Chapter 18 Sampling Distribution Models

Transcription

1 302 Part V From the Data at Had to the World at Large Chapter 18 Samplig Distributio Models 1. Coi tosses. a) The histogram of these proportios is expected to be smmetric, but ot because of the Cetral Limit Theorem. The sample of 16 coi flips is ot large. The distributio of these proportios is expected to be smmetric because the probabilit that the coi lads heads is the same as the probabilit that the coi lads tails. b) The histogram is expected to have its ceter at 0.5, the probabilit that the coi lads heads. c) The stadard deviatio of data displaed i this histogram should be approximatel equal pq to the stadard deviatio of the samplig distributio model, = (.)(.) = d) The expected umber of heads, p = 16(0.5) = 8, which is less tha 10. The Success/Failure coditio is ot met. The Normal model is ot appropriate i this case. 2. M&M s. a) The histogram of the proportios of gree cadies i the bags would probabl be skewed slightl to the right, for the simple reaso that the proportio of gree M&M s could ever fall below 0 o the left, but has the potetial to be higher o the right. b) The Normal model caot be used to approximate the histogram, sice the expected umber of gree M&M s is p = 50(0.10) = 5, which is less tha 10. The Success/Failure coditio is ot met. c) The histogram should be cetered aroud the expected proportio of gree M&M s, at about d) The proportio should have stadard deviatio 3. More cois. pq (.)(.) a) µ ˆp = p = 0.5 ad σ ( p ˆ) = = = About 68% of the sample proportios are expected to be betwee 0.4 ad 0.6, about 95% are expected to be betwee 0.3 ad 0.7, ad about 99.7% are expected to be betwee 0.2 ad 0.8. pq = (.)(.)

2 Chapter 18 Samplig Distributio Models 303 b) First of all, coi flips are idepedet of oe aother. There is o eed to check the 10% Coditio. Secod, p = q = 12.5, so both are greater tha 10. The Success/Failure coditio is met, so the samplig distributio model is N(0.5, 0.1). pq (.)(.) c) µ ˆp = p = 0.5 ad σ ( p ˆ) = = = About 68% of the sample proportios are expected to be betwee ad , about 95% are expected to be betwee ad 0.625, ad about 99.7% are expected to be betwee ad Coi flips are idepedet of oe aother, ad p = q = 32, so both are greater tha 10. The Success/Failure coditio is met, so the samplig distributio model is N(0.5, ). d) As the umber of tosses icreases, the samplig distributio model will still be Normal ad cetered at 0.5, but the stadard deviatio will decrease. The samplig distributio model will be less spread out. 4. Bigger bag. a) Radomizatio coditio: The 200 M&M s i the bag ca be cosidered represetative of all M&M s, ad the are thoroughl mixed. 10% coditio: 200 is certail less tha 10% of all M&M s. Success/Failure coditio: p = 20 ad q = 180 are both greater tha 10. b) The samplig distributio model is Normal, with: pq (.)(.) µ ˆp = p = 0.1 ad σ( p ˆ) = = About 68% of the sample proportios are expected to be betwee ad 0.121, about 95% are expected to be betwee ad 0.142, ad about 99.7% are expected to be betwee ad c) If the bags cotaied more cadies, the samplig distributio model would still be Normal ad cetered at 0.1, but the stadard deviatio would decrease. The samplig distributio model would be less spread out.

