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Chapter 1. What is Statistics? 1
Practice Final Questions for Statistics 225Introduction To Probability Models
Material Covered: Chapters 1-7 of Workbook and Text
This is a 2 hour final, worth 25% and marked out of 25 points. The total possiblepoints awarded for each question is given in square brackets at the beginning of eachquestion. Anything that can fit on two sides of an 81
2by 11 inch piece of paper may
be used as a reference during this quiz. A calculator may also be used. No other aidsare permitted.
1. What is Statistics?
(a) One hundred and twenty (120) pea plants are selected at random and the numberof pea pods produced per plant is measured (observed). From this group, anaverage number of pea pods per plant is computed. Match the columns: Allof the items in the first column will be used up in the matching procedure;however, one item in the second column will be left unmatched.
statistical terms pea pods example
(a) value of variable (a) average number of pea pods per plant for 120 pea plants(b) variable (b) all pea plants(c) parameter (c) number of pea pods per plant for all pea plants(d) population (d) number of pea pods for a pea plant(e) sample (e) average number of pods per plant for all pea plants(f) statistic (f) 120(g) sample size (g) number of pea pods per plant for 120 pea plants
(h) number of pea pods for a particular pea plant
terms (a) (b) (c) (d) (e) (f) (g)pea pod example
(b) Assume measurements for Ph levels in soil follow exactly a normal relative fre-quency distribution with population mean π = 5 and population standard de-viation π = 1.4. Use Empirical rule to determine percentage of Ph levels ininterval 3.6 to 6.4 (choose one).
(i) 0.68
(ii) 0.78
(iii) 0.95
(iv) 0.99
(v) 0.995
Chapter 2. Probability 2
2. Probability
(a) [1 point] Describe sample space associated with flipping a coin until either headsor tails occurs twice. Choose one.
(i) {π»π»π, ππ»π»,π»ππ», ππ,π»ππ, ππ»π}(ii) {π»π»,ππ»π»,π»ππ»π, ππ,π»ππ, ππ»π}(iii) {π»π»,ππ»π»,π»ππ», ππ,π»ππ, ππ»π}(iv) {π»π»,ππ»π»,π»ππ», ππ, πππ», ππ»π}(v) {π»π»,π»π»π,π»ππ», ππ, πππ», ππ»π}
(b) [1 point] Number of fourβdigit numbers that can be formed from digits 1, 2 and3, if each fourβdigit number must be odd is (choose one)
(i) 27
(ii) 35
(iii) 44
(iv) 54
(v) 67
(c) [1 point] In two rolls of a fair die, let event π΄ be the event that no fours, fives orsixes are rolled. Then, π (π΄) = (choose one)
(i) 836
(ii) 936
(iii) 1036
(iv) 1136
(v) 1336
(d) [1 point] Let πΈ and πΉ be two events of an experiment where π (πΈ) = 0.35,π (πΉ ) = 0.15 and π (πΈ β© πΉ ) = 0.03. Then π (οΏ½ΜοΏ½ βͺ πΉ ) =(i) 0.96
(ii) 0.97
(iii) 0.98
(iv) 0.99
(v) 1.00
Chapter 2. Probability 3
(e) [1 point] A survey was conducted comparing age with number of visits per yearto doctor. One person is chosen at random.
age β youth middleβaged elderly row totalsvisits 1 to 3 70 95 35 200
4 to 8 130 450 30 6109 to 11 90 30 70 190
column totals 290 575 135 1000
Chance person is a youth, given s/he makes 4β8 visits is (choose closest one):
(i) 0.112
(ii) 0.130
(iii) 0.183
(iv) 0.213
(v) 0.303
(f) [1 point] Two tickets drawn at random without replacement from following box.
