Sec 7.6 Inferences & Conclusions From Data Central Limit Theorem

Transcription

1 Sec 7. Ifereces & Coclusios From Data Cetral Limit Theorem Name: The Cetral Limit Theorem offers us the opportuity to make substatial statistical predictios about the populatio based o the sample. To better uderstad the relatioship betwee samples ad populatios let s cosider a cotrolled situatio with a populatio that is very small ad look at possible samples. Cosider the followig shoe sizes of studets to represet a etire populatio: 8,,, 12.The populatio mea () of these four data poits is. Cosider lookig at every possible sample of 2 from the group (allowig repetitio of data poits). The followig samples of 2 are possible. The populatio distributio is show at the right. Actual Data Values Fid the mea of each 2 umber samples (i.e. = 2) 1. {8,8}:. {,8}: 2. {8,}:. {,}: 3. {8,}:. {,}:. {8,12}: 12. {,12}: 5. {,8}: 13. {12,8}:. {,}: 1. {12,}: 7. {,}: 15. {12,}: 8. {,12}: 1. {12,12}: Create a frequecy histogram at the right, poits from the table above. usig data Sample Meas (Remider: I the histogram at the right the first class boudary would iclude the data poits i the rage 8.5 <.5 which suggests.5 would ot be icluded i the first bar of the histogram) 8 12 [7.5,8.5) [8.5,.5) [.5,.5) [.5,.5) [.5,12.5) 1. Fid the percetage of the 1 meas from problem #3 that are at most 1 uit from the mea (i.e. ). This would represet the percetage of the times a radom represetative sample would be withi 1 uit from the mea (or the percetage of the times a sample of 2 would be reasoably accurate for our populatio). 2. What is the mea( ) ad stadard deviatio( ) of the sample meas of size 2 from problem #3 ad verify that these formulas hold true for this data set ( ad ) M. Wikig Uit 7- page 15

2 3. Cosider usig the same populatio of 8,,,12 but this time usig all of the possible samples of size 3 would yield the followig. 1.{8,8,8}: 8.{8,,8}: 17.{,8,8}: 8 25.{,,8}: 33.{,8,8}: 1.{,,8}:.{12,81,8}: 57.{12,,8}: 2.{8,8,}: 8.{8,,}: 18.{,8,}: 8. 2.{,,}:. 3.{,8,}: 2.{,,}: 50.{12,8,}:. 58.{12,,}:. 3.{8,8,}:.{8,,}: 1.{,8,}: 27.{,,}: 35.{,8,}: 3.{,,}: 51.{12,8,}: 5.{12,,}:.{8,8,12}: 12.{8,,12}: 20.{,8,12}:. 28.{,,12}:. 3.{,8,12}:.{,,12}: 52.{12,8,12}:. 0.{12,,12}:. 5.{8,,8}: 8 13.{8,12,8}: 21.{,,8}: 8. 2.{,12,8}:. 37.{,8,8}: 5.{,12,8}: 53.{12,,8}:. 1.{12,12,8}:..{8,,}: 8. 1.{8,12,}:. 22.{,,}: 30.{,12,}: 38.{,8,}:..{,12,}:. 5.{12,,}: 2.{12,12,}: 7.{8,,}: 15.{8,12,}: 23.{,,}:. 31.{,12,}:. 3.{,8,}: 7.{,12,}: 55.{12,,}:. 3.{12,12,}:. 8.{8,,12}:. 1.{8,12,12}:. 2.{,,12}: 32.{,12,12}: 0.{,8,12}:. 8.{,12,12}:. 5.{12,,12}:.{12,12,12}: [8.5,.5) [.5,.5) [.5,.5) [.5,12.5) [12.5,13.5) Recreate your histogram from problem #3 o top of the completed histogram above.. Fid the percetage of the meas from problem # that are at most 1 uit from the mea (i.e. ). This would represet the percetage of the times a radom represetative sample would be withi 1 uit from the mea (or the percetage of the times a sample of 3 would be reasoably accurate for our populatio). 5. How does questio #7 compare with the questio #?. What is the mea( ) ad stadard deviatio( ) of the sample meas of size 3 from problem # usig the formulas ( ad )? 7. Describe ad compare the frequecy distributios betwee the populatio, the sample meas of size = 2, the sample meas of size = 3. M. Wikig Uit 7- page 157