3 304 Part V From the Data at Had to the World at Large 5. Just (u)luck. For 200 flips, the samplig distributio model is Normal with µ ˆp = p = 0.5 ad pq (.)(.) σ( p ˆ) = = Her sample proportio of p ˆ = 042. is about stadard deviatios below the expected proportio, which is uusual, but ot extraordiar. Accordig to the Normal model, we expect sample proportios this low or lower about 1.2% of the time. 6. Too ma gree oes? For 500 cadies, the samplig distributio model is Normal with µ ˆp = p = 0.1 ad pq (.)(.) σ( p ˆ) = = The sample proportio of p ˆ = 012. is about stadard deviatios above the expected proportio, which is ot at all uusual. Accordig to the Normal model, we expect sample proportios this high or higher about 6.8% of the time. 7. Speedig. a) µ ˆp = p = 0.70 pq ( 07. )( 03. ) σ( p ˆ) = = About 68% of the sample proportios are expected to be betwee ad 0.751, about 95% are expected to be betwee ad 0.802, ad about 99.7% are expected to be betwee ad b) Radomizatio coditio: The sample ma ot be represetative. If the flow of traffic is ver fast, the speed of the other cars aroud ma have some effect o the speed of each driver. Likewise, if traffic is slow, the police ma fid a smaller proportio of speeders tha the expect. 10% coditio: 80 cars represet less tha 10% of all cars Success/Failure coditio: p = 56 ad q = 24 are both greater tha 10. The Normal model ma ot be appropriate. Use cautio. (Ad do t speed!)

4 8. Smokig. Radomizatio coditio: 50 people are selected at radom 10% coditio: 50 is less tha 10% of all people. Success/Failure coditio: p = 13.2 ad q = 36.8 are both greater tha 10. The samplig distributio model is Normal, with: µ ˆp = p = pq ( )( ) σ( p ˆ) = = Chapter 18 Samplig Distributio Models 305 There is a approximate chace of 68% that betwee 20.2% ad 32.6% of 50 people are smokers, a approximate chace of 95% that betwee 14.0% ad 38.8% are smokers, ad a approximate chace of 99.7% that betwee 7.8% ad 45.0%are smokers. 9. Loas. a) µ ˆp = p = 7% pq 007)(. 093) σ ( p ˆ) = = 18.% 200 b) Radomizatio coditio: Assume that the 200 people are a represetative sample of all loa recipiets. 10% coditio: A sample of this size is less tha 10% of all loa recipiets. Success/Failure coditio: p = 14 ad q = 186 are both greater tha 10. Therefore, the samplig distributio model for the proportio of 200 loa recipiets who will ot make pamets o time is N(0.07, 0.018). c) Accordig to the Normal model, pˆ µ pˆ the probabilit that over 10% of pq these cliets will ot make timel pamets is approximatel )(. 093) 200 z

5 306 Part V From the Data at Had to the World at Large 10. Cotacts. a) Radomizatio coditio: 100 studets are selected at radom. 10% coditio: 100 is less tha 10% of all of the studets at the uiversit, provided the uiversit has more tha 1000 studets. Success/Failure coditio: p = 30 ad q = 70 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p =0.30 (. )(. ) σ ( pˆ) = pq. = b) Accordig to the Normal model, the probabilit that more tha oe-third of the studets i this sample wear cotacts is approximatel pˆ µ pˆ pq )(. 070) 100 z Back to school? Radomizatio coditio: We are cosiderig colleges with freshma classes of 400 studets. These are ot radom samples, ad ot all of the colleges cosidered ma be tpical of all colleges. We should be careful usig this samplig distributio model. 10% Coditio: 400 studets is less tha 10% of all college studets. Success/Failure coditio: p = 296 ad q = 104 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p =0.74 (. )(. ) σ ( pˆ) = pq. = Accordig to the samplig distributio model, about 68% of the colleges are expected to have retetio rates betwee ad 0.762, about 95% of the colleges are expected to have retetio rates betwee ad 0.784, ad about 99.7% of the colleges are expected to have retetio rates betwee ad However, the coditios for the use of this model ma ot be met. We should be cautious about makig a coclusios based o this model.

6 Chapter 18 Samplig Distributio Models Bige drikig. Radomizatio coditio: The studets were selected radoml. 10% coditio: 200 college studets are less tha 10% of all college studets. Success/Failure coditio: p = 88 ad q = 112 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p =0.44 (. )(. ) σ ( pˆ) = pq. = Accordig to the samplig distributio model, about 68% of samples of 200 studets are expected to have bige drikig proportios betwee ad 0.475, about 95% betwee ad 0.510, ad about 99.7% betwee ad Back to school, agai. Provided that the studets at this college are tpical, the samplig distributio model for the retetio rate, ˆp, is Normal with µ ˆp = p =0.74 ad stadard deviatio (. )(. ) σ ( pˆ) = pq. = This college has a right to brag about their retetio rate. 522/603 = 86.6% is over 7 stadard deviatios above the expected rate of 74%. 14. Bige sample. Sice the sample is radom ad the Success/Failure coditio is met, the samplig distributio model for the bige drikig rate, ˆp, is Normal with µ ˆp = p =0.44 ad (. )(. ) stadard deviatio σ ( pˆ) = pq. = The bige drikig rate at this college is lower tha the atioal result, but ot extremel low. 96/244 = 39.3% is ol about 1.5 stadard deviatios below the atioal rate of 44%.

