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Pertemuan 11 Peubah Acak Normal

Pertemuan 11 Peubah Acak Normal. Matakuliah : I0134-Metode Statistika Tahun : 2007. Outline Materi: Peluang sebaran normal. Basic Business Statistics (9 th Edition). The Normal Distribution and Other Continuous Distributions. Peluang sebaran normal. The Normal Distribution

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Pertemuan 11 Peubah Acak Normal

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  1. Pertemuan 11Peubah Acak Normal Matakuliah : I0134-Metode Statistika Tahun : 2007

  2. Outline Materi: • Peluang sebaran normal

  3. Basic Business Statistics (9th Edition) The Normal Distribution and Other Continuous Distributions

  4. Peluang sebaran normal • The Normal Distribution • The Standardized Normal Distribution • Evaluating the Normality Assumption • The Uniform Distribution • The Exponential Distribution

  5. Continuous Probability Distributions • Continuous Random Variable • Values from interval of numbers • Absence of gaps • Continuous Probability Distribution • Distribution of continuous random variable • Most Important Continuous Probability Distribution • The normal distribution

  6. The Normal Distribution • “Bell Shaped” • Symmetrical • Mean, Median and Mode are Equal • Interquartile RangeEquals 1.33 s • Random VariableHas Infinite Range f(X) X  Mean Median Mode

  7. The Mathematical Model

  8. Many Normal Distributions There are an Infinite Number of Normal Distributions Varying the Parameters  and , We Obtain Different Normal Distributions

  9. The Standardized Normal Distribution When X is normally distributed with a mean and a standard deviation , follows a standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. f(Z) f(X) X

  10. Finding Probabilities Probability is the area under the curve! f(X) X d c

  11. Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!

  12. Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5478 .5000 0.0 .5040 .5080 .5398 .5438 .5478 0.1 0.2 .5793 .5832 .5871 Probabilities Z = 0.12 0.3 .6179 .6217 .6255 Only One Table is Needed

  13. Standardizing Example Standardized Normal Distribution Normal Distribution

  14. Example Standardized Normal Distribution Normal Distribution

  15. Example (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .5832 .5000 0.0 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.21 0.3 .6179 .6217 .6255

  16. Example (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .4168 .3821 -0.3 .3783 .3745 -0.2 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 Z = -0.21 0.0 .5000 .4960 .4920

  17. Normal Distribution in PHStat • PHStat | Probability & Prob. Distributions | Normal … • Example in Excel Spreadsheet

  18. Example : Standardized Normal Distribution Normal Distribution

  19. Example:Example: (continued) Cumulative Standardized Normal Distribution Table (Portion) .02 Z .00 .01 .6179 .5000 0.0 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Z = 0.30 0.3 .6179 .6217 .6255

  20. Finding Z Values for Known Probabilities Cumulative Standardized Normal Distribution Table (Portion) What is Z Given Probability = 0.6217 ? .01 Z .00 0.2 0.0 .5040 .5000 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 .6179 .6255 .6217 0.3

  21. Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution

  22. More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam? 91 2.25

  23. (continued) More Examples of Normal Distribution Using PHStat What percentage of students scored between 65 and 89? 65 89 -1 2

  24. More Examples of Normal Distribution Using PHStat (continued) Only 5% of the students taking the test scored higher than what grade? ?=86.16 1.645

  25. Assessing Normality • Not All Continuous Random Variables are Normally Distributed • It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a Normal Distribution

  26. Assessing Normality (continued) • Construct Charts • For small- or moderate-sized data sets, do the stem-and-leaf display and box-and-whisker plot look symmetric? • For large data sets, does the histogram or polygon appear bell-shaped? • Compute Descriptive Summary Measures • Do the mean, median and mode have similar values? • Is the interquartile range approximately 1.33 s? • Is the range approximately 6 s?

  27. Assessing Normality (continued) • Observe the Distribution of the Data Set • Do approximately 2/3 of the observations lie between mean 1 standard deviation? • Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? • Do approximately 19/20 of the observations lie between mean 2 standard deviations? • Evaluate Normal Probability Plot • Do the points lie on or close to a straight line with positive slope?

  28. Assessing Normality (continued) • Normal Probability Plot • Arrange Data into Ordered Array • Find Corresponding Standardized Normal Quantile Values • Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis • Evaluate the Plot for Evidence of Linearity

  29. Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30 -2 -1 0 1 2 Look for Straight Line!

  30. Normal Probability Plot Left-Skewed Right-Skewed 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2 Rectangular U-Shaped 90 90 X X 60 60 Z Z 30 30 -2 -1 0 1 2 -2 -1 0 1 2

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