Normal DistributionNormal Distribution Zol (1)

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Normal DistributionNormal Distribution Zol (1)

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  • Taburan NormalGB6023 Kaedah Penyelidikan [email protected]

  • Objektif:Pengenalan kpd taburan normalCiri-ciri taburan normal standard.Pengenalan kepada Teorem limit memusat. (Central Limit Theorem)Penggunaan taburan normal dlm inferensi.Taburan t.

  • Taburan Theoretikal Taburan Empirikal Berdasarkan kepada dataTaburan Theoretikal Berdasarkan kepada matematikDiterbitkan dari model atau jangkaan dari data.

  • Taburan normalMengapa taburan normal sgt penting?Banyak pembolehubah bersandar selalunya diandaikan tertabur secara normal dlm populasi.Jika pembolehubah itu hampir tertabur secara normal, kita boleh membuat inferensi terhadap nilai pada pembolehubah itu.Cth: Taburan persampelan min.

  • Maka apa? So what?Taburan normal dan ciri-cirinya kita ketahui, dan jika pembolehubah yg kita minati itu tertabur secara normal, kita boleh mengaplikasikan apa yg kita tahu berkaitan dgn taburan normal itu dalam setiuasi penyelidikan kita.Kita boleh menentukan kebarangkalian sesuatu hasilan (outcomes)

  • Taburan normalSimetri, bentuk lengkuk locengJuga dikenali sebagai taburan Gaussian Poin titik perbubahan = 1 sisihan piwai dari minFormula matematik

  • Oleh kerana kita tahu bentuk lengkung, kita boleh kira luas kawasan dibawah lengkung. Peratusan kawasan dibawah lengkuk boleh digunakan untuk menentukan kebarangkalian sesuatu nilai yang di ambil dari taburan. Dengan kata lain, luas dibawah lengkung memberitahu kita berkenaan dgn kebarangkalian (nilai-p) untuk keputusan(data) dengan mengandaikan set data kita tertabur secara normal.

  • Kawasan utama dibawah lengkungUtk taburan normal + 1 SD ~ 68% + 2 SD ~ 95% + 3 SD ~ 99.9%

  • Cth min IQ = 100 sisihan piwai (s) = 15

  • masalah: Setiap taburan normal mempunyai nilai min m dan sisihan piawai s memerlukan pengiraannya sendiri untuk menentukan luas dibawah lengkung pada setiap titik.

  • Taburan Kebarangkalian NormalTaburan normal standard N(0,1)Kita setuju guna taburan normal standard.Bentuk loceng=0=1Nota: tidak semua taburan bentuk loceng adalah taburan normal

  • Normal Probability DistributionCan take on an infinite number of possible values.Curve has area or probability = 1

  • Taburan Normal Taburan normal standard membolehkan kita membuat hujah (claims) berkaitan dengan kebarangkalian sesuatu nilai berkaitan dengan data kita.Bagaimana kita mengaplikasi taburan normal standard kepada data kita?

  • KurtosisJika taburan adalah simetri, soalan seterusnya berkaitan dengan puncak tengah (central peak): Adakah ianya tinggi dan tajam, atau pendek dan lebar? Anda akan mendapat idea dari melihat bentuk histogram, tetapi pengukuran bernombor adalah lebih tepat.Ketinggian dan ketajaman puncak relatif kepada data-data yang lain yang diukur, dinamakan kurtosis.

  • Taburan normal standard mempunyai kurtosis = 3. A normal distribution has kurtosis exactly 3 Any distribution with kurtosis 3 (excess 0) is called mesokurtic.A distribution with kurtosis 3 is called leptokurtic. Compared to a normal distribution, its central peak is higher and sharper, and its tails are longer and fatter.

  • SkewnessThe first thing you usually notice about a distributions shape is whether it has one mode (peak) or more than one. If its unimodal (has just one peak), like most data sets, the next thing you notice is whether its symmetric or skewed to one side.If the bulk of the data is at the left and the right tail is longer, we say that the distribution is skewed right or positively skewed; if the peak is toward the right and the left tail is longer, we say that the distribution is skewed left or negatively skewed.

  • Skewness rule of thumb:If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data, so how can you interpret the skewness number? If skewness is less than 1 or greater than +1, the distribution is highly skewed.If skewness is between 1 and or between + and +1, the distribution is moderately skewed.If skewness is between and +, the distribution is approximately symmetric.