7 308 Part V From the Data at Had to the World at Large 15. Pollig. Radomizatio coditio: We must assume that the 400 voters were polled radoml. 10% coditio: 400 voters polled represet less tha 10% of potetial voters. Success/Failure coditio: p = 208 ad q = 192 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p =0.52 (. )(. ) σ ( pˆ) = pq. = Accordig to the Normal model, the probabilit that the ewspaper s sample will lead them to predict defeat (that is, predict budget support below 50%) is approximatel Seeds. Radomizatio coditio: We must assume that the 160 seeds i a pack are a radom sample. Sice seeds i each pack ma ot be a radom sample, proceed with cautio. 10% coditio: The 160 seeds represet less tha 10% of all seeds. Success/Failure coditio: p = ad q = 12.8 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p = 0.92 pq ( 092. )( 008. ) σ( p ˆ) = = Accordig to the Normal model, the probabilit that more tha 95% of the seeds will germiate is approximatel pˆ µ pˆ pq )(. 048) 400 z pˆ µ pˆ pq z )(. 008) 160

8 Chapter 18 Samplig Distributio Models Apples. Radomizatio coditio: A radom sample of 150 apples is take from each truck. 10% coditio: 150 is less tha 10% of all apples. Success/Failure Coditio: p = 12 ad q = 138 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p = 0.08 pq ( 008. )( 092. ) σ( p ˆ) = = Accordig to the Normal model, the probabilit that less tha 5% of the apples i the sample are usatisfactor is approximatel pˆ µ pˆ pq )(. 092) 150 z Geetic Defect. Radomizatio coditio: We will assume that the 732 ewbors are represetative of all ewbors. 10% coditio: These 732 ewbors certail represet less tha 10% of all ewbors. Success/Failure coditio: p = ad q = are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p = 0.04 pq ( 004. )( 096. ) σ( p ˆ) = = I order to get the 20 ewbors for the stud, the researchers hope to fid at least 20 p ˆ = as the proportio of ewbors i the sample with 732 juveile diabetes. pˆ Accordig to the Normal model, the probabilit that the researchers fid at least 20 ewbors with juveile diabetes is approximatel µ pˆ pq z )(. 096)

9 310 Part V From the Data at Had to the World at Large 19. Nosmokers. Radomizatio coditio: We will assume that the 120 customers (to fill the restaurat to capacit) are represetative of all customers. 10% coditio: 120 customers represet less tha 10% of all potetial customers. Success/Failure coditio: p = 72 ad q = 48 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p = 0.60 pq ( 060. )( 040. ) σ( p ˆ) = = Aswers ma var. We will use 3 stadard deviatios above the expected proportio of customers who demad osmokig seats to be ver sure. pq µ p ˆ +. (. ) Sice 120(0.734) = 88.08, the restaurat eeds at least 89 seats i the osmokig sectio. 20. Meals. Radomizatio coditio: We will assume that the 180 customers are represetative of all customers. 10% coditio: 180 customers represet less tha 10% of all potetial customers. Success/Failure coditio: p = 36 ad q = 144 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: µ ˆp = p = 0.20 pq ( 020. )( 080. ) σ( p ˆ) = = Aswers ma var. We will use 2 stadard deviatios above the expected proportio of customers who order the steak special to be prett sure. pq µ p ˆ +. (. ) Sice 180(0.2596) = , the chef eeds at least 47 steaks o had. 21. Samplig. a) The samplig distributio model for the sample mea is N µ, σ. b) If we choose a larger sample, the mea of the samplig distributio model will remai the same, but the stadard deviatio will be smaller.

