the answer to the problems in the end of chapter 7Full description
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CHAPTER 7 REVIEW 1) Suppose the probability that a light light bulb is defeti!e is is "#1# What is the probability probability that $ light bulbs are all defeti!e% A&S' ("#1)$ "#"""1 *) Co+essio+aires ,+o- that atte+da+e at a football stadiu. -ill be /"0""" o+ a lear day0 day0 $0""" if there is a light s+o-0 a+d 10""" if there is hea!y s+o-# 2urther.ore0 the probability of lear s,ies0 light s+o-0 or hea!y s+o- o+ a+y partiular day is 30 1450 a+d 14/0 respeti!ely# What a!erage atte+da+e should be e6peted for the seaso+% A&S' 1 *
(/"0""") +
1 5
( $0""") +
1 /
(10""")
= $70""
5) I+ a lottery0 lottery0 1"0""" ti,ets are sold sold at 1 eah -ith a pri8e of 70"" for o+e -i++er# What is the a!erage result for eah bettor% A&S' The atual -i++i+g payoff is 7$99 beause the -i++er paid 1 for a ti,et0 so -e ha!e :uto.e' Wi+ ;ose Probability' 141"0""" 9099941"0""" Ra+do. Variable' 7$9 9 <1
Thus0 the a!erage result for eah perso+ betti+g o+ the lottery is a "#* "# * loss# $) A .a+ager .ust hoose a.o+g three optio+s# optio+s# :ptio+ A has a 1"= ha+e of resulti+g resulti+g i+ a *"0""" gai+ but other-ise -ill result i+ a 1"0""" loss# loss# :ptio+ > has a "= ha+e of gai+i+g $"0""" a+d a "= ha+e of losi+g *"""# 2i+ally0 2i+ally0 optio+ C has a = ha+e of gai+i+g ?""0""" but other-ise -ill -ill result i+ a loss o *"0"""# Whih optio+ should the .a+ager .a+ager hoose% A&S' E(A) "#1"(*"0""") @ "#9"(<1"0""") 1/0""" E(>) "#"($"0""") @ "#"(<*""") 190""" E(C) "#"(?""0""") @ "#9(<*"0""") *10""" The .a+ager should hoose optio+ C# ) A high-ay e+gi+eer ,+o-s that his re- a+ lay .iles of high-ay o+ a lear day0 * .iles .iles o+ a rai+y day0 a+d o+ly 1 .ile o+ a s+o-y day# day# Suppose the probabilities are "#/0 "#50 a+d "#10 respeti!ely0 -hat are the .ea+ (e6peted !alue) a+d the !aria+e% A&S' µ x *
σ x
= ∑ xi p i = ("#/) + *("#5) + 1("#1) = 5#7 = ∑ ( xi − µ ) * p i = ( − 5#7) * ("#/) + (* − 5#7) * ("#5) + (1 − 5#7) * ("#1) = *#/1
/) A tele!isio+ ga.e sho- has three payoffs -ith -ith the follo-i+g probabilities'
Payoff ()' " 1""" 1"0""" Probability' "#/ "#5 "#1 What are the .ea+ a+d sta+dard de!iatio+ for the payoff !ariable' A&S' µ σ σ
7) At a -arehouse sale 1"" usto.ers are i+!ited to hoose o+e of 1"" ide+tial bo6es# 2i!e bo6es o+tai+ 7"" olor tele!isio+ sets0 * bo6es o+tai+ $" a.orders0 a+d the re.ai+i+g bo6es o+tai+ */" a.eras# What should a usto.er be -illi+g to pay to partiipate i+ the sale% A&S' E(6) 7""("#") @ $"("#*) @ */"("#7") 5* ?) The a!erage a++ual i+o.es of high shool a+d ollege graduates i+ a .id<-ester+ to-+ are *10""" a+d 50"""0 respeti!ely# If a o.pa+y hires perso++el -ith at least a high shool diplo.a a+d *"= of its e.ployees ha!e bee+ through ollege0 -hat is the .ea+ i+o.e of the o.pa+y e.ployees% A&S' E(6) *10"""("#?) @ 50"""("#*) *50?"" 