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 𝜇 = 5 and population standard de-viation 𝜎 = 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) {𝐻𝐻𝑇, 𝑇𝐻𝐻,𝐻𝑇𝐻, 𝑇𝑇,𝐻𝑇𝑇, 𝑇𝐻𝑇}(ii) {𝐻𝐻,𝑇𝐻𝐻,𝐻𝑇𝐻𝑇, 𝑇𝑇,𝐻𝑇𝑇, 𝑇𝐻𝑇}(iii) {𝐻𝐻,𝑇𝐻𝐻,𝐻𝑇𝐻, 𝑇𝑇,𝐻𝑇𝑇, 𝑇𝐻𝑇}(iv) {𝐻𝐻,𝑇𝐻𝐻,𝐻𝑇𝐻, 𝑇𝑇, 𝑇𝑇𝐻, 𝑇𝐻𝑇}(v) {𝐻𝐻,𝐻𝐻𝑇,𝐻𝑇𝐻, 𝑇𝑇, 𝑇𝑇𝐻, 𝑇𝐻𝑇}

(b) [1 point] Number of four–digit numbers that can be formed from digits 1, 2 and3, if each four–digit 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 𝐴 be the event that no fours, fives orsixes are rolled. Then, 𝑃 (𝐴) = (choose one)

(i) 836

(ii) 936

(iii) 1036

(iv) 1136

(v) 1336

(d) [1 point] Let 𝐸 and 𝐹 be two events of an experiment where 𝑃 (𝐸) = 0.35,𝑃 (𝐹 ) = 0.15 and 𝑃 (𝐸 ∩ 𝐹 ) = 0.03. Then 𝑃 (�̄� ∪ 𝐹 ) =(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 → youth middle–aged 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 4–8 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.

1𝑎 2𝑎 1𝑏 3𝑏 2𝑐 3𝑐

Probability first ticket is a “1” and second card is a “2” 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, 𝑌 , Whirlpool representativeDarlene makes in a day is given by following probability distribution.

𝑦 0 1 2 3 4 5𝑝(𝑦) 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, 𝑌 , Whirlpool representativeDarlene makes in a day is given by following probability distribution.

𝑦 0 1 2 3 4 5𝑝(𝑦) 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 𝑉 (𝑌 ) = 6, then 𝑉 (2𝑌 − 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 × 10−5(ii) 6.7926 × 10−5(iii) 7.7926 × 10−5(iv) 8.7926 × 10−5(v) 9.7926 × 10−5

(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 𝜆 = 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

𝑚(𝑡) =14𝑒𝑡

1− 34𝑒𝑡.

(i) binomial, 𝝁 = 4

(ii) binomial, 𝝈 = 4

(iii) geometric, 𝝁 = 4

(iv) geometric, 𝝈 = 4

(v) poisson, 𝝁 = 4

(j) [1 point] According to Tchebysheff’s Theorem, if 𝜇 = 2 and 𝜎 = 0.5 for randomvariable 𝑌 , then 𝑃 (1 < 𝑌 < 3) ≥ 𝑎 where 𝑎 = (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 𝑌 be a continuous random variable where

𝑓(𝑦) =

{𝑘𝑦 + 5 if 0 ≤ 𝑦 ≤ 100 otherwise

Then constant 𝑘 is (choose one)

(i) −4750

(ii) −4850

(iii) −4950

(iv) −5050

(v) does not exist

(b) [1 point] Let 𝑌 be a continuous random variable where

𝑓(𝑦) =

{1𝑘

if −3 ≤ 𝑦 ≤ 150 otherwise

Then constant 𝑘 = (choose one)

(i) 3

(ii) 9

(iii) 12

(iv) 15

(v) 18

(c) [1 point] Let 𝑌 be a continuous random variable where

𝑓(𝑦) =

{118

if −3 ≤ 𝑦 ≤ 150 otherwise

Then, for −3 ≤ 𝑦 ≤ 15, distribution 𝐹 (𝑦) =(i) 𝒚−3

18

(ii) 𝒚15

(iii) 𝒚−315

(iv) 𝒚18

(v) 𝒚+318

Chapter 4. Continuous Variables and Their Probability Distributions 8

(d) [1 point] Let 𝑌 be a continuous random variable where

𝐹 (𝑦) =

⎧⎨⎩0, 𝑦 < −3,𝑦+318

, −3 ≤ 𝑦 < 15,1, 𝑦 ≥ 15.

