Chapter 17 Sampling Distribution Models

Transcription

1 Chapter 17 Samplig Distributio Models 353 Chapter 17 Samplig Distributio Models 1. Sed moey. All of the histograms are cetered aroud p As gets larger, the shape of the histograms get more uimodal ad symmetric, approachig a Normal model, while the variability i the sample proportios decreases. 2. Character recogitio. All of the histograms are cetered aroud p As gets larger, the shapes of the histograms get more uimodal ad symmetric, approachig a Normal model, while the variability i the sample proportios decreases. 3. AP Exam. a) Class mea = 3.4 Class stadard deviatio = b) c) Sample 5,4 5,4 5,3 5,1 4,4 4,3 4,1 4,3 4,1 3,1 Mea Sample Meas d) The mea of the samplig distributio of sample meas is 3.4, the same as the mea of the populatio, ad the stadard deviatio is 0.876, smaller tha the stadard deviatio of the populatio. The mea of the samplig distributio is a ubiased estimator of the populatio mea. 4. AP Exam II. a) b) Sample 5,4,4 5,4,3 5,4,1 5,4,3 5,4,1 5,3,1 4,4,3 4,4,1 4,3,1 4,3,1 Mea Sample Meas c) The mea of the samplig distributio of sample meas is still 3.4, the same as the mea of the populatio, ad the stadard deviatio is 0.582, smaller tha the stadard deviatio of the populatio ad smaller tha the previous stadard deviatio of the sample meas for samples of size 2.

2 354 Part V From the Data at Had to the World at Large 5. Marriage. These may be cosidered Beroulli trials. There are oly two possible outcomes, beig pessimistic about the future of marriage ad family, or ot. The probability of beig pessimistic about the future of marriage ad family is costat, p = We are radomly samplig less tha 10% of all Americas. Let Y the umber of people who are pessimistic about the future of marriage ad family i = 20 people. a) Use Biom(20, 0.27). 25% of 20 people is 5 people. P(25% or fewer) PY ( 5) PY ( 0) PY ( 1) PY ( 5) (0.27) (0.73) (0.27) (0.73) (0.27) (0.73) Accordig to the Biomial model, the probability that, i a sample of 20 adults, 25% or fewer of the people i the sample are pessimistic about the future of marriage ad family is approximately b) EY ( ) p20(0.27) 5.4 people. SD( Y) pq 20(0.27)(0.73) people. Use N(5.4, 1.986): y z z z Accordig to the Normal model, the probability that, i a sample of 20 adults, 25% or fewer of the people i the sample are pessimistic about the future of marriage ad family is approximately There is a differece of about i the estimated probabilities. c) Let Y the umber of people who are pessimistic about the future of marriage ad family i = 700 people. Use Biom(700, 0.27). 25% of 700 people is 175 people.

3 P(25% or fewer) PY ( 175) PY ( 0) PY ( 1) PY ( 175) Chapter 17 Samplig Distributio Models 355 (0.27) (0.73) (0.27) (0.73) (0.27) (0.73) Accordig to the Biomial model, the probability that, i a sample of 700 adults, 25% or fewer of the people i the sample are pessimistic about the future of marriage ad family is approximately d) EY ( ) p700(0.27) 189 people. SD( Y) pq 700(0.27)(0.73) people. Use N(189, ): y z z z Accordig to the Normal model, the probability that, i a sample of 700 adults, 25% or fewer of the people i the sample are pessimistic about the future of marriage ad family is approximately There is a differece of about i the estimated probabilities. e) For part b, p = 5.4 ad q = This situatio does t meet the Success/Failure coditio. The probability approximated by the Normal model is ot close to the probability calculated usig the biomial model. I part d, p = 189 ad q = 511. The Success/Failure coditio is easily met, ad the probabilities are quite close. It is importat to ote, however, that whe the Success/Failure coditio is met, the Normal model is still oly a approximatio. It s just a reasoable approximatio. 6. Wow. Just, wow. These may be cosidered Beroulli trials. There are oly two possible outcomes, believig that shape-shiftig reptilia people cotrol our world, or ot. The probability of such a belief is costat, p = We are radomly samplig less tha 10% of all Americas.

