Parameter, Statistic and Random Samples

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Clifford Hicks
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1 Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e., it is a quatity whose value ca be calculated from the sample data. It is a radom variable with a distributio fuctio. Statistics are used to make iferece about ukow populatio parameters. The radom variables X, X 2,, X are said to form a (simple) radom sample of size if the X i s are idepedet radom variables ad each X i has the sample probability distributio. We say that the X i s are iid. STA286 week 8

2 Example Sample Mea ad Variace Suppose X, X 2,, X is a radom sample of size from a populatio with mea μ ad variace σ 2. The sample mea is defied as X i X i. The sample variace is defied as 2 S 2 ( X i X ). The sample stadard deviatio, S, is the square root of the sample variace. i STA286 week 8 2

3 Quatiles A quatile of a sample, x p, is the value for which a specific fractio, p, of the data values is less tha or equal to it, ad (-p) is greater tha it. The most kow quatile is the media which is the 50th quatile. Quatiles are ofte described as percetiles ad represets a estimate of a characteristic of the theoretical distributio. If a data set cotais observatios, the the pth percetile is the th p ( + ) value i the ordered data set. 00 We ca describe the spread or variability of a distributio by givig several percetiles. STA286 week 8 3

4 Quartiles The 25th percetile is called the first quartile (Q ). The 75th percetile is called the third quartile (Q 3 ). Note, the media is the secod quartile Q 2. The distace betwee the first ad third quartiles is called the Iterquartile rage (IQR) i.e. IQR Q 3 Q. The IQR is aother measure of spread that is less sesitive to the ifluece of extreme values. STA286 week 8 4

5 The five-umber summary Thefive-umber summary of a set of observatios cosists of the smallest observatio, the first quartile, the media, the third quartile ad the largest observatio. These five umbers give a reasoably complete descriptio of both the ceter ad the spread of the distributio. MINITAB commads: Stat > Basic Statistics > Display Descriptive Statistics STA286 week 8 5

6 Example The highway mileages of 20 cars, arraged i icreasig order are: Give the five umber summary. Aswer We have, mi 3, Q 8, media 23, Q 3 27, max 32. The MINITAB output usig the above commads is as follows: Variable N Miimum Q Media Q3 Maximum mileage STA286 week 8 6

7 Box-plot A box-plot is a graph of the five-umber summary. Example: Make a box-plot for the data i the above example. Boxplot of Mileages 30 Mileages MINITAB commads: Graph > Boxplot STA286 week 8 7

8 Quatile Plots A quatile plot is a plot of the data values o the vertical axis agaist a empirical assessmet of the fractio of observatios exceeded by the data value. A very useful quatile plot is the Normal-Quatile-Quatile plot. It is ofte used by aalysts to determie whether a data set came from a ormal distributio. A Normal Quatile Quatile plot is a plot of the empirical (data) quatiles agaist the correspodig quatiles of the ormal distributio STA286 week 8 8

9 Iterpretig Normal Quatile Plots If the data comes form ay ormal distributio, the NQQ plot produces a straight lie o the plot. If the poits o a ormal quatile plot lie close to a straight lie, the plot idicates that the data are ormal. Systematic deviatios from a straight lie idicate a oormal distributio. Outliers appear as poits that are far away from the overall patter of the plot. STA286 week 8 9

10 Histogram, the scores plot ad the ormal quatile plot for data geerated from a ormal distributio (N(500, 20)) Frequecy 5 value value Normal Probability Plot for value cores 99 ML Estimates Mea: StDev: Percet STA286 week Data

11 Histogram, the scores plots ad the ormal quatile plot for data geerated from a right skewed distributio 0 Frequecy value 0 value cores 2 STA286 week 8

12 2 cores value Norm al Probability Plot for value 99 M L Estim ates M ea: StDev: Percet STA286 Data week 8 2

13 Histogram, the scores plots ad the ormal quatile plot for data geerated from a left skewed distributio 0 Frequecy value value score STA286 week 8 3

14 2 score value Normal Probability Plot for value 99 ML Estimates M ea: StDev: Percet Data STA286 week 8 4

15 Histogram, the scores plots ad the ormal quatile plot for data geerated from a uiform distributio (0,5) Frequecy value 5 4 value cores STA286 week 8 5

16 2 cores value Normal Probability Plot for value 99 M L Estim ates M ea: StDev: Percet STA286 week 8 6 Data

17 Samplig Distributio of a Statistic The samplig distributio of a statistic is the distributio of values take by the statistic i all possible samples of the same size from the same populatio. The distributio fuctio of a statistic is NOT the same as the distributio of the origial populatio that geerated the origial sample. The form of the theoretical samplig distributio of a statistic will deped upo the distributio of the observable radom variables i the sample. STA286 week 8 7

18 Samplig from Normal populatio Ofte we assume the radom sample X, X 2, X is from a ormal populatio with ukow mea μ ad variace σ 2. Suppose we are iterested i estimatig μ ad testig whether it is equal to a certai value. For this we eed to kow the probability distributio of the estimator of μ. STA286 week 8 8

19 Samplig Distributio of Sample Mea Suppose X, X 2, X are i.i.d ormal radom variables with ukow mea μ ad variace σ 2 the X ~ 2 σ N μ, Proof: STA286 week 8 9