𝑃 (−2 < 𝑌 < 9) ≈ (choose closest one)(i) 0.61

(ii) 0.68

(iii) 0.73

(iv) 0.79

(v) 0.81

(e) [1 point] Let 𝑌 be a continuous random variable where

𝑓(𝑦) =

{118

if −3 ≤ 𝑦 ≤ 150 otherwise

Then expected value 𝜇 = (choose closest one)

(i) 3

(ii) 6

(iii) 9

(iv) 15

(v) 18

(f) [1 point] Let 𝑌 be a continuous random variable where

𝑓(𝑦) =

{118

if −3 ≤ 𝑦 ≤ 150 otherwise

Then variance 𝜎2 = (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 𝑌 be a continuous random variable where

𝑓(𝑦) =

{118

if −3 ≤ 𝑦 ≤ 150 otherwise

Then 𝐸[2𝑌 3 − 𝑌 2] = (choose closest one)(i) 1241

(ii) 1341

(iii) 1441

(iv) 1541

(v) 1641

(h) [1 point] Let 𝑍 be a standard normal variable.𝑃 (−2.3 < 𝑍 < 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 Γ(6) = (choose one)

(i) 6

(ii) 24

(iii) 120

(iv) 720

(v) 5040

(j) [1 point] A chi–squared random variable is a special case of a gamma randomvariable with parameters (𝛼, 𝛽) = (choose one)

(i)(𝝂2, 0)

(ii)(𝝂2, 12

)(iii)

(𝝂2, 1)

(iv)(𝝂2, 2)

(v)(𝝂3, 2)

Chapter 4. Continuous Variables and Their Probability Distributions 10

(k) [1 point] Memoryless property of exponential distribution is (choose one)

(i) 𝑷 (𝒀 > 𝒔 + 𝒕∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒕); 𝒔, 𝒕 ≥ 0(ii) 𝑷 (𝒀 > 𝒔 + 𝒕∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒔); 𝒔, 𝒕 ≥ 0(iii) 𝑷 (𝒀 > 𝒔∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒔); 𝒔, 𝒕 ≥ 0(iv) 𝑷 (𝒀 > 𝒔 + 𝒕∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒔 + 𝒕); 𝒔, 𝒕 ≥ 0(v) 𝑷 (𝒀 > 𝒔∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒕)𝑷 (𝒀 > 𝒔); 𝒔, 𝒕 ≥ 0

(l) [1 point] For a Beta random variable, parameters (𝛼, 𝛽) = (4.5, 6.5),𝜇 = (choose closest one)

(i) 0.409

(ii) 0.419

(iii) 0.429

(iv) 0.439

(v) 0.449

(m) [1 point] Moment–generating function for normal random variable 𝑌 is 𝑒𝜇𝑡+𝜎2𝑡2/2

and so, for 𝑚(𝑡) = 𝑒−5𝑡+6𝑡2, 𝑃 (𝑌 ≤ −7) ≈ (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 𝑝(𝑦1, 𝑦2)

𝑦2 ↓ 𝑦1 → 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0

The marginal density for 𝑌2 is (choose one)

(i)𝑦1 -1 -2

𝑝(𝑦1) 0.5 0.5

(ii)𝑦1 1 2 3

𝑝(𝑦1) 0.3 0.6 0.1

(iii)𝑦2 -1 -2

𝑝(𝑦2) 0.5 0.5

(iv)𝑦2 1 2 3

𝑝(𝑦2) 0.3 0.6 0.1

(v)𝑦2 -1 -2

𝑝(𝑦1) 0.5 0.5

(b) [1 point] Consider joint density 𝑝(𝑦1, 𝑦2)