4 356 Part V From the Data at Had to the World at Large Let Y the umber of people who believe that shape-shiftig reptilia people cotrol our world i = 100 people. a) Use Biom(100, 0.04). 6% of 100 people is 6 people. P(at least 6 people) PY ( 6) PY ( 6) PY ( 7) PY ( 100) (0.04) (0.96) (0.04) (0.96) (0.04) (0.96) Accordig to the Biomial model, the probability that, i a sample of 100 Americas, at least 6% of the people i the sample believe that shape-shiftig reptilia people cotrol our world is approximately b) EY ( ) p100(0.04) 4 people. SD( Y) pq 100(0.04)(0.96) people. Use N(4, 1.960): y z 6 4 z z Accordig to the Normal model, the probability that, i a sample of 100 adults, at least 6% of the people i the sample believe that shape-shiftig reptilia people cotrol our world is approximately There is a differece of about 0.06 i the estimated probabilities. c) Let Y the umber of people who believe i the JFK cospiracy theory i = 100 people. Use Biom(100, 0.51). 57% of 100 people is 57 people. P(at least 57%) PY ( 57) PY ( 57) PY ( 58) PY ( 100) (0.51) (0.49) (0.51) (0.49) (0.51) (0.49)

5 Chapter 17 Samplig Distributio Models 357 Accordig to the Biomial model, the probability that, i a sample of 100 adults, at least 57% of the people i the sample believe i the JFK cospiracy is approximately d) EY ( ) p100(0.51) 51 people. SD( Y) pq 100(0.51)(0.49) people. Use N(51, 4.999): y z z z Accordig to the Normal model, the probability that, i a sample of 100 adults, at least 57% of the people i the sample believe i the JFK cospiracy is approximately There is a differece of about 0.02 i the estimated probabilities. e) For part b, p = 4 ad q = 96. This situatio does t meet the Success/Failure coditio. The probability approximated by the Normal model is ot close to the probability calculated usig the biomial model. I part d, p = 51 ad q = 49. The Success/Failure coditio is met, ad the probabilities are close. It is importat to ote, however, that whe the Success/Failure coditio is met, the Normal model is still oly a approximatio. It s just a reasoable approximatio. 7. Sed moey, agai. a) Observed mea Theoretical mea Observed st. dev. Theoretical stadard deviatio (0.05)(0.95) / (0.05)(0.95) / (0.05)(0.95) / (0.05)(0.95) / b) All of the values seem very close to what we would expect from theory. c) The histogram for = 200 looks quite uimodal ad symmetric. We should be able to use the Normal model.

6 358 Part V From the Data at Had to the World at Large d) The success/failure coditio requires p ad q to both be at least 10, which is ot satisfied util = 200 for p = The theory supports the choice i part c. 8. Character recogitio, agai. a) Observed mea Theoretical mea Observed st. dev. Theoretical stadard deviatio (0.85)(0.15) / (0.85)(0.15) / (0.85)(0.15) / (0.85)(0.15) / b) All of the values seem very close to what we would expect from theory. c) Certaily, the histogram for = 100 is uimodal ad symmetric, but the histogram for = 75 looks early Normal, too. We should be able to use the Normal model for either. d) The success/failure coditio requires p ad q to both be at least 10, which is satisfied for both = 75 ad = 100 whe p = The theory supports the choice i part c. 9. Sample maximum. a) A Normal model is ot appropriate for the samplig distributio of the sample maximum. The histogram is skewed strogly to the left. b) No. The 95% rule is based o the Normal model, ad the Normal model is ot appropriate here. 10. Soup. a) A Normal model is ot appropriate for the samplig distributio of the sample variaces. The histogram is skewed to the right. b) No. The 95% rule is based o the Normal model, ad the Normal model is ot appropriate here. 11. Coi tosses. a) The histogram of these proportios is expected to be symmetric, 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 symmetric because the probability that the coi lads heads is the same as the probability that the coi lads tails. b) The histogram is expected to have its ceter at 0.5, the probability that the coi lads heads.

7 Chapter 17 Samplig Distributio Models 359 c) The stadard deviatio of data displayed i this histogram should be approximately equal to the stadard deviatio of the samplig distributio pq (0.5)(0.5) 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. 12. M&M s. a) The histogram of the proportios of gree cadies i the bags would probably be skewed slightly 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 13. More cois. pq (0.1)(0.9) a) ˆp (0.5)(0.5) p 0.5 ad SD( pˆ ) pq 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. b) Coi flips are idepedet of oe aother. There is o eed to check the 10% Coditio. 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).

8 360 Part V From the Data at Had to the World at Large c) ˆp pq (0.5)(0.5) p 0.5 ad SD( pˆ ) 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. 14. Bigger bag. a) Radomizatio coditio: The 200 M&M s i the bag ca be cosidered represetative of all M&M s, ad they are thoroughly mixed. 10% coditio: 200 is certaily 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: ˆp SD( pˆ ) pq (0.1)(0.9) 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 p 0.1 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.