1π 2π 1π 3π 2π 3π
Probability first ticket is a β1β and second card is a β2β is (choose closest one)
(i) 0.1333
(ii) 0.2163
(iii) 0.2566
(iv) 0.3777
(v) 0.4333
(g) [1 point] Urn A has 10 red and 9 blue marbles; urn B has 10 red and 10 bluemarbles. A fair coin is tossed. If coin comes up heads, a marble from urn A ischosen, otherwise a marble from urn B is chosen. Chance coin is flipped headsgiven a red marble is chosen is (choose closest one)
(i) 1739
(ii) 1839
(iii) 1939
(iv) 2039
(v) 2139
Chapter 3. Discrete Random Variables and Their Probability Distributions 4
3. Discrete Random Variables and Their Probability Distributions
(a) [1 point] Number of sales of household appliances, π , Whirlpool representativeDarlene makes in a day is given by following probability distribution.
π¦ 0 1 2 3 4 5π(π¦) 0.10 0.28 0.18 0.11 0.16 0.17
Expected number of sales she makes is (choose closest one):
(i) 0.41
(ii) 1.45
(iii) 2.46
(iv) 3.45
(v) 3.76
(b) [1 point] Number of sales of household appliances, π , Whirlpool representativeDarlene makes in a day is given by following probability distribution.
π¦ 0 1 2 3 4 5π(π¦) 0.10 0.28 0.18 0.11 0.16 0.17
Standard deviation in number of sales she makes is (choose closest one):
(i) 0.37
(ii) 0.40
(iii) 1.66
(iv) 2.75
(v) 3.76
(c) [1 point] If π (π ) = 6, then π (2π β 4) = (choose one)(i) 8
(ii) 16
(iii) 20
(iv) 24
(v) 32
Chapter 3. Discrete Random Variables and Their Probability Distributions 5
(d) [1 point] On a multiple choice exam with 5 possible answers for each of 10 ques-tions, what is probability a student gets 8 or more correct answers just byguessing? Choose closest one. [Hint: binomial.]
(i) 5.7926 Γ 10β5(ii) 6.7926 Γ 10β5(iii) 7.7926 Γ 10β5(iv) 8.7926 Γ 10β5(v) 9.7926 Γ 10β5
(e) [1 point] There is a 43% chance of making a basket on a free throw and eachthrow is independent of each other throw. What is expected number of throwsto make first basket? Choose one. [Hint: geometric.]
(i) 2.33
(ii) 4.65
(iii) 6.11
(iv) 8.39
(v) 10.42
(f) [1 point] There is a 95% chance of passing any exam. What is variance in numberof attempts until third exam is passed? Choose closest one. [Hint: negativebinomial.]
(i) 0.146
(ii) 0.156
(iii) 0.166
(iv) 0.176
(v) 0.186
(g) [1 point] Eight journalists randomly picked from a pack of 240 of which 15 arealso photographers. Chance 3 of 8 picked are photographers is (choose one)
(i)
(83
)(2325
)(
2408
) (ii)(
153
)(2255
)(
2258
) (iii)(
153
)(2255
)(
2408
)
(iv)
(155
)(2253
)(
2408
) (v)(
153
)(55
)(
158
)
Chapter 3. Discrete Random Variables and Their Probability Distributions 6
(h) [1 point] Average of π = 7 particles hit a magnetic detection field per microsec-ond. What is probability at most 5 particles hit in one microsecond? Chooseclosest one. [Hint: poisson.]
(i) 0.231
(ii) 0.254
(iii) 0.273
(iv) 0.293
(v) 0.301
(i) [1 point] Identify the moment generating function
π(π‘) =14ππ‘
1β 34ππ‘.
