is connected directly and the two after encounter a busy signal?
(a)0.008
(b)0.032
(c)0.488
(d)0.512
(e)0.992
A die is to be rolled twice and we are to observe the numbers that falls face up. Let A be the event of observing the same number, B be the event of observing different numbers, C be the event of observing the total 4. Answer the following 3 questions using the information here.
- Which of the following is the best answer if you are researching the mutually exclusive events?
(a)A, B and C are not mutually exclusive
(b)Only A and B are mutually exclusive
(c)Only A and C are mutually exclusive
(d)Only B and C are mutually exclusive
- Which of the following is the best answer if you are researching the independent events?
(a)None of them are pairwise independent
(b)A and B are pairwise independent
(c)A and C are pairwise independent
(d)B and C are pairwise independent
(e)A, B and C are mutually independent
- Which of the following is P(C|B)?
(a)6/36=0.1667
(b)3/36=0.0833
(c)3/30=0.10
(d)2/36=0.0556
(e)2/30=0.0667
The following table shows the number of panelists in each category of consumer test group.
Family Income
Occupation / Low / Medium / High / TotalHomemaker / 8 / 26 / 6 / 40
Blue-collar worker / 16 / 40 / 14 / 70
White-collar worker / 6 / 62 / 12 / 80
Professional / 0 / 2 / 8 / 10
Total / 30 / 130 / 40 / 200
One person is selected at random, Answer the following 3 questions using this information.
- Find the probability that the selected person is a professional?
(a)0
(b)0.05
(c)0.10
(d)0.15
(e)0.20
- Find the probability that the selected person is a professional with a medium income?
(a)0
(b)0.010
(c)0.015
(d)0.200
(e)0.250
- If the selected person is a professional, find the probability of the selected person having the medium income?
(a)0
(b)0.010
(c)0.015
(d)0.200
(e)0.250
- The following mixed system is established with five units. Each unit has the same failure probability of 0.02.
___
|------| 2 |------|
| ____ |
___|------| 3 |------| _____
------| 1 |-----| ____ |------| 5 |-----
|------| 4 |------|
Which of the following is the system reliability (success probability for the system)?
(a)0.0004
(b)0.02
(c)0.396
(d)0.9604
(e)0.98
An accountant randomly select accounts and check balances. Let X be the number of selected accounts having incorrect balances. The following is the probability mass functions for X.
x01234otherwise
p(x)1/4210/4220/4210/421/420
Answer the following 5 questions using this information.
- What is the probability that the selected number of accounts having incorrect balances is at least 2?
(a)1/42
(b)11/42
(c)31/42
(d)41/42
(e)1
- What is the expected number of selected accounts having incorrect balances? (Hint:E(X))
(a)0
(b)1
(c)2
(d)3
(e)4
- What is the variance for the number of selected accounts having incorrect balances? (Hint:Var(X))
(a)2
(b)0.6667
(c)2.6667
(d)4.6667
(e)6.6667
- The decision maker defines a new random variable, Y=X+2. What is the E(Y)?
(a)Same as E(X)
(b)E(X)+2
(c)E(X)-2
(d)2E(X)
(e)E(X)/2
- The decision maker defines a new random variable, Y=X+2. What is the Var(Y)?
(a)Same as Var(X)
(b)Var(X)+2
(c)Var(X)-2
(d)2Var(X)
(e)4Var(X)
Some of the formulas that you may need
Relative Frequency = Frequency / sample size. Density = Relative Frequency / class width
The Range: R = the largest data - the smallest data The Interquartile range: IQR = Q3 - Q1
Coefficient of variation: CV=100()Coefficient of Skewness: SK=
The sample mean of X's: . The sample variance of X's:
The sample standard deviation of X's:
Mutually exclusive or Disjoint: Two events, A and B have no outcomes in common.
Conditional Probability: For any two events A and B with P(B)>0, the conditional probability of A given B has occurred is defined by P(A | B ) = P(AB) / P(B).
Independence: Two events A and B are independent if P(AB) = P(A)P(B)
For any two events A and B, P(AB) = P(A) + P(B) - P(AB).
For any three events, P(ABC)=P(A) + P(B) + P(C) - P(AB) - P(AC)- P(BC) +P(ABC).
For any three events, P(AB)=P(ABC)+ P(ABC’).
Predicting the Reliability of Systems with n components:
Ri(t): The reliability of individual component = P(component i survives beyond time t)
Rs(t): The reliability of the system = P(system works)
Series SystemRs(t)=R1(t)R2(t)…….Rn(t)
Parallel System Rs(t) = 1 - [1-R1(t)][1-R2(t)]……[1-Rn(t)]
Number of ways of randomly selecting k from n:
Discrete probability distribution, p(x) is legitimate if
- 0 p(x)=P(X=x) 1 for all x where X is a discrete random variable (r.v).
Cumulative Distribution Function, (CDF) for discrete X:
where a and b are
The expected value of a discrete variable, X :
The expected value of a function of discrete variables, h(x) :
If h(x)=aX+b then E(h(X))=aE(X)+b where a and b are constants
The variance of a discrete variable, X:2 = Var(X) = where
If h(x)=aX+b then Var(h(X))=a2Var(X) where a and b are constants
The standard deviation of numerical variable, X: =