10 Chapter 18 Samplig Distributio Models Samplig, part II. a) The samplig distributio model for the sample mea will be skewed to the left as well, cetered at µ, with stadard deviatio σ. b) Whe the sample size is icreased, the samplig distributio model becomes more Normal i shape, cetered at µ, with stadard deviatio σ. c) As we make the sample larger the distributio of data i the sample should more closel resemble the shape, ceter, ad spread of the populatio. 23. GPAs. Radomizatio coditio: Assume that the studets are radoml assiged to semiars. Idepedece assumptio: It is reasoable to thik that GPAs for radoml selected studets are mutuall idepedet. 10% coditio: The 25 studets i the semiar certail represet less tha 10% of the populatio of studets. Large Eough Sample coditio: The distributio of GPAs is roughl uimodal ad smmetric, so the sample of 25 studets is large eough. The mea GPA for the freshme was µ = 34., with stadard deviatio σ = Sice the coditios are met, the Cetral Limit Theorem tells us that we ca model the samplig distributio of the mea GPA with a Normal model, with µ = 34. ad stadard deviatio 035. σ( ) = The samplig distributio model for the sample mea GPA is approximatel N( 34., 007. ). 24. Home values. Radomizatio coditio: Homes were selected at radom. Idepedece assumptio: It is reasoable to thik that assessmets for radoml selected homes are mutuall idepedet. 10% coditio: The 100 homes i the sample certail represet less tha 10% of the populatio of all homes i the cit. A small cit will likel have more tha 1,000 homes. Large Eough Sample coditio: A sample of 100 homes is large eough.

11 312 Part V From the Data at Had to the World at Large The mea home value was µ = $ 140, 000, with stadard deviatio σ = $ 60, 000. Sice the coditios are met, the Cetral Limit Theorem tells us that we ca model the samplig distributio of the mea home value with a Normal model, with µ = $ 140, 000 ad stadard deviatio 60, 000 σ( ) = =$ The samplig distributio model for the sample mea home values is approximatel N( , 6000 ). 25. Pregac. a) µ σ µ σ Accordig to the Normal model, approximatel 21.1% of all pregacies are expected to last betwee 270 ad 280 das. b) µ σ = das Accordig to the Normal model, the logest 25% of pregacies are expected to last approximatel das or more. c) Radomizatio coditio: Assume that the 60 wome the doctor is treatig ca be cosidered a represetative sample of all pregat wome. Idepedece assumptio: It is reasoable to thik that the duratios of the patiets pregacies are mutuall idepedet. 10% coditio: The 60 wome that the doctor is treatig certail represet less tha 10% of the populatio of all wome. Large Eough Sample coditio: The sample of 60 wome is large eough. I this case, a sample would be large eough, sice the distributio of pregacies is Normal.

12 Chapter 18 Samplig Distributio Models 313 The mea duratio of the pregacies was µ = 266 das, with stadard deviatio σ = 16 das. Sice the distributio of pregac duratios is Normal, we ca model the samplig distributio of the mea pregac duratio with a Normal model, with 16 µ = 266 das ad stadard deviatio σ( ) = 207. das. 60 d) Accordig to the Normal model, the probabilit that the mea pregac duratio is less tha 260 das is Raifall. a) Accordig to the Normal model, Ithaca is expected to get more tha 40 iches of rai i approximatel 13.7% of ears. b) Accordig to the Normal model, Ithaca is expected to get less tha 31.9 iches of rai i driest 20% of ears. µ σ = c) Radomizatio coditio: Assume that the 4 ears i which the studet was i Ithaca ca be cosidered a represetative sample of all ears. Idepedece assumptio: It is reasoable to thik that the raifall totals for the ears are mutuall idepedet. 10% coditio: The 4 ears i which the studet was i Ithaca certail represet less tha 10% of all ears. Large eough sample coditio: The sample of 4 ears is large eough. I this case, a sample would be large eough, sice the distributio of aual raifall is Normal. The mea raifall was µ = iches, with stadard deviatio σ = 42. iches. Sice the distributio of earl raifall is Normal, we ca model the samplig distributio of the mea aual raifall with a Normal model, with µ = iches ad stadard deviatio 42. σ( ) = = 21. iches. 4