𝑦2 ↓ 𝑦1 → 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0

𝐹 (3,−1) = (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 𝑌1 and 𝑌2

𝑓(𝑦1, 𝑦2) =

{14(3𝑦1 + 5𝑦2), 0 ≤ 𝑦1 ≤ 1, 0 ≤ 𝑦2 ≤ 1

0, otherwise

and also marginal densities for 𝑌1 and 𝑌2

𝑓1(𝑦1) =

{34𝑦1 +

58

0 < 𝑦1 < 10 elsewhere

and

𝑓2(𝑦2) =

{38+ 5

4𝑦2 0 < 𝑦2 < 1

0 elsewhere

Then 𝑓(𝑦1∣𝑦2) = (choose one)

(i)14(3𝒚1+5𝒚2)38+5

4𝒚1

(ii)34𝒚1+

58

14(3𝒚1+5𝒚2)

(iii)14(3𝒚1+5𝒚2)34𝒚1+

58

(iv)38+5

4𝒚2

14(3𝒚1+5𝒚2)

(v)14(3𝒚1+5𝒚2)38+5

4𝒚2

(d) [1 point] Random variables 𝑌1 and 𝑌2 independent if (choose one)

(i) 𝑓(𝑦1, 𝑦2) = 𝑓1(𝑦1)𝑓2(𝑦2)

(ii) 𝑓(𝑦1, 𝑦2) ∕= 𝑓1(𝑦1)𝑓2(𝑦2)(iii) 𝑓(𝑦1, 𝑦2) = 𝑓1(𝑦1) + 𝑓2(𝑦2)

(iv) 𝑓(𝑦1, 𝑦2) ∕= 𝑓1(𝑦1) + 𝑓2(𝑦2)(v) 𝑓(𝑦1, 𝑦2) ∕= 𝑓1(𝑦1)𝑓2(𝑦2)

Chapter 5. Multivariate Probability Distributions 13

(e) [1 point] Consider joint density 𝑝(𝑦1, 𝑦2)

𝑦2 ↓ 𝑦1 → 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0

𝑉 (𝑌1) = (choose one)

(i) 0.16

(ii) 0.22

(iii) 0.28

(iv) 0.32

(v) 0.36

(f) [1 point] Let

𝑓(𝑦1, 𝑦2) =

{6(1− 𝑦2), 0 ≤ 𝑦1 ≤ 𝑦2 ≤ 10, otherwise

Then 𝐸(𝑌1𝑌2) = (circle one)

(i) 120

(ii) 220

(iii) 320

(iv) 420

(v) 520

Chapter 5. Multivariate Probability Distributions 14

(g) [1 point] If

𝑓(𝑦1, 𝑦2) =

{6(1− 𝑦2), 0 ≤ 𝑦1 ≤ 𝑦2 ≤ 10, otherwise

and two marginal densities are

𝑓1(𝑦1) = 3− 6𝑦1 + 3𝑦21and

𝑓2(𝑦2) = 6𝑦2 − 6𝑦22and 𝐸(𝑌1𝑌2) =

320, then Cov(𝑌1, 𝑌2) = (choose one)

(i) 140

(ii) 240

(iii) 340

(iv) 440

(v) 540

(h) [1 point] Consider joint density 𝑝(𝑦1, 𝑦2)

𝑦2 ↓ 𝑦1 → 1 2 3-1 0 0.4 0.1-2 0.3 0.2 0

Cov(3𝑌1, 4𝑌2) = (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

𝑓(𝑦1, 𝑦2) =

{18𝑦1𝑒

− 𝑦1+𝑦22 , 𝑦1 > 0, 𝑦2 > 0

0, otherwise

then 𝑌1 and 𝑌2 are independent, where (choose one)