  • Skor-Zjika kita tahu min populasi dan sisihan piawai populasi, untuk sebarang nilai X kita boleh mengira skor z dengan menggunakan rumus:

  • Info penting skor-zSkor z memberitahu kita berapa jauh samada di atas atau di bawah min nilai tersebut dalam unit sisihan piawai. Ianya adalah transformasi linear dari skor asal.mendarab(atau bahagi) dan tambah/tolak X dengan satu pemalar (constant)Kedudukan ranking skor tidak akan berubah.Z = (X-m)/s oleh ituX = sZ + m

  • Kebarangkalian dan skor z: berdasarkan jadual z Luas total = 1Hanya mempunyai kebarangkalian dari lebarUtk satu nombor skor z yang infinitekebarangkalian adalah 0 ( untuk satu titik poin)Dlm jadual tiada nilai negatifSimetri, oleh itu kawasan bawah negatif = kawasan di atas positif.Lakaran sangat membantu!!!!

  • Kebarangkalian yg digambarkan kawasan dibawah lengkungLuas keseluruhan dibawah lengkung = 1Kawasan berwarna merah p(z > 1)Kawasan berwarna biru p(-1< z
  • Strategi mencari kebarangkalian utk pembolehubah rawak dari taburan normalLakarkan gambaran dalam kawasan yang anda inginkan dalam taburan normal.Ekpresikan kawasan berdasarkan kepada kawasan dalam jadual.Lihat kawasan dengan menggunakan jadualJika perlu lakukan operasi + dan -

  • Suppose Z has standard normal distribution Find p(0
  • Find p(-1.57
  • Find p(Z>.78)

  • Z is standard normalCalculate p(-1.2
  • ExampleData come from distribution: m = 10, s = 3What proportion fall beyond X=13?Z = (13-10)/3 = 1=normsdist(1) or table 0.158715.9% fall above 13

  • Example: IQA common example is IQIQ scores are theoretically normally distributed. Mean of 100 Standard deviation of 15

  • IQs are normally distributed with mean 100 and standard deviation 15. Find the probability that a randomly selected person has an IQ between 100 and 115

  • Say we have GRE scores are normally distributed with mean 500 and standard deviation 100. Find the probability that a randomly selected GRE score is greater than 620.We want to know whats the probability of getting a score 620 or beyond.

    p(z > 1.2)Result: The probability of randomly getting a score of 620 is ~.12 or (12%)

  • Work time...(DIY)What is the area for scores less than z = -1.5?What is the area between z =1 and 1.5?What z score cuts off the highest 30% of the distribution?What two z scores enclose the middle 50% of the distribution?If 500 scores are normally distributed with mean = 50 and SD = 10, and an investigator throws out the 20 most extreme scores, what are the highest and lowest scores that are retained?

  • Standard ScoresZ is not the only transformation of scores to be usedFirst convert whatever score you have to a z score.New score = new s.d.(z) + new mean

    Example- T scores = mean of 50 s.d. 10Then T = 10(z) + 50.Examples of standard scores: IQ, GRE, SAT

  • Wrap upAssuming our data is normally distributed allows for us to use the properties of the normal distribution to assess the likelihood of some outcomeThis gives us a means by which to determine whether we might think one hypothesis is more plausible than another (even if we dont get a direct likelihood of either hypothesis)

  • Central limit theoremBased on probability theoryTwo stepsTake a given population and draw random samples again and againPlot the means from the results of Step 1 and it will be a normal curve where the center of the curve is the mean and the variation represents the standard errorEven if the population distribution is skewed, the distribution from Step 2 will be normal!

  • *The t Distributions and Degrees of FreedomYou can think of the t statistic as an "estimated z-score." With a large sample, the estimation is very good and the t statistic will be very similar to a z-score. With small samples, however, the t statistic will provide a relatively poor estimate of z.

  • *The t Distributions and Degrees of Freedom (cont.)The value of degrees of freedom, df = n - 1, is used to describe how well the t statistic represents a z-score. Also, the value of df will determine how well the distribution of t approximates a normal distribution. For large values of df, the t distribution will be nearly normal, but with small values for df, the t distribution will be flatter and more spread out than a normal distribution.

  • Sekian , Terima Kasih

    If you worried about falling off the bike, you'd never get on.~ Lance Armstrong.

    **Figure 9.1Distributions of the t statistic for different values of degrees of freedom are compared to a normal z-score distribution. Like the normal distribution, t distributions are bell-shaped and symmetrical and have a mean of zero. However, t distributions have more variability, indicated by the flatter and more spread-out shape. The larger the value of df is, the more closely the t distribution approximates a normal distribution.