13 314 Part V From the Data at Had to the World at Large d) Accordig to the Normal model, the probabilit that those four ears averaged less tha 30 iches of rai is Pregat agai. a) The distributio of pregac duratios ma be skewed to the left sice there are more premature births tha ver log pregacies. Moder practice of medicie stops pregacies at about 2 weeks past ormal due date b iducig labor or performig a Caesarea sectio. b) We ca o loger aswer the questios posed i parts a ad b. The Normal model is ot appropriate for skewed distributios. The aswer to part c is still valid. The Cetral Limit Theorem guaratees that the samplig distributio model is Normal whe the sample size is large. 28. At work. a) The distributio of legth of time people work at a job is likel to be skewed to the right, because some people sta at the same job for much loger tha the mea plus two or three stadard deviatios. Additioall, the left tail caot be log, because a perso caot work at a job for less tha 0 ears. b) The Cetral Limit Theorem guaratees that the distributio of the mea time is Normall distributed for large sample sizes, as log as the assumptios ad coditios are satisfied. The CLT does t help us with the distributio of idividual times. 29. Dice ad dollars. a) Let X = the umber of dollars wo i oe pla. µ = EX ( ) = = $ σ = Var( X) = ( 0 2) ( ) 6 + ( ) 6 = σ = SD( X) = Var( X) = 13 $ b) X + X = the total wiigs for two plas. µ = EX ( + X) = EX ( ) + EX ( ) = 2+ 2= $ 4 σ = SD( X + X) = Var( X) + Var( X) = $. 5 10

14 Chapter 18 Samplig Distributio Models 315 c) I order to wi at least $100 i 40 plas, ou must average at least = $. 250per pla. The expected value of the wiigs is µ = $2, with stadard deviatio σ = $ Rollig a die is radom ad the outcomes are mutuall idepedet, so the Cetral Limit Theorem guaratees that the samplig distributio model is Normal with µ x = $2 ad stadard $. 361 deviatio σ( x ) = $ Accordig to the Normal model, the probabilit that ou wi at least $100 i 40 plas is approximatel (This is equivalet to usig N(80, 22.83) to model our total wiigs.) 30. New game. a) Let X = the amout of moe wo. X $40 $0 $10 P(X) = = 36 b) µ = EX ( ) = $ σ = Var( X) = ( 40 ( 0. 28)) ( (. )) 36 + ( (. )) 36. σ = SD( X) = Var( X) = $ c) µ = E( X + X + X + X + X) = 5E( X) = 5( 0. 28) = $ σ = SD( X + X + X + X + X) = 5( Var( X)) = 5( ) $ d) I order for the perso ruig the game to make a profit, the average wiigs of the 100 people must be less tha $0. The expected value of the wiigs is µ = $. 028, with stadard deviatio σ = $ Rollig a die is radom ad the outcomes are mutuall idepedet, so the Cetral Limit Theorem guaratees that the samplig distributio model is Normal with µ x = $. 028 ad stadard deviatio σ( x ) = $ Accordig to the Normal model, the probabilit that the perso ruig the game makes a profit is approximatel