(i) binomial, π = 4
(ii) binomial, π = 4
(iii) geometric, π = 4
(iv) geometric, π = 4
(v) poisson, π = 4
(j) [1 point] According to Tchebysheffβs Theorem, if π = 2 and π = 0.5 for randomvariable π , then π (1 < π < 3) β₯ π where π = (choose one)(i) 0.75
(ii) 0.80
(iii) 0.85
(iv) 0.90
(v) 0.95
Chapter 4. Continuous Variables and Their Probability Distributions 7
4. Continuous Variables and Their Probability Distributions
(a) [1 point] Let π be a continuous random variable where
π(π¦) =
{ππ¦ + 5 if 0 β€ π¦ β€ 100 otherwise
Then constant π is (choose one)
(i) β4750
(ii) β4850
(iii) β4950
(iv) β5050
(v) does not exist
(b) [1 point] Let π be a continuous random variable where
π(π¦) =
{1π
if β3 β€ π¦ β€ 150 otherwise
Then constant π = (choose one)
(i) 3
(ii) 9
(iii) 12
(iv) 15
(v) 18
(c) [1 point] Let π be a continuous random variable where
π(π¦) =
{118
if β3 β€ π¦ β€ 150 otherwise
Then, for β3 β€ π¦ β€ 15, distribution πΉ (π¦) =(i) πβ3
18
(ii) π15
(iii) πβ315
(iv) π18
(v) π+318
Chapter 4. Continuous Variables and Their Probability Distributions 8
(d) [1 point] Let π be a continuous random variable where
πΉ (π¦) =
β§β¨β©0, π¦ < β3,π¦+318
, β3 β€ π¦ < 15,1, π¦ β₯ 15.
π (β2 < π < 9) β (choose closest one)(i) 0.61
(ii) 0.68
(iii) 0.73
(iv) 0.79
(v) 0.81
(e) [1 point] Let π be a continuous random variable where
π(π¦) =
{118
if β3 β€ π¦ β€ 150 otherwise
Then expected value π = (choose closest one)
(i) 3
(ii) 6
(iii) 9
(iv) 15
(v) 18
(f) [1 point] Let π be a continuous random variable where
π(π¦) =
{118
if β3 β€ π¦ β€ 150 otherwise
Then variance π2 = (choose closest one)
(i) 23
(ii) 24
(iii) 25
(iv) 26
(v) 27
Chapter 4. Continuous Variables and Their Probability Distributions 9
(g) [1 point] Let π be a continuous random variable where
π(π¦) =
{118
if β3 β€ π¦ β€ 150 otherwise
Then πΈ[2π 3 β π 2] = (choose closest one)(i) 1241
(ii) 1341
(iii) 1441
(iv) 1541
(v) 1641
(h) [1 point] Let π be a standard normal variable.π (β2.3 < π < 0.14) = (choose closest one)(i) 0.4449
(ii) 0.5449
(iii) 0.6449
(iv) 0.7449
(v) 0.8449
(i) [1 point] Gamma function evaluated at 6 is Ξ(6) = (choose one)
(i) 6
(ii) 24
(iii) 120
(iv) 720
(v) 5040
(j) [1 point] A chiβsquared random variable is a special case of a gamma randomvariable with parameters (πΌ, π½) = (choose one)
(i)(π2, 0)
(ii)(π2, 12
)(iii)
(π2, 1)
(iv)(π2, 2)
(v)(π3, 2)
Chapter 4. Continuous Variables and Their Probability Distributions 10
(k) [1 point] Memoryless property of exponential distribution is (choose one)
(i) π· (π > π + πβ£π > π) = π· (π > π); π, π β₯ 0(ii) π· (π > π + πβ£π > π) = π· (π > π); π, π β₯ 0(iii) π· (π > πβ£π > π) = π· (π > π); π, π β₯ 0(iv) π· (π > π + πβ£π > π) = π· (π > π + π); π, π β₯ 0(v) π· (π > πβ£π > π) = π· (π > π)π· (π > π); π, π β₯ 0