9) Bou a+ hoose o+e of the three bo6es# >o6 A has four bills a+d a si+gle 1"" bill0 bo6 > has $"" bills a+d 1"" 1"" bills0 a+d bo6 C has *$ 1 bills# Bou a+ ha!e all of bo6 C or bli+dly pi, o+e bill out of either bo6 A or bo6 ># Whih offers the greatest e6peted -i++i+g% A&S' E(A) ($4) @ 1""(14) *$ E(>) ($""4"") @ 1""(1""4"") *$ E(C) *$ All offer the sa.e e6peted -i++i+g# 1") >obs o..uti+g ti.es to -or, ha!e a +or.al distributio+ -ith a .ea+ of $ .i+utes a+d sta+dard de!iatio+ of 1" .i+utes# Ho- ofte+ does >ob get to -or, i+ 5" to $ .i+utes% A&S' +or.aldf(5"0 $0 $0 1") $5#5*= 11) Ti.es u+til ser!ie at a restaura+t ha!e a +or.al distributio+ -ith .ea+ of 1" .i+utes a+d sta+dard de!iatio+ of 5 .i+utes# Whats the ha+e of it ta,i+g lo+ger tha+ 1 .i+utes to get ser!ie% Whats the ha+e of it ta,i+g +o .ore tha+ 1 .i+utes% A&S' +or.aldf(10 1E990 1"0 5) $#$/=D +or.aldf(<1E990 10 1"0 5) 9#$= 1*) :+e states a++ual rai+fall has a +or.al distributio+ -ith a .ea+ of 1"" i+hes a+d sta+dard de!iatio+ * i+hes# Suppose or+ gro-s best -he+ the a++ual rai+fall is bet-ee+ 1"" a+d 1" i+hes# Whats the ha+e of ahie!i+g this a.ou+t of rai+fall% A&S' +or.aldf(1""0 1"0 1""0 *) $7#75=
15) oes the follo-i+g table represe+t the probability distributio+ for a disrete ra+do. !ariable% F 1 * 5 $ P(F) "#* "#5 "#5 "#$ A&S' &o0 beause Gp 1#* 1$) A ra+do. !ariable F has &(150 "#$)# esribe the distributio+ of * $F (that is0 eah data poi+t i+ the distributio+ is .ultiplied by $0 a+d that !alue is subtrated fro. *#) A&S' We are gi!e+ that the distributio+ of F is +or.al -ith µ x = 15 a+d σ x = "#$ # >eause µ a bx = a ± b µ x 0 µ * $ x = * − $ µ x = * − $(15) = −"# Also0 beause σ a bx = bσ x 0 σ * $ x = $σ x = $("#$) = 1#? # ±
−
±
−
1) The PA of stude+ts -ho ta,e the AP Statistis e6a. is appro6i.ately +or.ally distributed -ith a .ea+ of 5#$ -ith a sta+dard de!iatio+ of "#5# What is the probability that a stude+t seleted at ra+do. fro. this group has a PA lo-er tha+ 5#"% A&S' P(F J 5#") +or.aldf(<1E990 50 5#$0 "#5) "#"91? 1/) Co+sider a set of 9""" sores o+ a +atio+al test that is ,+o-+ to be appro6i.ately +or.ally distributed -ith a .ea+ of "" a+d a sta+dard de!iatio+ of 9"# a) What is the probability that a ra+do.ly seleted stude+t has a sore greater tha+ /""% b) Ho- .a+y sore are there bet-ee+ $" a+d /""% ) Rahel +eeds to be i+ the top 1= of the sores o+ this test of Kualify for a sholarship# What is the .i+i.u. sore Rahel +eeds% A&S' a) &(""0 9") P(F L /"") +or.aldf(/""0 1E990 ""0 9") "#1555 b) P($" J F J /"") +or.aldf($"0 /""0 ""0 9") "#79" There are "#79"(9""") 197 sores# ) i+!&or.("#990 ""0 9") 7"9#57 *
17) Co+sider a ra+do. !ariable F -ith µ x = 50 σ x = "#* # 2i+d a) µ 5 / b) σ 5 / A&S' a) µ 5 / x = 5 + / µ x = 5 + /(5)= *1 * * * * * * b) >eause σ a bx = b σ 0 σ 5 / x = / σ x = 5/("#*) = 9 +
+
x
x
+
+
Thus0
+
σ 5+ / x
=
* σ 5+ / x
=
9
=5
1?) Co+sider t-o disrete0 i+depe+de+t0 ra+do. !ariables F a+d B -ith µ x 2i+d µ a+d σ # A&S' x + y
µ x + y = µ x + µ y =
x + y
5+ = ?
=
*
50 σ x
= 10 µ y =
*
0 σ y
= 1#5#
>eause F a+d B are i+depe+de+t0 -e ha!eσ x + y = σ x* + σ y* = 1 + 1#5 = 1#* 19) Co+sider a probability de+sity ur!e defi+ed by the li+e y *6 o+ the i+ter!al M"01N (the area u+der y *6 o+ M"01N is 1#) 2i+d P("#* O F O "#7)# A&S' (b + b* )h M*("#*) + *("#7)N("#7 − "#*) = = "#$ The shaded area is a trape8oid -hose area is 1 * * *") Co+sider the follo-i+g t-o probability distributio+s for i+depe+de+t disrete ra+do. !ariable F a+d B' F * 5 $ P(F) "#5 "# % B P(B)