15 316 Part V From the Data at Had to the World at Large 31. AP Stats. a) µ = 5( ) + 4( ) + 3( ) + 2( ) + 1( ) σ = ( )( ) + ( )( ) + ( )( ) ( )( ) + ( )( ) The calculatio for stadard deviatio is based o a rouded mea. Use techolog to calculate the mea ad stadard deviatio to avoid iaccurac. b) The distributio of scores for 40 radoml selected studets would ot follow a Normal model. The distributio would resemble the populatio, mostl uiform for scores 1 4, with about half as ma 5s. c) Radomizatio coditio: The scores are selected radoml. Idepedece assumptio: It is reasoable to thik that the radoml selected scores are idepedet of oe aother. 10% coditio: The 40 scores represet less tha 10% of all scores. Large Eough Sample coditio: A sample of 40 scores is large eough. Sice the coditios are satisfied, the samplig distributio model for the mea of 40 radoml selected AP Stat scores is Normal, with µ = µ ad stadard deviatio σ. σ ( ) =. = Museum membership. a) µ = 50(. 0 41) + 100(. 0 37) + 250(. 0 14) + 500(. 0 07) (. 0 01) $ σ = ( )(. 0 41) + ( )(. 0 37) + ( )(. 0 14) ( )(. 0 07) + ( )(. 0 01) $ The calculatio for stadard deviatio is based o a rouded mea. Use techolog to calculate the mea ad stadard deviatio to avoid iaccurac. b) The distributio of doatios for 50 ew members would ot follow a Normal model. The ew members would probabl make doatios tpical of the curret member populatios, so the distributio would resemble the populatio, skewed to the right. c) Radomizatio coditio: The members are ot selected radoml. The are simpl the ew members that da. However, the doatios the make are probabl tpical of the doatios made b curret members. Idepedece assumptio: It is reasoable to thik that the doatios for the ew members would ot affect oe aother. 10% coditio: The 50 doatios represet less tha 10% of all doatios. Large Eough Sample coditio: The sample of 50 doatios is large eough.

16 Chapter 18 Samplig Distributio Models 317 Sice the coditios are satisfied, the samplig distributio model for the mea of 50 doatios is Normal, with µ = µ $ ad stadard deviatio σ σ ( ) = = AP Stats, agai. Sice the teacher cosiders his 63 studets tpical, ad 63 is less tha 10% of all studets, the samplig distributio model for the mea AP Stat score for 63 studets is Normal, with mea µ = µ ad stadard deviatio σ σ ( ) = = µ σ( ) Accordig to the samplig distributio model, the probabilit that the class of 63 studets achieves a average of 3 o the AP Stat exam is about 21.4%. 34. Joiig the museum. If the ew members ca be cosidered a radom sample of all museum members, ad the 80 ew members are less tha 10% of all members, the the samplig distributio model for the mea doatio of 80 members is Normal, with µ = µ $ ad stadard deviatio σ ( ) = =$ µ σ ( ) Accordig to the samplig distributio model, there is a 98.8% probabilit that the average doatio for 80 ew members is at least $ Pollutio. a) Radomizatio coditio: Assume that the 80 cars ca be cosidered a represetative sample of all cars of this tpe. Idepedece assumptio: It is reasoable to thik that the CO emissios for these cars are mutuall idepedet. 10% coditio: The 80 cars i the fleet certail represet less tha 10% of all cars of this tpe. Large Eough Sample coditio: A sample of 80 cars is large eough.

17 318 Part V From the Data at Had to the World at Large The mea CO level was µ = 29. gm/mi, with stadard deviatio σ = 04. gm/mi. Sice the coditios are met, the CLT allows us to model the samplig distributio of the with a 04. Normal model, with µ = 29. gm/mi ad stadard deviatio σ( ) = = gm/mi. 80 b) Accordig to the Normal model, the probabilit that is betwee 3.0 ad 3.1 gm/mi is approximatel c) Accordig to the Normal model, there is ol a 5% chace that the fleet s mea CO level is greater tha approximatel 2.97 gm/mi. µ σ( ) = Potato chips. a) Accordig to the Normal model, ol about 4.78% of the bags sold are uderweight. b) P(oe of the 3 bags are uderweight) = ( ) c) Radomizatio coditio: Assume that the 3 bags ca be cosidered a represetative sample of all bags. Idepedece assumptio: It is reasoable to thik that the weights of these bags are mutuall idepedet. 10% coditio: The 3 bags certail represet less tha 10% of all bags. Large Eough Sample coditio: Sice the distributio of bag weights is believed to be Normal, the sample of 3 bags is large eough. The mea weight is µ = ouces, with stadard deviatio σ = 012. ouces. Sice the coditios are met, we ca model the samplig distributio of with a Normal model, with µ = ouces ad stadard deviatio 012. σ( ) = ouces. 3 Accordig to the Normal model, the probabilit that the mea weight of the 3 bags is less tha 10 ouces is approximatel