9 15. Just (u)lucky. Chapter 17 Samplig Distributio Models 361 For 200 flips, the samplig distributio model is Normal with ˆp p 0.5 ad pq (0.5)(0.5) SD( pˆ ) Her sample proportio of pˆ 0.42 is about stadard deviatios below the expected proportio, which is uusual, but ot extraordiary. Accordig to the Normal model, we expect sample proportios this low or lower about 1.2% of the time. 16. Too may gree oes? For 500 cadies, the samplig distributio model is Normal with ˆp p 0.1 ad pq (0.1)(0.9) SD( pˆ ) The sample proportio of pˆ 0.12 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. 17. Speedig. a) ˆp p 0.70 SD( pˆ ) pq (0.7)(0.3) 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 may ot be represetative. If the flow of traffic is very fast, the speed of the other cars aroud may have some effect o the speed of each driver. Likewise, if traffic is slow, the police may fid a smaller proportio of speeders tha they 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 may ot be appropriate. Use cautio. (Ad do t speed!) 18. Smokig. Radomizatio coditio: 50 people are selected at radom 10% coditio: 50 is less tha 10% of all people. Success/Failure coditio: p = 10.3 ad q = 39.7 are both greater tha 10.

10 362 Part V From the Data at Had to the World at Large The samplig distributio model is Normal, with: SD( pˆ ) pq (0.206)(0.794) ˆp p There is a approximate chace of 68% that betwee 14.9% ad 26.3% of 50 people are smokers, a approximate chace of 95% that betwee 9.2% ad 32.0% are smokers, ad a approximate chace of 99.7% that betwee 3.5% ad 37.7%are smokers. 19. Visio. a) Radomizatio coditio: Assume that the 170 childre are a represetative sample of all childre. 10% coditio: A sample of this size is less tha 10% of all childre. Success/Failure coditio: p = 20.4 ad q = are both greater tha 10. Therefore, the samplig distributio model for the proportio of 170 childre who are earsighted is N 0.12, or N(0.12, 0.025). 170 b) The Normal model is to the right. c) They might expect that the proportio of earsighted studets to be withi 2 stadard deviatios of the mea. Accordig to the Normal model, this meas they might expect betwee 7% ad 17% of the studets to be earsighted, or betwee about 12 ad 29 studets. 20. Mortgages. a) Radomizatio coditio: Assume that the 1731 mortgages are a represetative sample of all mortgages. 10% coditio: A sample of this size is less tha 10% of all mortgages. Success/Failure coditio: p = ad q = are both greater tha 10. Therefore, the samplig distributio model for the proportio of foreclosures o mortgages is N 0.124, or N(0.124, 0.008). 1731

11 b) The Normal model is to the right. c) They might expect that the proportio of mortgage foreclosures to be withi 2 stadard deviatios of the mea. Accordig to the Normal model, this meas they might expect betwee 10.8% ad 14.0% of the mortgages to udergo foreclosure, or betwee about 187 ad 242 foreclosures. 21. Loas. a) ˆp p 0.07 SD( pˆ ) pq (0.07)(0.93) Chapter 17 Samplig Distributio Models 363 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 paymets o time is N(0.07, 0.018). c) Accordig to the Normal model, the probability that over 10% of these cliets will ot make timely paymets is approximately Tees with phoes. a) Radomizatio coditio: 100 studets are selected at radom. 10% coditio: 100 is less tha 10% of tees. Success/Failure coditio: p = 78 ad q = 22 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: p 0.78 ˆp pˆ p ˆ z pq z (0.07)(0.93) 200 z pq (0.78)(0.22) SD( pˆ )

12 364 Part V From the Data at Had to the World at Large b) Accordig to the Normal model, the probability that less tha three-fourths of the studets i this sample have a cell phoe is approximately pˆ pˆ z pq z z Back to school? Radomizatio coditio: We are cosiderig radom samples of 400 studets who took the ACT. 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: p 0.74 ˆp pq (0.74)(0.26) SD( pˆ ) 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 may ot be met. We should be cautious about makig ay coclusios based o this model. 24. Bige drikig. Radomizatio coditio: The studets were selected radomly. 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: p 0.44 ˆp pq (0.44)(0.56) SD( pˆ )