(i) 𝑌1 is gamma where 𝛼 = 2 and 𝛽 = 2, and 𝑌2 is exponential where 𝜈 = 1

(ii) 𝑌1 is gamma where 𝛼 = 2 and 𝛽 = 2, and 𝑌2 is exponential where 𝜈 = 3

(iii) 𝑌1 is gamma where 𝛼 = 2 and 𝛽 = 3, and 𝑌2 is exponential where 𝜈 = 2

(iv) 𝑌1 is gamma where 𝛼 = 2 and 𝛽 = 2, and 𝑌2 is exponential where 𝜈 = 2

(v) 𝑌1 is gamma where 𝛼 = 3 and 𝛽 = 2, and 𝑌2 is exponential where 𝜈 = 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 𝑌1, 𝑌2 and𝑌3 represent number of faculty teaching mathematics, statistics and physics,respectively. Then 𝑉 (𝑌1 + 3𝑌2) = (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 𝑌 be a continuous random variable where

𝑓(𝑦) =

{32𝑦2, −1 ≤ 𝑦 ≤ 1

0 elsewhere

Determine density for 𝑈 = 3− 𝑌 . Choose one.(i) 𝒇𝑼(𝒖) =

12(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 4

(ii) 𝒇𝑼(𝒖) =22(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 3

(iii) 𝒇𝑼(𝒖) =32(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 4

(iv) 𝒇𝑼(𝒖) =42(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 4

(v) 𝒇𝑼(𝒖) =52(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 6

(b) [1 point] Let 𝑌 be a continuous random variable where

𝑓(𝑦) =

{12, 9 ≤ 𝑦 ≤ 11

0 elsewhere

If 𝑈 = 2𝑌 2, then 𝑓𝑈(𝑢) = 𝑓𝑌 (ℎ−1(𝑢))

∣∣∣ 𝑑𝑑𝑢ℎ−1(𝑢)

∣∣∣ = (choose one)(i) 𝒇𝒀

[(𝒖2

)12

] ∣∣∣∣14 (𝒖2)−13∣∣∣∣ = 18 (𝒖2)−13

(ii) 𝒇𝒀

[(𝒖2

)12

] ∣∣∣∣13 (𝒖2)−12∣∣∣∣ = 16 (𝒖2)−12

(iii) 𝒇𝒀

[(𝒖3

)12

] ∣∣∣∣14 (𝒖2)−12∣∣∣∣ = 112 (𝒖2)−13

(iv) 𝒇𝒀

[(𝒖2

)12

] ∣∣∣∣14 (𝒖2)−12∣∣∣∣ = 18 (𝒖2)−12

(v) 𝒇𝒀

[(𝒖2

)13

] ∣∣∣∣14 (𝒖2)−13∣∣∣∣ = 112 (𝒖3)−13

Chapter 6. Functions of Random Variables 17

(c) [1 point] Consider independent geometric variables 𝑌1, 𝑌2, 𝑌3, all with parameter𝑝, 𝑖 = 1, 2, 3, and so all with moment generating function,

𝑚𝑌𝑖(𝑡) =

[𝑝𝑒𝑡

1− (1− 𝑝)𝑒𝑡], 𝑖 = 1, 2, 3.

Calculate moment generating function of 𝑈 = 𝑌1 + 𝑌2 + 𝑌3 to determine distri-bution of 𝑈 (choose one):

(i) binomial, with parameters (𝑟 =∑3

𝑖=1 𝑛𝑖, 𝑝)

(ii) negative binomial with parameters (𝑟 = 3, 𝑝)

(iii) negative binomial with parameters (𝑟 =∑3

𝑖=1 𝑛𝑖, 𝑝)

(iv) geometric with parameter 𝑝

(v) geometric with parameter 3

(d) [1 point] Consider 𝑌1, . . . , 𝑌𝑛 independent beta with 𝛼 = 4 and 𝛽 = 1,[Γ(𝛼+ 𝛽)

Γ(𝛼)Γ(𝛽)

]𝑦𝛼−1(1− 𝑦)𝛽−1 =

[Γ(5)

Γ(4)Γ(1)

]𝑦4−1(1− 𝑦)1−1 = 4𝑦3,

with distribution function

𝐹𝑌 (𝑦) =∫ 𝑦04𝑡3 𝑑𝑡 = 𝑦4.