(l) [1 point] For a Beta random variable, parameters (πΌ, π½) = (4.5, 6.5),π = (choose closest one)
(i) 0.409
(ii) 0.419
(iii) 0.429
(iv) 0.439
(v) 0.449
(m) [1 point] Momentβgenerating function for normal random variable π is πππ‘+π2π‘2/2
and so, for π(π‘) = πβ5π‘+6π‘2, π (π β€ β7) β (choose closest one)
(i) 0.104
(ii) 0.211
(iii) 0.233
(iv) 0.254
(v) 0.282
Chapter 5. Multivariate Probability Distributions 11
5. Multivariate Probability Distributions
(a) [1 point] Consider joint density π(π¦1, π¦2)
π¦2 β π¦1 β 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0
The marginal density for π2 is (choose one)
(i)π¦1 -1 -2
π(π¦1) 0.5 0.5
(ii)π¦1 1 2 3
π(π¦1) 0.3 0.6 0.1
(iii)π¦2 -1 -2
π(π¦2) 0.5 0.5
(iv)π¦2 1 2 3
π(π¦2) 0.3 0.6 0.1
(v)π¦2 -1 -2
π(π¦1) 0.5 0.5
(b) [1 point] Consider joint density π(π¦1, π¦2)
π¦2 β π¦1 β 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0
πΉ (3,β1) = (choose closest one)(i) 0.1
(ii) 0.2
(iii) 0.3
(iv) 0.4
(v) 0.5
Chapter 5. Multivariate Probability Distributions 12
(c) [1 point] Consider joint density of π1 and π2
π(π¦1, π¦2) =
{14(3π¦1 + 5π¦2), 0 β€ π¦1 β€ 1, 0 β€ π¦2 β€ 1
0, otherwise
and also marginal densities for π1 and π2
π1(π¦1) =
{34π¦1 +
58
0 < π¦1 < 10 elsewhere
and
π2(π¦2) =
{38+ 5
4π¦2 0 < π¦2 < 1
0 elsewhere
Then π(π¦1β£π¦2) = (choose one)
(i)14(3π1+5π2)38+5
4π1
(ii)34π1+
58
14(3π1+5π2)
(iii)14(3π1+5π2)34π1+
58
(iv)38+5
4π2
14(3π1+5π2)
(v)14(3π1+5π2)38+5
4π2
(d) [1 point] Random variables π1 and π2 independent if (choose one)
(i) π(π¦1, π¦2) = π1(π¦1)π2(π¦2)
(ii) π(π¦1, π¦2) β= π1(π¦1)π2(π¦2)(iii) π(π¦1, π¦2) = π1(π¦1) + π2(π¦2)
(iv) π(π¦1, π¦2) β= π1(π¦1) + π2(π¦2)(v) π(π¦1, π¦2) β= π1(π¦1)π2(π¦2)
Chapter 5. Multivariate Probability Distributions 13
(e) [1 point] Consider joint density π(π¦1, π¦2)
π¦2 β π¦1 β 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0
π (π1) = (choose one)
(i) 0.16
(ii) 0.22
(iii) 0.28
(iv) 0.32
(v) 0.36
(f) [1 point] Let
π(π¦1, π¦2) =
{6(1β π¦2), 0 β€ π¦1 β€ π¦2 β€ 10, otherwise
Then πΈ(π1π2) = (circle one)
(i) 120
(ii) 220
(iii) 320
(iv) 420
(v) 520
Chapter 5. Multivariate Probability Distributions 14
(g) [1 point] If
π(π¦1, π¦2) =
{6(1β π¦2), 0 β€ π¦1 β€ π¦2 β€ 10, otherwise
and two marginal densities are
π1(π¦1) = 3β 6π¦1 + 3π¦21and
π2(π¦2) = 6π¦2 β 6π¦22and πΈ(π1π2) =
320, then Cov(π1, π2) = (choose one)
(i) 140
(ii) 240
(iii) 340
(iv) 440
(v) 540
(h) [1 point] Consider joint density π(π¦1, π¦2)
π¦2 β π¦1 β 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0
Cov(3π1, 4π2) = (choose one)
(i) 2.1
(ii) 2.2
(iii) 2.3
(iv) 2.4
(v) 2.5
Chapter 5. Multivariate Probability Distributions 15
(i) [1 point] Consider density
π(π¦1, π¦2) =
{18π¦1π
β π¦1+π¦22 , π¦1 > 0, π¦2 > 0
0, otherwise
then π1 and π2 are independent, where (choose one)
(i) π1 is gamma where πΌ = 2 and π½ = 2, and π2 is exponential where π = 1