13 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. Chapter 17 Samplig Distributio Models 365 Provided that the studets at this college are typical, the samplig distributio model for the retetio rate, ˆp, is Normal with ˆp p 0.74 ad stadard pq (0.74)(0.26) deviatio ( pˆ ) 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%. 26. 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 pq (0.44)(0.56) ˆp p 0.44 ad stadard deviatio ( pˆ ) The bige drikig rate at this college is lower tha the atioal result, but ot extremely low. 96/244 = 39.3% is oly about 1.5 stadard deviatios below the atioal rate of 44%. 27. Pollig. Radomizatio coditio: We must assume that the 400 voters were polled radomly. 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: p 0.52 ˆp pq (0.52)(0.48) SD( pˆ )

14 366 Part V From the Data at Had to the World at Large Accordig to the pˆ pˆ Normal model, the z pq probability that the ewspaper s sample will lead them to z predict defeat (that is, (0.52)(0.48) predict budget support 400 below 50%) is z approximately Seeds. Radomizatio coditio: We must assume that the 160 seeds i a pack are a radom sample. Sice seeds i each pack may 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: ˆp pq (0.92)(0.08) p 0.92 SD( pˆ ) Accordig to the Normal model, the probability that more tha 95% of the seeds will germiate is approximately pˆ pˆ z pq z (0.92)(0.08) 160 z Gaydar. Radomizatio coditio: We must assume that the 100 photographs were radomized before they were show to the wome. 10% coditio: This study does t ivolve a sample, just radomizatio of a set of photos. The 10% coditio does ot apply. Success/Failure coditio: p = 65 ad q = 35 are both greater tha 10. Therefore, the samplig distributio model for ˆp is Normal, with: ˆp pq (0.65)(0.35) p 0.65 SD( pˆ )

15 Chapter 17 Samplig Distributio Models 367 Accordig to the Normal model, the probability of correctly classifyig 80 of 100 me is approximately pˆ pˆ z pq z (0.65)(0.35) 100 z Geetic Defect. Radomizatio coditio: We will assume that the 732 ewbors are represetative of all ewbors. 10% coditio: These 732 ewbors 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: pq (0.04)(0.96) ˆp p 0.04 SD( pˆ ) I order to get the 20 ewbors for the study, the researchers hope to fid at least 20 pˆ as the proportio of ewbors i the sample with juveile 732 diabetes. Accordig to the Normal model, the probability that the researchers fid at least 20 ewbors with juveile diabetes is approximately pˆ pˆ z pq z (0.04)(0.96) 732 z No Childre sectio. Radomizatio coditio: We will assume that the 120 customers (to fill the restaurat to capacity) are represetative of all customers. 10% coditio: 120 customers represet less tha 10% of all potetial customers. Success/Failure coditio: p = 36 ad q = 84 are both greater tha 10.

16 368 Part V From the Data at Had to the World at Large Therefore, the samplig distributio model for ˆp is Normal, with: pq (0.30)(0.70) ˆp p 0.30 SD( pˆ ) Aswers may vary. We will use 3 stadard deviatios above the expected proportio of customers with childre to be very sure. pq pˆ (0.0418) Sice 120(0.4254) = , the restaurat eeds about 51 seats i the familyfriedly sectio. 32. 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: ˆp p 0.20 pq (0.20)(0.80) SD( pˆ ) Aswers may vary. We will use 2 stadard deviatios above the expected proportio of customers who order the steak special to be pretty sure. pq pˆ (0.0298) Sice 180(0.2596) = , the chef eeds at least 47 steaks o had. 33. Samplig. a) The samplig distributio model for the sample mea is N,. 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. 34. Samplig, part II. a) The samplig distributio model for the sample mea will be skewed to the left as well, cetered at, with stadard deviatio.