Expected value of 𝑌(𝑛) = max(𝑌1, . . . , 𝑌𝑛) is (choose one)

(i) 𝒏4𝒏+1

(ii) 4𝒏4𝒏+1

(iii) 4𝒏4𝒏+3

(iv) 𝒏4𝒏+5

(v) 𝒏4𝒏+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, 𝑌 , at a lake on any trip, is a randomvariable with following distribution.

𝑦 1 2 3𝑝(𝑦) 0.1 0.8 0.1

Two parameters, 𝜇�̄� and 𝜎2�̄� , 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 𝑇 , follows a 𝑡 distribution where 𝑛 = 15.If 𝑃 (𝑇 ≤ 𝜙0.75) = 0.75, 𝜙0.75 = (choose one)(i) 0.61

(ii) 0.65

(iii) 0.69

(iv) 0.73

(v) 0.77

(c) [1 point] Suppose lake level, 𝑌 , on any given day in Lake Michigan is normallydistributed, variance in lake level, 𝑆21 , is measured over 𝑛1 = 5 random days atSt. Joseph harbor, variance in lake level, 𝑆22 is measured over 𝑛2 = 7 random

days at South Haven harbor. If 𝜎21 = 3𝜎22 and 𝑃

(𝑆21𝑆22

< 𝑏)= 0.95, then 𝑏 =

(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 𝑝 = 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) {𝐻𝐻,𝑇𝐻𝐻,𝐻𝑇𝐻, 𝑇𝑇,𝐻𝑇𝑇, 𝑇𝐻𝑇}(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 × 10−5

(e) (i) 2.33

(f) (iii) 0.166

(g) (iii)

(153

)(2255

)(

2408

)

(h) (v) 0.301

(i) (iii) geometric, 𝝁 = 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) 𝒚+318

(d) (i) 0.61

(e) (ii) 6

(f) (v) 27

(g) (ii) 1341

(h) (ii) 0.5449

(i) (iii) 120

(j) (iv)(𝝂2, 2)

(k) (ii) 𝑷 (𝒀 > 𝒔 + 𝒕∣𝒀 > 𝒕) = 𝑷 (𝒀 > 𝒔); 𝒔, 𝒕 ≥ 0(l) (i) 0.409

(m) (v) 0.282

Chapter 5 Practice Final Answers. Multivariate Probability Distributions 24

5. Multivariate Probability Distributions

(a) (iii)

𝑦2 -1 -2𝑝(𝑦2) 0.5 0.5

(b) (v) 0.5

(c) (v)14(3𝒚1+5𝒚2)38+5

4𝒚2

(d) (i) 𝑓(𝑦1, 𝑦2) = 𝑓1(𝑦1)𝑓2(𝑦2)

(e) (v) 0.36

(f) (iii) 320

(g) (i) 140

(h) (iv) 2.4

(i) (iv) 𝑌1 is gamma where 𝛼 = 2 and 𝛽 = 2, and 𝑌2 is exponential where 𝛽 = 2

(j) (iii) 9.2475

Chapter 6 Practice Final Answers. Functions of Random Variables 25

6. Functions of Random Variables

(a) (iii) 𝒇𝑼(𝒖) =32(3 − 𝒖)2 , 2 ≤ 𝒖 ≤ 4

(b) (iv) 𝒇𝒀

[(𝒖2

)12

] ∣∣∣∣14 (𝒖2)−12∣∣∣∣

(c) [1 point] (ii) negative binomial with parameters (𝑟 = 3, 𝑝)

(d) (ii) 4𝒏4𝒏+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