(6) Fundamental Sampling Distribution and Data Discription

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1 34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye

2 8.1 Radom Samplig: Populatio: A populatio cosists of the total observatios with which we are cocered at a particular time. Sample: A sample is a subset of a populatio. Radom Sample Let be idepedet radom variable; havig the, 1,.. same probability distributio f(), the, 1,.. defied to be a radom sample of size from the populatio f(x) Statistic : Ay fuctio of the radom sample is called a statistic. is

3 8. Some importat Statistics: 1- Locatio Measures of a Sample: The Sample Mea,Media ad Mode (a) Sample Mea: If, 1,.. represet a radom sample of size, the the sample mea is defied by the statistic: i i 1 (1)

4 Properties of the Mea (Readig): 1. The mea is the most commoly used measure of certai locatio i statistics.. It employs all available iformatio. 3. The mea is affected by extreme values. 4. It is easy to calculate ad to uderstad. 5. It has a uique value give a set of data.

5 E 1: The legth of time, i miutes, that 10 patiets waited i a doctor's office before receivig treatmet were recorded as follows: 5, 11, 9, 5, 10, 15, 6, 10, 5 ad 10. Fid the mea. Solutio: 10, x i x i See Ex 8.4 pg 8

6 (b) Sample Media: If 1... represet a radom sample of size, arraged i icreasig order of magitude, the the sample media, which is deoted by Q is defied by the statistic: Q ( 1) if is odd ( ) 1 if is eve ()

7 Properties of the Media(Readig): 1. The media is easy to compute if the umber of observatios is relatively small.. It is ot affected by extreme values.

8 E : The umber of foreig ships arrivig at a east cost port o 7 radomly selected days were 8, 3, 9, 5, 6, 8 ad 5. Fid the sample media. Solutio: The arraged values are: , 4 Q 6

9 E 3: The icotie cotets for a radom sample of 6 cigarettes of a certai brad are foud to be.3,.7,.5,.9, 3.1 ad 1.9 milligrams. Fid the media. Solutio: The arraged values are: 1.9,.3,.5,.7,.9, 3.1. Q 6 6 6, 3,

10 (c)sample Mode: The sample mode is the value of the sample that occurs most ofte.

11 Properties of the Mode(Readig): 1. The value of the mode for small sets of data is almost useless.. It requires o calculatio.

12 E(4): The umbers of icorrect aswers o a true false test for a radom sample of 14 studets were recorded as follows:, 1, 3, 0, 1, 3, 6, 0, 3, 3,, 1, 4, ad, fid the mode. Solutio: mode=3 H.w. Fid the mea ad media

13 Notes: 1. The sample meas usually will ot vary as much from sample to sample as will the media.. The media (whe the data is ordered) ad the mode ca be used for qualitative as well as quatitative data.

14 - Variability Measures of a Sample: The Sample Variace, Stadard deviatio ad Rage: The Rage: (a)the rage of a radom sample defied by the statistic, where ad ( ) (1) respectively the largest ad the smallest observatios which is deoted by R, the: ( ) (1) 1... is R Max Mi x x ( ) (1)

15 E 5: Let a radom sample of five members of a sorority are 108,11,17,118 ad 113. Fid the rage. Solutio: R=17-108=19

16 (b) Sample Variace:... If represet a radom sample of size, 1 the the sample variace, which is deoted by is defied by the statistic: S S ( i ) i i 1 i 1 i 1 1 i 1 1 [ ] 1 [ i ] 1 i 1

17 Ex 8. pg 9 A compariso of coffee prices at 4 radomly selected grocery stores i Sa Diego showed icreases from the previous moth of 1, 15, 17, 0, cets for a 00 gram jar. Fid the variace of this radom sample of price icreases. Solutio: S i i 1 i ( i ) (1 16) (15 16) (17 16) (0 16) 1 3

18 (b) Sample Stadard Deviatio: S S

19 E 6 : The grade poit average of 0 college seiors selected at radom from a graduatig class are as follows: 3., 1.9,.7,.4,.8,.9, 3.8, 3.0,.5, 3.3, 1.8,.5, 3.7,.8,.0, 3.,.3,.1,.5, 1.9. Calculate the variace ad the stadard deviatio. Solutio: i 53.3, i i 1 i 1 (aswer: ).665, S 0.585, S 0.34 See Ex 8.3 pg 30

20 8.3 Samplig Distributios Defiitio: The probability distributio of a statistic is called a samplig distributio.