17 Chapter 17 Samplig Distributio Models 369 b) Whe the sample size is icreased, the samplig distributio model becomes more Normal i shape, cetered at, with stadard deviatio. c) As we make the sample larger the distributio of data i the sample should more closely resemble the shape, ceter, ad spread of the populatio. 35. Waist size. a) The distributio of waist size of 250 me i Utah is uimodal ad slightly skewed to the right. A typical waist size is approximately 36 iches, ad the stadard deviatio i waist sizes is approximately 4 iches. b) All of the histograms show distributios of sample meas cetered ear 36 iches. As gets larger the histograms approach the Normal model i shape, ad the variability i the sample meas decreases. The histograms are fairly Normal by the time the sample reaches size CEO compesatio. a) The distributio of total compesatio for the CEOs for the 800 largest U.S. compaies is uimodal, but strogly skewed to the right with several large outliers. b) All of the histograms are cetered ear $10,000,000. As gets larger, the variability i sample meas decreases, ad histograms approach the Normal shape. However, they are still visibly skewed to the right, with the possible exceptio of the histogram for = 200. c) This rule of thumb does t seem to be true for highly skewed distributios. 37. Waist size revisited. a) Observed mea Theoretical mea Observed st. dev. Theoretical stadard deviatio / / / / b) The observed values are all very close to the theoretical values. c) For samples as small as 5, the samplig distributio of sample meas is uimodal ad symmetric. The Normal model would be appropriate. d) The distributio of the origial data is early uimodal ad symmetric, so it does t take a very large sample size for the distributio of sample meas to be approximately Normal.

18 370 Part V From the Data at Had to the World at Large 38. CEOs revisited. a) Observed mea Theoretical mea Observed st. dev. Theoretical stadard deviatio 30 10, , , / , , , / , , , / , , , / b) The observed values are all very close to the theoretical values. c) All the samplig distributios are still quite skewed, with the possible exceptio of the samplig distributio for = 200, which is still somewhat skewed. The Normal model would ot be appropriate. d) The distributio of the origial data is strogly skewed, so it will take a very large sample size before the distributio of sample meas is approximately Normal. 39. GPAs. Radomizatio coditio: Assume that the studets are radomly assiged to semiars. Idepedece assumptio: It is reasoable to thik that GPAs for radomly selected studets are mutually idepedet. 10% coditio: The 25 studets i the semiar certaily represet less tha 10% of the populatio of studets. Large Eough Sample coditio: The distributio of GPAs is roughly uimodal ad symmetric, so the sample of 25 studets is large eough. The mea GPA for the freshme was 3.4, with stadard deviatio 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 0.35 y 3.4 ad stadard deviatio SD( y) The samplig distributio model for the sample mea GPA is approximately N (3.4,0.07).

19 Chapter 17 Samplig Distributio Models Home values. Radomizatio coditio: Homes were selected at radom. Idepedece assumptio: It is reasoable to thik that assessmets for radomly selected homes are mutually idepedet. 10% coditio: The 100 homes i the sample certaily represet less tha 10% of the populatio of all homes i the city. A small city will likely have more tha 1000 homes. Large Eough Sample coditio: A sample of 100 homes is large eough. The mea home value was $140,000, with stadard deviatio $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 y $140,000 ad stadard 60, 000 deviatio SD( y) $ The samplig distributio model for the sample mea home values is approximately N (140000,6000). 41. Lucky spot? a) Smaller outlets have more variability tha the larger outlets, just as the Cetral Limit Theorem predicts. b) If the lottery is truly radom, all outlets are equally likely to sell wiig tickets. 42. Safe cities. The stadard deviatio of the samplig model for the mea is. So, cities i which the average is based o a smaller umber of drivers will have greater variatio i their averages ad will be more likely to be both safest ad least safe.

20 372 Part V From the Data at Had to the World at Large 43. Pregacy. a) y z z 16 z 0.25 y z z 16 z Accordig to the Normal model, approximately 21.1% of all pregacies are expected to last betwee 270 ad 280 days. b) y z y y days Accordig to the Normal model, the logest 25% of pregacies are expected to last approximately days 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 mutually idepedet. 10% coditio: The 60 wome that the doctor is treatig certaily 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, ay sample would be large eough, sice the distributio of pregacies is Normal.

21 Chapter 17 Samplig Distributio Models 373 The mea duratio of the pregacies was 266 days, with stadard deviatio 16 days. Sice the distributio of pregacy duratios is Normal, we ca model the samplig distributio of the mea pregacy duratio with a Normal model, with y 266 days ad stadard deviatio 16 SD( y) 2.07 days. 60 d) Accordig to the y Normal model, with z mea 266 days ad stadard deviatio 2.07 z 16 days, the probability that the mea 20 pregacy duratio is z less tha 260 days is Raifall. a) Accordig to the Normal model, Ithaca is expected to get more tha 40 iches of rai i approximately 13.7% of years. b) Accordig to the Normal model, Ithaca is expected to get less tha 31.9 iches of rai i driest 20% of years. y z y y 31.9 c) Radomizatio coditio: Assume that the 4 years i which the studet was i Ithaca ca be cosidered a represetative sample of all years. Idepedece assumptio: It is reasoable to thik that the raifall totals for the years are mutually idepedet. 10% coditio: The 4 years i which the studet was i Ithaca certaily represet less tha 10% of all years.