(ii) π1 is gamma where πΌ = 2 and π½ = 2, and π2 is exponential where π = 3
(iii) π1 is gamma where πΌ = 2 and π½ = 3, and π2 is exponential where π = 2
(iv) π1 is gamma where πΌ = 2 and π½ = 2, and π2 is exponential where π = 2
(v) π1 is gamma where πΌ = 3 and π½ = 2, and π2 is exponential where π = 2
(j) [1 point] There are 9 different faculty members and 3 subjects: mathematics,statistics and physics. There is a 60%, 35% and 15% chance a faculty mem-ber teaches mathematics, statistics and physics, respectively. Let π1, π2 andπ3 represent number of faculty teaching mathematics, statistics and physics,respectively. Then π (π1 + 3π2) = (choose closest one)
(i) 9.0475
(ii) 9.1475
(iii) 9.2475
(iv) 9.3475
(v) 9.4475
Chapter 6. Functions of Random Variables 16
6. Functions of Random Variables
(a) [1 point] Let π be a continuous random variable where
π(π¦) =
{32π¦2, β1 β€ π¦ β€ 1
0 elsewhere
Determine density for π = 3β π . Choose one.(i) ππΌ(π) =
12(3 β π)2 , 2 β€ π β€ 4
(ii) ππΌ(π) =22(3 β π)2 , 2 β€ π β€ 3
(iii) ππΌ(π) =32(3 β π)2 , 2 β€ π β€ 4
(iv) ππΌ(π) =42(3 β π)2 , 2 β€ π β€ 4
(v) ππΌ(π) =52(3 β π)2 , 2 β€ π β€ 6
(b) [1 point] Let π be a continuous random variable where
π(π¦) =
{12, 9 β€ π¦ β€ 11
0 elsewhere
If π = 2π 2, then ππ(π’) = ππ (ββ1(π’))
β£β£β£ πππ’ββ1(π’)
β£β£β£ = (choose one)(i) ππ
[(π2
)12
] β£β£β£β£14 (π2)β13β£β£β£β£ = 18 (π2)β13
(ii) ππ
[(π2
)12
] β£β£β£β£13 (π2)β12β£β£β£β£ = 16 (π2)β12
(iii) ππ
[(π3
)12
] β£β£β£β£14 (π2)β12β£β£β£β£ = 112 (π2)β13
(iv) ππ
[(π2
)12
] β£β£β£β£14 (π2)β12β£β£β£β£ = 18 (π2)β12
(v) ππ
[(π2
)13
] β£β£β£β£14 (π2)β13β£β£β£β£ = 112 (π3)β13
Chapter 6. Functions of Random Variables 17
(c) [1 point] Consider independent geometric variables π1, π2, π3, all with parameterπ, π = 1, 2, 3, and so all with moment generating function,
πππ(π‘) =
[πππ‘
1β (1β π)ππ‘], π = 1, 2, 3.
Calculate moment generating function of π = π1 + π2 + π3 to determine distri-bution of π (choose one):
(i) binomial, with parameters (π =β3
π=1 ππ, π)
(ii) negative binomial with parameters (π = 3, π)
(iii) negative binomial with parameters (π =β3
π=1 ππ, π)
(iv) geometric with parameter π
(v) geometric with parameter 3
(d) [1 point] Consider π1, . . . , ππ independent beta with πΌ = 4 and π½ = 1,[Ξ(πΌ+ π½)
Ξ(πΌ)Ξ(π½)
]π¦πΌβ1(1β π¦)π½β1 =
[Ξ(5)
Ξ(4)Ξ(1)
]π¦4β1(1β π¦)1β1 = 4π¦3,
with distribution function
πΉπ (π¦) =β« π¦04π‘3 ππ‘ = π¦4.
Expected value of π(π) = max(π1, . . . , ππ) is (choose one)
(i) π4π+1
(ii) 4π4π+1
(iii) 4π4π+3
(iv) π4π+5
(v) π4π+6
Chapter 7. Sampling Distributions and the Central Limit Theorem 18
7. Sampling Distributions and the Central Limit Theorem
(a) [1 point] Assume number of fish caught, π , at a lake on any trip, is a randomvariable with following distribution.