21 8.4 Samplig Distributios of Meas ad the Cetral Limit Theorem ( ), V ( ) (3) E

22 Cetral Limit Theorem: If is the mea of a radom sample of size take from a populatio with mea µ ad fiite variace σ², the the limitig form of the distributio of: Z as / (4) is approximately the stadard ormal distributio

23 Notes f ( Z ) ~ N (0,1) E ( ), V ( ), ~ N (, ) (5)

24 E(8.4 pg 34): A electrical firm maufactures light bulbs that have a legth of life that is approximately ormally distributed with mea equal to 800 hours ad a stadard deviatio of 40 hours. Fid the probability that a radom sample of 16 bulbs will have a average life of less tha 775 hours.

25 Solutio: Let be the legth of life ad life; 16, 800, 40 is the average P ( 775) P ( Z P ( Z.5) / 16 See Ex 8.5 pg 37

26 Samplig distributio of the differece betwee two meas Theorem If idepedet samples of size 1 ad are draw at radom from populatios, discrete or cotiuous with meas 1 ad variaces ad respectively, the the samplig distributio of the differece of meas is approximately ormally 1 distributed with mea ad variace give by: Hece ad 1 1 ( 1 1 ) ( 1 ) 1 is approximately a stadard ormal variable. 1 E ( ), (6) Z ( 1 ) ( 1 ) 1 1 (7)

27 E(10): A sample of size 1 =5 is draw at radom from a populatio that is ormally distributed with mea ad variace 1 9 ad the sample mea is 1 recorded. A secod radom sample of size =4 is selected idepedet of the first sample from a differet populatio that is also ormally distributed with mea 40 ad variace 4 ad the sample mea is recorded. Fid P ( 1 8.) 1 50

28 Solutio: See Case Study 8. pg P ( 1 8.) P ( Z ) P ( Z 1.08)

29 E(11): (H.W) The televisio picture tubes of maufacturer A have a mea lifetime of 6.5 years ad a stadard deviatio of 0.9 year, while those of maufacturer B have a mea lifetime of 6 years ad a stadard deviatio of 0.8 year. What is the probability that a radom sample of 36 tubes from maufacture A will have a mea lifetime that at least 1 year more tha the mea lifetime of a sample of 49 from tubes maufacturer B?

30 Solutio: Populatio 1 Populatio P[( 1 ) 1) P ( Z ) P ( Z.65) P( Z.65)

31 Samplig Distributio of the sample Proportio (Readig ): Let = o. of elemets of type A i the sample P= populatio proportio = o. of elemets ˆp of type A i the populatio / N = sample proportio = o. of elemets of type A i the sample / = x/

32 x ~ biomial (, p) E ( x ) p, V ( x ) pq x 1. E ( pˆ ) E ( ) p x pq. V ( pˆ ) V ( ), q 1 p 3. For large, we have: pq pˆ ~ N ( p, ) Z pˆ pq p ~ N (0,1)

33 8.6 t Distributio (pg 46): * t distributio has the followig properties: 1. It has mea of zero.. It is symmetric about the mea. 3. It rages from to. 4. Compared to the ormal distributio, the t distributio is less peaked i the ceter ad has higher tails. 5. It depeds o the degrees of freedom (-1). 6. The t distributio approaches the ormal distributio as (-1) approaches.

34 Notes Sice the t-distributio is symmetric about zero we have t 1- = - t Table A4 pg represet the critical values of t-distributio Where t leavig a area of to The right. -t t

35 Corollary 8.1 s

36 E(1): Fid: ( a) t whe v ( b) t whe v ( c) t whe v

37 (a) t 0.05 at v = 14 t =.1448 c t at v = 7 t = t = 3.499

38 E(13): Fid: ( a) P ( T.365) whe v 7 ( b) P ( T 1.318) whe v 4 ( c) P ( T.179) whe v 1 ( d ) P ( T.567) whe v 17

39 solutio 0.975

(7 One- and Two-Sample Estimation Problem )

(7 One- and Two-Sample Estimation Problem ) 34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate: