Some probability distribution The Normal Distribution 9/4/1435هـ Noha Hussein Elkhidir

Objectives Introduce the Normal Distribution Properties of the Standard Normal Distribution Introduce the Central Limit Theorem 9/4/1435هـ Noha Hussein Elkhidir

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 9/4/1435هـ Noha Hussein Elkhidir

Normal Distribution Symmetrical, bell-shaped curve Also known as Gaussian distribution Point of inflection = 1 standard deviation from mean Mathematical formula 9/4/1435هـ Noha Hussein Elkhidir

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. 9/4/1435هـ Noha Hussein Elkhidir

Key Areas under the Curve For normal distributions + 1 SD ~ 68% + 2 SD ~ 95% + 3 SD ~ 99.9% 9/4/1435هـ Noha Hussein Elkhidir

Example IQ mean = 100 s = 15 9/4/1435هـ Noha Hussein Elkhidir

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 9/4/1435هـ Noha Hussein Elkhidir

Normal Probability Distributions Standard 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 9/4/1435هـ Noha Hussein Elkhidir

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 9/4/1435هـ Noha Hussein Elkhidir

How do we apply the standard normal distribution to our data? 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? 9/4/1435هـ Noha Hussein Elkhidir

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 9/4/1435هـ Noha Hussein Elkhidir

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] 9/4/1435هـ Noha Hussein Elkhidir

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! 9/4/1435هـ Noha Hussein Elkhidir

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. 9/4/1435هـ Noha Hussein Elkhidir

Look up the areas using the table. 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. 9/4/1435هـ Noha Hussein Elkhidir

Suppose Z has standard normal distribution Find p(0<Z<1.23) 9/4/1435هـ Noha Hussein Elkhidir

Find p(-1.57<Z<0) 9/4/1435هـ Noha Hussein Elkhidir

Find p(Z>.78) 9/4/1435هـ Noha Hussein Elkhidir

Z is standard normal Calculate p(-1.2<Z<.78) 9/4/1435هـ Noha Hussein Elkhidir

Table I: P(0<Z<z) 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

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

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

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

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

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

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

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 9/4/1435هـ Noha Hussein Elkhidir

Example data: Mean of data is 100 Standard deviation of data is 15 9/4/1435هـ Noha Hussein Elkhidir

The data are normally distributed with mean 100 and standard deviation 15. Find the probability that a randomly selected data between 100 and 115 9/4/1435هـ Noha Hussein Elkhidir

Result: The probability of randomly getting a score of 620 is ~.12 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 9/4/1435هـ Noha Hussein Elkhidir

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? 9/4/1435هـ Noha Hussein Elkhidir