π¦ 1 2 3π(π¦) 0.1 0.8 0.1
Two parameters, ποΏ½ΜοΏ½ and π2οΏ½ΜοΏ½ , for sampling distribution of average number of
fish caught on two trips to lake are given by, respectively, (choose closest pair)
(i) (2, 0.1)
(ii) (2, 0.2)
(iii) (2, 0.3)
(iv) (2, 0.4)
(v) (2, 0.5)
(b) [1 point] Consider π , follows a π‘ distribution where π = 15.If π (π β€ π0.75) = 0.75, π0.75 = (choose one)(i) 0.61
(ii) 0.65
(iii) 0.69
(iv) 0.73
(v) 0.77
(c) [1 point] Suppose lake level, π , on any given day in Lake Michigan is normallydistributed, variance in lake level, π21 , is measured over π1 = 5 random days atSt. Joseph harbor, variance in lake level, π22 is measured over π2 = 7 random
days at South Haven harbor. If π21 = 3π22 and π
(π21π22
< π)= 0.95, then π =
(choose one)
(i) 9.34
(ii) 10.24
(iii) 10.75
(iv) 10.85
(v) 11.03
Chapter 7. Sampling Distributions and the Central Limit Theorem 19
(d) [1 point] We want to know fraction of times a measuring instrument is incorrect.How many measurements should be taken by instrument if we want samplefraction incorrect to within 0.05 of population fraction incorrect with probability0.80? (Hint: maximum number occurs at π = 0.5.) Choose one.
(i) 160
(ii) 164
(iii) 170
(iv) 174
(v) 180
Chapter 1 Practice Final Answers. What is Statistics? 20
1. What is Statistics?
(a) h, d, e, c, g, a, f
(b) (i) 0.68
Chapter 2 Practice Final Answers. Probability 21
2. Probability
(a) (iii) {π»π»,ππ»π»,π»ππ», ππ,π»ππ, ππ»π}(b) (iv) 54
(c) (ii) 936
(d) (ii) 0.97
(e) (iv) 0.213
(f) (i) 0.1333
(g) (iv) 2039
Chapter 3 Practice Final Answers. Discrete Random Variables and Their Probability Distributions22
3. Discrete Random Variables and Their Probability Distributions
(a) (iii) 2.46
(b) (iii) 1.66
(c) (iv) 24
(d) (iii) 7.7926 Γ 10β5
(e) (i) 2.33
(f) (iii) 0.166
(g) (iii)
(153
)(2255
)(
2408
)
(h) (v) 0.301
(i) (iii) geometric, π = 4
(j) (i) 0.75
Chapter 4 Practice Final Answers. Continuous Variables and Their Probability Distributions23
4. Continuous Variables and Their Probability Distributions
(a) (v) does not exist
(b) (v) 18
(c) (v) π+318
(d) (i) 0.61
(e) (ii) 6
(f) (v) 27
(g) (ii) 1341
(h) (ii) 0.5449
(i) (iii) 120
(j) (iv)(π2, 2)
(k) (ii) π· (π > π + πβ£π > π) = π· (π > π); π, π β₯ 0(l) (i) 0.409
(m) (v) 0.282
Chapter 5 Practice Final Answers. Multivariate Probability Distributions 24
5. Multivariate Probability Distributions
(a) (iii)
π¦2 -1 -2π(π¦2) 0.5 0.5
(b) (v) 0.5
(c) (v)14(3π1+5π2)38+5
4π2
(d) (i) π(π¦1, π¦2) = π1(π¦1)π2(π¦2)
(e) (v) 0.36
(f) (iii) 320
(g) (i) 140
(h) (iv) 2.4
(i) (iv) π1 is gamma where πΌ = 2 and π½ = 2, and π2 is exponential where π½ = 2
(j) (iii) 9.2475
Chapter 6 Practice Final Answers. Functions of Random Variables 25
6. Functions of Random Variables
(a) (iii) ππΌ(π) =32(3 β π)2 , 2 β€ π β€ 4
(b) (iv) ππ
[(π2
)12
] β£β£β£β£14 (π2)β12β£β£β£β£
(c) [1 point] (ii) negative binomial with parameters (π = 3, π)
(d) (ii) 4π4π+1
Chapter 7 Practice Final Answers. Sampling Distributions and the Central Limit Theorem26
7. Sampling Distributions and the Central Limit Theorem
(a) (i) (2, 0.1)
(b) (iii) 0.69
(c) (v) 11.03
(d) (ii) 164