3 THE CENTRAL LIMIT THEOREM If samples of size, where 30, are draw from ay populatio with a mea (μ) ad a stadard deviatio (σ), the the samplig distributio of sample meas approimates a ormal distributio. The greater the sample size, the better the approimatio. If the populatio itself is ormally distributed, the samplig distributio of the sample meas is ormally distributed for ay sample size. For all distributios o The mea of the sample meas is equal to the populatio mea (i.e. ) o The stadard deviatio of the sample meas ca be described by the formula ( ). 8. The followig shows the populatio distributio of the umber of hours of sleep elemetary childre were gettig at oe school. Assume sample size of 81 is draw from the populatio. Decide which of the graphs would most closely resemble the samplig distributio of the sample meas for the graph. Eplai your reasoig. (A) (B) (C). Use the cetral limit theorem if possible. For a sample of = 3, fid the probability of a sample mea beig less tha 15 if μ = 1 ad σ = 3. Nothig is kow about the populatio distributio. 3.1% 3.1% M. Wikig Uit 7- page 158

4 . Use the cetral limit theorem if possible. For a sample of =, fid the probability of a sample mea beig greater tha.2 if μ = 8 ad σ = 2. The populatio distributio is uiformly distributed. 3.1% 3.1%. Use the cetral limit theorem if possible. For a sample of = 1, fid the probability of a sample mea beig greater tha 1.3 if μ = 1 ad σ = 1. The populatio distributio is ormally distributed. 3.1% 3.1% 12. True / False. The distributio of sample meas is always ormally distributed. Regardless of the sample size the mea of the sample meas is always the same. The rage of the iterval estimate icreases as the level of cofidece icreases. As the sample size decreases for a fied level of cofidece the iterval size also stays fied. 13. Cosider the followig situatios ad usig the CENTRAL LIMIT THEOREM fid the MEAN of the sample meas (μ ) ad the STANDARD DEVIAITON of the sample meas also kow as Stadard Error(σ ): I. Data for the parkig lot of Phoei H.S. showed that the mea year model of a car is with a stadard deviatio of 2.5 a. If radom samples of 30 cars at a time were take, b. If radom samples of 30 cars at a time were take, what what would be the mea of all of the samples? would be the stadard deviatio of all of the samples? II. The mea GPA for seiors this year is 2.75 with a stadard deviatio of 0.2 ad is approimately ormally distributed. a. If radom samples of 1 seiors at a time were take, what would be the mea of all of the samples? b. If radom samples of 1 seiors at a time were take, what would be the stadard deviatio of all of the samples? III. The mea speed of computer processors i the school is 2.8 ghz with stadard deviatio of 0. ghz. The data is NOT ormally distributed. a. If radom samples of 3 computers at a time were take, what would be the mea of all of the samples? b. If radom samples of 3 computers at a time were take, what would be the stadardd deviatio of all of the samples? M. Wikig Uit 7- page 15

5 1. Cosider the followig situatios ad usig the CENTRAL LIMIT THEOREM fid the requested PROBABILITY. I. The mea legth of wait time to see a tech specialist at the APPLE store is miutes with a stadard deviatio of 5.3 miutes. (Assumig the populatio data is ormally distributed) a. If those parameters are correct, what is the probability that 3 frieds would have to wait more tha a total of a 0 miutes to see a tech specialist? (hit: = 3 ad their average wait time was 0/3 miutes) b. If those parameters are correct, what is the probability of havig a tech specialist log that of his cliets had a average wait time betwee ad miutes? II. May times baks look at the mea value of a savigs accout over the periods of time. Over the past year, the mea value of a particular perso s savigs accout is $3200 with a stadard deviatio of $300. (Assumig the populatio data is ormally distributed) a. If those parameters are correct, what is the probability that for 7 radomly selected days of the year, the average value over those 7 days is less tha $3000? b. If those parameters are correct, what is the probability that for 5 radomly selected days of the year, the average value over those 5 days is betwee $3300 ad $3500? M. Wikig Uit 7- page 10