22 374 Part V From the Data at Had to the World at Large Large eough sample coditio: The sample of 4 years is large eough. I this case, ay sample would be large eough, sice the distributio of aual raifall is Normal. The mea raifall was 35.4 iches, with stadard deviatio 4.2 iches. Sice the distributio of yearly raifall is Normal, we ca model the samplig distributio of the mea aual raifall with a Normal model, with 4.2 y 35.4 iches ad stadard deviatio SD( y) 2.1 iches. 4 d) Accordig to the Normal model, with mea 35.4 iches ad stadard deviatio 2.4 iches, the probability that those four years averaged less tha 30 iches of rai is Pregat agai. a) The distributio of pregacy duratios may be skewed to the left sice there are more premature births tha very log pregacies. Moder practice of medicie stops pregacies at about 2 weeks past ormal due date by 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. 46. At work. a) The distributio of legth of time people work at a job is likely to be skewed to the right, because some people stay at the same job for much loger tha the mea plus two or three stadard deviatios. Additioally, the left tail caot be log, because a perso caot work at a job for less tha 0 years. b) The Cetral Limit Theorem guaratees that the distributio of the mea time is Normally 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.

23 47. Dice ad dollars. a) Let X = the umber of dollars wo i oe play EX ( ) $ Var( X) (0 2) (12) (10 2) SD( X) Var( X) 13 $3.61 b) X + X = the total wiigs for two plays. EX ( X) EX ( ) EX ( ) 22 $4 SD( X X) Var( X) Var( X) $5.10 Chapter 17 Samplig Distributio Models 375 c) I order to wi at least $100 i 40 plays, you must average at least 100 $ per play. The expected value of the wiigs is $2, with stadard deviatio $3.61. Rollig a die is radom ad the outcomes are mutually idepedet, so the Cetral Limit Theorem guaratees that the samplig distributio model is Normal with x $2 ad stadard $3.61 deviatio SD( x) $ Accordig to the Normal model, the probability that you wi at least $100 i 40 plays is approximately (This is equivalet to usig N(80, 22.83) to model your total wiigs.) 48. New game. a) Let X = the amout of moey wo. X $40 $0 $10 P(X)

24 376 Part V From the Data at Had to the World at Large b) EX ( ) $ Var( X) (40 ( 0.28)) (0 ( 0.28)) ( 10 ( 0.28)) SD( X) Var( X) $18.33 c) EX ( XXXX) 5 EX ( ) 5( 0.28) $1.40 SD( XXXXX) 5( Var( X)) 5( ) $40.99 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 $0.28, with stadard deviatio $ Rollig a die is radom ad the outcomes are mutually idepedet, so the Cetral Limit Theorem guaratees that the samplig distributio model is Normal with x $0.28 ad stadard deviatio SD( x) $ Accordig to the Normal model, the probability that the perso ruig the game makes a profit is approximately AP Stats a) 5(0.125) 4(0.211) 3(0.256) 2(0.180) 1(0.228) ( ) (0.125) ( ) (0.211) ( ) (0.256) 2 2 ( ) (0.180) ( ) (0.228) b) The distributio of scores for 40 radomly selected studets would ot follow a Normal model. The distributio would resemble the populatio, mostly uiform for scores 1 4, with about half as may 5s. c) Radomizatio coditio: The scores are selected radomly. Idepedece assumptio: It is reasoable to thik that the radomly 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.

25 Chapter 17 Samplig Distributio Models 377 Sice the coditios are satisfied, the samplig distributio model for the mea of 40 radomly selected AP Stat scores is Normal, with y ad stadard deviatio SD( y) Museum membership. a) 50(0.41) 100(0.37) 250(0.14) 500(0.07) 1000(0.01) $ ( ) (0.41) ( ) (0.37) ( ) (0.14) 2 2 ( ) (0.07) ( ) (0.01) $ The calculatio for stadard deviatio is based o a rouded mea. Use techology to calculate the mea ad stadard deviatio to avoid iaccuracy. b) The distributio of doatios for 50 ew members would ot follow a Normal model. The ew members would probably make doatios typical of the curret member populatios, so the distributio would resemble the populatio, skewed to the right. c) Radomizatio coditio: The members are ot selected radomly. They are simply the ew members that day. However, the doatios they make are probably typical of the doatios made by 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. Sice the coditios are satisfied, the samplig distributio model for the mea of 50 doatios is Normal, with y $ ad stadard deviatio SD( y) AP Stats 2012, agai. Sice the teacher cosiders his 63 studets typical, 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 y ad stadard deviatio ( y)