18 Chapter 18 Samplig Distributio Models 319 d) For 24 bags, the stadard deviatio of the samplig distributio model is 012. σ( ) = ouces. Now, a average of 10 ouces is over 8 stadard deviatios 24 below the mea of the samplig distributio model. This is extremel ulikel. 37. Tips. a) Sice the distributio of tips is skewed to the right, we ca t use the Normal model to determie the probabilit that a give part will tip at least $20. b) No. A sample of 4 parties is probabl ot a large eough sample for the CLT to allow us to use the Normal model to estimate the distributio of averages. c) A sample of 10 parties ma ot be large eough to allow the use of a Normal model to describe the distributio of averages. It would be risk to attempt to estimate the probabilit that his ext 10 parties tip a average of $15. However, sice the distributio of tips has µ = $. 960, with stadard deviatio σ = $. 540, we still kow that the mea of the 540. samplig distributio model is µ = $. 960 with stadard deviatio σ( ) = $ We do t kow the exact shape of the distributio, but we ca still assess the likelihood of specific meas. A mea tip of $15 is over 3 stadard deviatios above the expected mea tip for 10 parties. That s ot ver likel to happe. 38. Groceries. a) Sice the distributio of purchases is skewed, we ca t use the Normal model to determie the probabilit that a give purchase is at least $40. b) A sample of 10 customers ma ot be large eough for the CLT to allow the use of a Normal model for the samplig distributio model. If the distributio of purchases is ol slightl skewed, the sample ma be large eough, but if the distributio is heavil skewed, it would be ver risk to attempt to determie the probabilit. c) Radomizatio coditio: Assume that the 50 purchases ca be cosidered a represetative sample of all purchases. Idepedece assumptio: It is reasoable to thik that the purchases are mutuall idepedet, uless there is a sale or other icetive to purchase more. 10% coditio: The 50 purchases certail represet less tha 10% of all purchases. Large Eough Sample coditio: The sample of 50 purchases is large eough. The mea purchase is µ = $32, with stadard deviatio σ = $20. Sice the coditios are met, the CLT allows us to model the samplig distributio of with a Normal model, with µ = $32 ad stadard deviatio 20 σ( ) = $ Accordig to the Normal model, the probabilit that the mea purchase of the 50 customers is at least $40 is approximatel

19 320 Part V From the Data at Had to the World at Large 39. More tips. a) Radomizatio coditio: Assume that the tips from 40 parties ca be cosidered a represetative sample of all tips. Idepedece assumptio: It is reasoable to thik that the tips are mutuall idepedet, uless the service is particularl good or bad durig this weeked. 10% coditio: The tips of 40 parties certail represet less tha 10% of all tips. Large Eough Sample coditio: The sample of 40 parties is large eough. The mea tip is µ = $. 960, with stadard deviatio σ = $ Sice the coditios are satisfied, the CLT allows us to model the samplig distributio of with a Normal model, with µ = $. 960 ad stadard deviatio 540. σ( ) = $ I order to ear at least $500, the waiter would have to average 500 = $ per part. 40 Accordig to the Normal model, the probabilit that the waiter ears at least $500 i tips i a weeked is approximatel b) Accordig to the Normal model, µ the waiter ca expect to have a σ( ) mea tip of about $ , which correspods to about = $ for 40 parties, i the best % of such weekeds More groceries. a) Assumptios ad coditios for the use of the CLT were verified i Exercise 38. The mea purchase is µ = $32, with stadard deviatio σ = $20. Sice the sample is large, the CLT allows us to model the samplig distributio of with a Normal model, with 20 µ = $32 ad stadard deviatio σ( ) = $ I order to have reveues of at least $10,000, the mea purchase must be at least 10, 000 = $ Accordig to the Normal model, the probabilit of havig a mea purchase at least that high (ad therefore at total reveue of at least $10,000) is