20 The Cetral Limit Theorem Let X, X 2, be a sequece of i.i.d radom variables with mea E(X i ) μ < ad Var(X i ) σ 2 <. Let S μ The, Z coverges i distributio to Z ~ N(0,). σ Also, Z coverges i distributio to Z ~ N(0,). σ Example X μ S X i i STA286 week 8 20

21 Example Suppose that the weights of airlie passegers are kow to have a distributio with a mea of 75kg ad a std. dev. of 0kg. A certai plae has a passeger weight capacity of 7700kg. What is the probability that a flight of 00 passegers will exceed the capacity? week 8 2

22 Questio State whether the followig statemets are true or false. (i) As the sample size icreases, the mea of the samplig distributio of the sample mea X decreases. (ii) As the sample size icreases, the stadard deviatio of the samplig distributio of the sample mea X decreases. (iii) The mea X of a radom sample of size 4 from a egatively skewed distributio is approximately ormally distributed. (iv) The distributio of the proportio of successes X i a sufficietly large sample is approximately ormal with mea p ad stadard deviatio p ( p) where p is the populatio proportio ad is the sample size. (v) If X is the mea of a simple radom sample of size 9 from N(500, 8) distributio, the X has a ormal distributio with mea 500 ad variace 36. week 8 22

23 Questio State whether the followig statemets are true or false. o A large sample from a skewed populatio will have a approximately ormal shaped histogram. o The mea of a populatio will be ormally distributed if the populatio is quite large. o The average blood cholesterol level recorded i a SRS of 00 studets from a large populatio will be approximately ormally distributed. o The proportio of people with icomes over $ , i a SRS of 0 people, selected from all Caadia icome tax filers will be approximately ormal. week 8 23

24 Exercise A parkig lot is patrolled twice a day (morig ad afteroo). I the morig, the chace that ay particular spot has a illegally parked car is If the spot cotaied a car that was ticketed i the morig, the probability the spot is also ticketed i the afteroo is 0.. If the spot was ot ticketed i the morig, there is a chace the spot is ticketed i the afteroo. a) Suppose tickets cost $0. What is the expected value of the tickets for a sigle spot i the parkig lot. b) Suppose the lot cotais 400 spots. What is the distributio of the value of the tickets for a day? c) What is the probability that more tha $200 worth of tickets are writte i a day? week 8 24

25 Law of Large Numbers - Example Toss a coi times. Suppose X i 0 if i if i th th toss came up H toss came up T X i s are Beroulli radom variables with p ½ ad E(X i ) ½. The proportio of heads is X X i. X i Ituitively approaches ½ as. STA286 week 8 25

26 STA286 week 8 26 Law of Large Numbers Iterested i sequece of radom variables X, X 2, X 3, such that the radom variables are idepedet ad idetically distributed (i.i.d). Let Suppose E(X i ) μ, V(X i ) σ 2, the ad Ituitively, as, so i X i X ( ) ( ) μ i i i i X E X E X E ( ) ( ) X V X V X V i i i i 2 2 σ ( ) 0 X V ( ) μ X E X

27 Formally, the Weak Law of Large Numbers (WLLN) states the followig: Suppose X, X 2, X 3, are i.i.d with E(X i ) μ <, V(X i ) σ 2 <, the for ay positive umber a as. ( X a) 0 P μ This is called Covergece i Probability. STA286 week 8 27

28 Recall - The Chi Square distributio If Z ~ N(0,) the, X Z 2 has a Chi-Square distributio with parameter, i.e., X χ ~ 2 (). Ca proof this usig chage of variable theorem for uivariate radom variables. The momet geeratig fuctio of X is m X () t 2t / If X χ, X ~ χ, K, X χ, all idepedet the Proof ~ 2 k ( v ) 2 ( v ) k ( v ) ~ k T ~ χ i X i 2 Σ k v i STA286 week 8 28

29 Claim Suppose X, X 2, X are i.i.d ormal radom variables with mea μ ad variace σ 2 X. The, i μ Z are idepedet stadard ormal i σ variables, where i, 2,, ad Proof: i Z 2 i i 2 X i μ σ 2 ~ χ ( ) STA286 week 8 29

30 Samplig Distributio of S 2 Suppose X, X 2, X are i.i.d ormal radom variables with mea μ ad variace σ 2. The, ( ) 2 σ s 2 2 σ 2 2 ( X i X ) ~ χ( ) i Further, it ca be show that X ad s 2 are idepedet. STA286 week 8 30

31 t distributio Suppose Z ~ N(0,) idepedet of X ~ χ 2 (). The, T Z X / v ~ t ( ). v Proof: usig oe dimesioal chage of variables theorem. The desity fuctio of the t-distributio is give by STA286 week 8 3

32 Claim Suppose X, X 2, X are i.i.d ormal radom variables with mea μ ad variace σ 2. The, Proof: X μ ~ t S / ( ) STA286 week 8 32

33 F distributio Suppose X ~ χ 2 () idepedet of Y ~ χ 2 (m). The, X / Y / m ~ F (, m) The desity fuctio of the F distributio is give by STA286 week 8 33

34 Properties of the F distributio The F-distributio is a right skewed distributio. F( ) i.e. m, F ( < a) P F(, m) (, m) P F (, m) > a P F( m, ) > a Ca use Table A.6 i appedix to fid percetile of the F- distributio. Example STA286 week 8 34

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,