26 378 Part V From the Data at Had to the World at Large y - m y z = SD( y) z = z = Accordig to the samplig distributio model, the probability that the class of 63 studets achieves a average of 3 o the AP Stat exam is about 20%. 52. 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 y $ ad stadard deviatio ( y) $ y y z SD( y) z z Accordig to the samplig distributio model, there is a 98.8% probability 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 type. Idepedece assumptio: It is reasoable to thik that the CO emissios for these cars are mutually idepedet. 10% coditio: The 80 cars i the fleet certaily represet less tha 10% of all cars of this type. Large Eough Sample coditio: A sample of 80 cars is large eough. The mea CO level was 2.9 gm/mi, with stadard deviatio 0.4 gm/mi. Sice the coditios are met, the CLT allows us to model the samplig distributio of the y with a Normal model, with y 2.9 gm/mi ad stadard 0.4 deviatio SD( y) gm/mi. 80

27 Chapter 17 Samplig Distributio Models 379 b) Accordig to the Normal model, the probability that y is betwee 3.0 ad 3.1 gm/mi is approximately c) Accordig to the Normal model, there is oly a 5% chace that the fleet s mea CO level is greater tha approximately 2.97 gm/mi. y y z SD( y) y y Potato chips. a) Accordig to the Normal model, oly about 4.78% of the bags sold are uderweight. b) P (oe of the 3 bags are uderweight) 3 ( ) 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 mutually idepedet. 10% coditio: The 3 bags certaily 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 10.2 ouces, with stadard deviatio 0.12 ouces. Sice the coditios are met, we ca model the samplig distributio of y with a Normal model, with y 10.2 ouces ad stadard deviatio 0.12 ( y) ouces. 3

28 380 Part V From the Data at Had to the World at Large Accordig to the Normal model, the probability that the mea weight of the 3 bags is less tha 10 ouces is approximately d) For 24 bags, the stadard deviatio of the samplig distributio model is 0.12 SD( y) ouces. Now, a average of 10 ouces is over 8 stadard 24 deviatios below the mea of the samplig distributio model. This is extremely ulikely. 55. Tips. a) Sice the distributio of tips is skewed to the right, we ca t use the Normal model to determie the probability that a give party will tip at least $20. b) No. A sample of 4 parties is probably 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 may ot be large eough to allow the use of a Normal model to describe the distributio of averages. It would be risky to attempt to estimate the probability that his ext 10 parties tip a average of $15. However, sice the distributio of tips has $9.60, with stadard deviatio $5.40, we still kow that the mea of the samplig distributio model is y $ with stadard deviatio SD( y) $ 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 very likely to happe. 56. Groceries. a) Sice the distributio of Suday purchases is skewed, we ca t use the Normal model to determie the probability that a give purchase is at least $40. b) A sample of 10 customers may ot be large eough for the CLT to allow the use of a Normal model for the samplig distributio model. If the distributio of Suday purchases is oly slightly skewed, the sample may be large eough, but if the distributio is heavily skewed, it would be very risky to attempt to determie the probability.

29 Chapter 17 Samplig Distributio Models 381 c) Radomizatio coditio: Assume that the 50 Suday purchases ca be cosidered a represetative sample of all purchases. Idepedece assumptio: It is reasoable to thik that the Suday purchases are mutually idepedet, uless there is a sale or other icetive to purchase more. 10% coditio: The 50 purchases certaily represet less tha 10% of all purchases. Large Eough Sample coditio: The sample of 50 purchases is large eough. The mea Suday purchase is $32, with stadard deviatio $20. Sice the coditios are met, the CLT allows us to model the samplig distributio of y with a Normal model, with y $32 ad stadard deviatio 20 ( y) $ Accordig to the Normal model, the probability the mea Suday purchase of 50 customers is at least $40 is about 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 mutually idepedet, uless the service is particularly good or bad durig this weeked. 10% coditio: The tips of 40 parties certaily represet less tha 10% of all tips. Large Eough Sample coditio: The sample of 40 parties is large eough. The mea tip is $9.60, with stadard deviatio $5.40. Sice the coditios are satisfied, the CLT allows us to model the samplig distributio of y with a Normal model, with y $9.60 ad stadard deviatio 5.40 SD( y) $ I order to ear at least $500, the waiter would have to average 500 $ per party.

