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Some probability distribution The Normal Distribution

Some probability distribution The Normal Distribution. Objectives. Introduce the Normal Distribution Properties of the Standard Normal Distribution Introduce the Central Limit Theorem. Normal Distribution. Why are normal distributions so important?

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Some probability distribution The Normal Distribution

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  1. Some probability distributionThe Normal Distribution Noha Hussein Elkhidir

  2. Objectives • Introduce the Normal Distribution • Properties of the Standard Normal Distribution • Introduce the Central Limit Theorem Noha Hussein Elkhidir

  3. Normal Distribution Why are normal distributions so important? • Many dependent variables are commonly assumed to be normally distributed in the population • If a variable is approximately normally distributed we can make inferences about values of that variable • Example: Sampling distribution of the mean Noha Hussein Elkhidir

  4. Normal Distribution • Symmetrical, bell-shaped curve • Also known as Gaussian distribution • Point of inflection = 1 standard deviation from mean • Mathematical formula Noha Hussein Elkhidir

  5. Since we know the shape of the curve, we can calculate the area under the curve • The percentage of that area can be used to determine the probability that a given value could be pulled from a given distribution • The area under the curve tells us about the probability- in other words we can obtain a p-value for our result (data) by treating it as a normally distributed data set. Noha Hussein Elkhidir

  6. Key Areas under the Curve • For normal distributions+ 1 SD ~ 68%+ 2 SD ~ 95%+ 3 SD ~ 99.9% Noha Hussein Elkhidir

  7. Example IQ mean = 100 s = 15 Noha Hussein Elkhidir

  8. Problem: • Each normal distribution with its own values of m and s would need its own calculation of the area under various points on the curve Noha Hussein Elkhidir

  9. Normal Probability DistributionsStandard Normal Distribution – N(0,1) • We agree to use the standard normal distribution • Bell shaped • =0 • =1 • Note: not all bell shaped distributions are normal distributions Noha Hussein Elkhidir

  10. Normal Probability Distribution • Can take on an infinite number of possible values. • The probability of any one of those values occurring is essentially zero. • Curve has area or probability = 1 Noha Hussein Elkhidir

  11. Normal Distribution • The standard normal distribution will allow us to make claims about the probabilities of values related to our own data • How do we apply the standard normal distribution to our data? Noha Hussein Elkhidir

  12. Z-score If we know the population mean and population standard deviation, for any value of X we can compute a z-score by subtracting the population mean and dividing the result by the population standard deviation Noha Hussein Elkhidir

  13. Important z-score info • Z-score tells us how far above or below the mean a value is in terms of standard deviations • It is a linear transformation of the original scores • Multiplication (or division) of and/or addition to (or subtraction from) X by a constant • Relationship of the observations to each other remains the same Z = (X-m)/s then X = sZ + m [equation of the general form Y = mX+c] Noha Hussein Elkhidir

  14. Probabilities and z scores: z tables • Total area = 1 • Only have a probability from width • For an infinite number of z scores each point has a probability of 0 (for the single point) • Typically negative values are not reported • Symmetrical, therefore area below negative value = Area above its positive value • Always helps to draw a sketch! Noha Hussein Elkhidir

  15. Probabilities are depicted by areas under the curve • Total area under the curve is 1 • The area in red is equal to p(z > 1) • The area in blue is equal to p(-1< z <0) • Since the properties of the normal distribution are known, areas can be looked up on tables or calculated on computer. Noha Hussein Elkhidir

  16. Strategies for finding probabilities for the standard normal random variable. • Draw a picture of standard normal distribution depicting the area of interest. • Re-express the area in terms of shapes like the one on top of the Standard Normal Table • Look up the areas using the table. • Do the necessary addition and subtraction. Noha Hussein Elkhidir

  17. Suppose Z has standard normal distribution Find p(0<Z<1.23) Noha Hussein Elkhidir

  18. Find p(-1.57<Z<0) Noha Hussein Elkhidir

  19. Find p(Z>.78) Noha Hussein Elkhidir

  20. Z is standard normalCalculate p(-1.2<Z<.78) Noha Hussein Elkhidir

  21. Table I: P(0<Z<z) z .00 .01 .02 .03 .04 .05 .06 0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1404 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 …………………… 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770

  22. Examples P(0<Z<1) = 0.3413 • Example P(1<Z<2) =P(0<Z<2)–P(0<Z<1) =0.4772–0.3413 =0.1359

  23. Examples P(Z≥1) =0.5–P(0<Z<1) =0.5–0.3413 =0.1587

  24. Examples P(Z ≥ -1) =0.3413+0.50 =0.8413

  25. Examples P(-2<Z<1) =0.4772+0.3413 =0.8185

  26. Examples P(Z ≤ 1.87) =0.5+P(0<Z ≤1.87) =0.5+0.4693 =0.9693

  27. Examples P(Z<-1.87) = P(Z>1.87) = 0.5–0.4693 = 0.0307

  28. Example • Data come from distribution: m = 0, s = 3 • What proportion fall beyond X=13? • Z = (13-10)/3 = 1 • =normsdist(1) or table  0.1587 • 15.9% fall above 13 Noha Hussein Elkhidir

  29. Example data: • Mean of data is 100 • Standard deviation of data is 15 Noha Hussein Elkhidir

  30. The data are normally distributed with mean 100 and standard deviation 15. Find the probability that a randomly selected data between 100 and 115 Noha Hussein Elkhidir

  31. 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 what’s 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 Noha Hussein Elkhidir

  32. homework: • 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? Noha Hussein Elkhidir

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