20 Chapter 18 Samplig Distributio Models 321 b) Accordig to the Normal model, the mea purchase o the worst 10% of such das is approximatel $ , so 312 customers are expected to sped about $ µ σ( ) = IQs. a) Accordig to the Normal model, the probabilit that the IQ of a studet from East State is at least 125 is approximatel b) First, we will eed to geerate a model for the differece i IQ betwee the two schools. Sice we are choosig at radom, it is reasoable to believe that the studets IQs are idepedet, which allows us to calculate the stadard deviatio of the differece. µ = EE ( W) = EE ( ) EW ( ) = = 10 σ = SD( E W) = Var( E) + Var( W) 2 2 = Sice both distributios are Normal, the distributio of the differece is N(10, ). Accordig to the Normal model, the probabilit that the IQ of a studet at ESU is at least 5 poits higher tha a studet at WSU is approximatel c) Radomizatio coditio: Studets are radoml sampled from WSU. Idepedece assumptio: It is reasoable to thik that the IQs are mutuall idepedet. 10% coditio: The 3 studets certail represet less tha 10% of studets. Large Eough Sample coditio: The distributio of IQs is Normal, so the distributio of sample meas of samples of a size will be Normal, so a sample of 3 studets is large eough.

21 322 Part V From the Data at Had to the World at Large The mea IQ is µ w = 120, with stadard deviatio σ w = 10. Sice the distributio IQs is Normal, we ca model the samplig distributio of w with a Normal model, with µ w = 120 with stadard 10 deviatio σ( w ) = Accordig to the Normal model, the probabilit that the mea IQ of the 3 WSU studets is above 125 is approximatel d) As i part c, the samplig distributio of e, the mea IQ of 3 ESU studets, ca be 8 modeled with a Normal model, with µ e = 130 with stadard deviatio σ( e ) = The distributio of the differece i mea IQ is Normal, with the followig parameters: µ = Ee ( w) = Ee ( ) Ew ( ) = = 10 σ e w e w = SD( e w) = Var( e) + Var( w) = Accordig to the Normal model, the probabilit that the mea IQ of 3 ESU studets is at least 5 poits higher tha the mea IQ of 3 WSU studets is approximatel Milk. a) Accordig to the Normal model, the probabilit that a Arshire averages more tha 50 pouds of milk per da is approximatel

22 Chapter 18 Samplig Distributio Models 323 b) First, we will eed to geerate a model for the differece i milk productio betwee the two cows. Sice we are choosig at radom, it is reasoable to believe that the cows milk productios are idepedet, which allows us to calculate the stadard deviatio of the differece. µ = EA ( J) = EA ( ) EJ ( ) = = 4 pouds σ = SD( A J) = Var( A) + Var( J) 2 2 = pouds Sice both distributios are Normal, the distributio of the differece is N(4, ). Accordig to the Normal model, the probabilit that the Arshire gives more milk tha the Jerse is approximatel c) Radomizatio coditio: Assume that the farmer s 20 Jerses are a represetative sample of all Jerses. Idepedece assumptio: It is reasoable to thik that the cows have mutuall idepedet milk productio. 10% coditio: The 20 cows certail represet less tha 10% of cows. Large Eough Sample coditio: Sice the distributio of dail milk productio is Normal, the sample meas of samples of a size are Normall distributed, so the sample of 20 cows is certail large eough. The mea milk productio is µ j = 43 pouds, with stadard deviatio σ j = 5. Sice the distributio of milk productio is Normal, we ca model the samplig distributio of j with a Normal model, with µ j = 43 pouds with 5 stadard deviatio σ( j ) = Accordig to the Normal model, the probabilit that the mea milk productio of the 20 Jerses is above 45 pouds of milk per da is approximatel

23 324 Part V From the Data at Had to the World at Large d) As i part c, the samplig distributio of a, the mea milk productio of 10 Arshires, ca be modeled with a Normal model, with µ a = 47 pouds with stadard deviatio 6 σ( a ) = pouds. 10 The distributio of the differece i mea milk productio is Normal, with the followig parameters: µ a j = Ea ( j) = Ea ( ) Ej ( ) = = 4 pouds σ a j = SD( a j) = Var( a) + Var( j) = pouds Accordig to the Normal model, the probabilit that the mea milk productio of 10 Arshires is at least 5 pouds higher tha the mea milk productio of 20 Jerses is approximatel