30 382 Part V From the Data at Had to the World at Large Accordig to the Normal model, the probability that the waiter ears at least $500 i tips i a weeked is approximately b) Accordig to the Normal model, the waiter ca expect to have a mea tip of about $ , which correspods to about $ for 40 parties, i the best 10% of such weekeds. y y z SD( y) y y More groceries. a) Assumptios ad coditios for the use of the CLT were verified i a previous exercise. The mea purchase is $32, with stadard deviatio $20. Sice the sample is large, the CLT allows us to model the samplig distributio of y with 20 a Normal model, with y $32 ad stadard deviatio SD( y) $ I order to have reveues of at least $10,000, the mea Suday purchase must be at least 10,000 $ Accordig to the Normal model, the probability of havig a mea Suday purchase at least that high (ad therefore at total reveue of at least $10,000) is

31 Chapter 17 Samplig Distributio Models 383 b) y y z ( y) y y Accordig to the Normal model, the mea Suday purchase o the worst 10% of such days is approximately $ , so 312 customers are expected to sped about $ IQs. a) Accordig to the Normal model, the probability that the IQ of a studet from East State is at least 125 is approximately 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. EE ( W) EE ( ) EW ( ) SD( E W ) Var( E) Var( W ) Sice both distributios are Normal, the distributio of the differece is N(10, ). Accordig to the Normal model, the probability that the IQ of a studet at ESU is at least 5 poits higher tha a studet at WSU is approximately

32 384 Part V From the Data at Had to the World at Large c) Radomizatio coditio: Studets are radomly sampled from WSU. Idepedece assumptio: It is reasoable to thik that the IQs are mutually idepedet. 10% coditio: The 3 studets certaily 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 ay size will be Normal, so a sample of 3 studets is large eough. The mea IQ is w 120, with stadard deviatio w 10. Sice the distributio IQs is Normal, we ca model the samplig distributio of w with a Normal model, with w 120 with stadard deviatio 10 SD( w) Accordig to the Normal model, the probability that the mea IQ of the 3 WSU studets is above 125 is approximately d) As i part c, the samplig distributio of e, the mea IQ of 3 ESU studets, ca be modeled with a Normal model, with e 130 with stadard deviatio 8 SD( e ) The distributio of the differece i mea IQ is Normal, with the followig parameters: ew Ee ( w) Ee ( ) Ew ( ) SD( e w) Var( e ) Var( w) ew Accordig to the Normal model, the probability that the mea IQ of 3 ESU studets is at least 5 poits higher tha the mea IQ of 3 WSU studets is approximately

33 54. Milk. a) Accordig to the Normal model, the probability that a Ayrshire averages more tha 50 pouds of milk per day is approximately Chapter 17 Samplig Distributio Models 385 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. EA ( J) EA ( ) EJ ( ) pouds SD( A J) Var( A) Var( J) pouds Sice both distributios are Normal, the distributio of the differece is N(4, ). Accordig to the Normal model, the probability that the Ayrshire gives more milk tha the Jersey is approximately c) Radomizatio coditio: Assume that the farmer s 20 Jerseys are a represetative sample of all Jerseys. Idepedece assumptio: It is reasoable to thik that the cows have mutually idepedet milk productio. 10% coditio: The 20 cows certaily represet less tha 10% of cows. Large Eough Sample coditio: Sice the distributio of daily milk productio is Normal, the sample meas of samples of ay size are Normally distributed, so the sample of 20 cows is certaily large eough.

34 386 Part V From the Data at Had to the World at Large The mea milk productio is 43 pouds, with stadard deviatio 5. j Sice the distributio of milk productio is Normal, we ca model the samplig distributio of j with a Normal model, with 43 pouds with stadard j 5 deviatio SD( j ) Accordig to the Normal model, the probability that the mea milk productio of the 20 Jerseys is above 45 pouds of milk per day is approximately j d) As i part c, the samplig distributio of a, the mea milk productio of 10 Ayrshires, ca be modeled with a Normal model, with a 47 pouds with 6 stadard deviatio SD( a) pouds. 10 The distributio of the differece i mea milk productio is Normal, with the followig parameters: a j a j Ea ( j) Ea ( ) Ej ( ) pouds SD( a j ) Var( a) Var( j ) pouds Accordig to the Normal model, the probability that the mea milk productio of 10 Ayrshires is at least 5 pouds higher tha the mea milk productio of 20 Jerseys is approximately