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

Some probability distributionSome probability distributionThe Normal DistributionThe Normal Distribution

هـ9/4/1435 Noha Hussein Elkhidir

ObjectivesObjectivesIntroduce the Normal Distribution

Properties of the Standard Normal Distribution

Introduce the Central Limit Theorem


Normal DistributionNormal DistributionWhy 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


Normal DistributionNormal Distribution

Symmetrical, bell-shaped curveAlso known as Gaussian

distributionPoint of inflection = 1 standard

deviation from meanMathematical formula


f (X ) 1

2(e)

(X )2

2 2

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.


Key Areas under the CurveKey Areas under the Curve

For normal distributions

+ 1 SD ~ 68%+ 2 SD ~ 95%

+ 3 SD ~ 99.9%


Example IQ mean = 100 s Example IQ mean = 100 s = 15= 15


Problem :◦Each normal distribution with its own

values of and would need its own calculation of the area under various

points on the curve


Normal Probability Normal Probability DistributionsDistributions

Standard Normal Distribution – N(0,1)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


Normal Probability Normal Probability DistributionDistribution

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


Normal DistributionNormal DistributionThe standard normal distribution

will allow us to make claims about the probabilities of values

related to our own dataHow do we apply the standard

normal distribution to our data?


Z-scoreZ-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


zX

Important z-score infoImportant z-score infoZ-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-)/then

X = Z + [equation of the general form Y = mX+c]


Probabilities and z scores: z Probabilities and z scores: z tablestables

Total area = 1Only 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!هـ9/4/1435 Noha Hussein Elkhidir

Probabilities are depicted by areas under the Probabilities are depicted by areas under the curvecurve

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.


Strategies for finding probabilities Strategies for finding probabilities for the standard normal random for the standard normal random

variablevariable..

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.


Suppose Z has standard normal Suppose Z has standard normal distribution Find p(0<Z<1.23)distribution Find p(0<Z<1.23)


Find p(-1.57<Z<0)Find p(-1.57<Z<0)


Find p(Z>.78)Find p(Z>.78)


Z is standard normalZ is standard normalCalculate p(-1.2<Z<.78)Calculate p(-1.2<Z<.78)


Table I: P(0<Z<z)Table I: P(0<Z<z)

z .00 .01 .02 .03 .04 .05 .06 0.0. 0000. 0040. 0080. 0120. 0160. 0199. 02390.1. 0398. 0438. 0478. 0517. 0557. 0596. 0636 0.2. 0793. 0832. 0871. 0910. 0948. 0987. 10260.3. 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

ExamplesExamples

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

ExamplesExamples

P(Z≥1)

= 0.5–P(0<Z<1)= 0.5–0.3413

= 0.1587

ExamplesExamples

P(Z ≥ -1)=0.3413+0.50

=0.8413

ExamplesExamples

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

=0.8185

ExamplesExamples

P(Z ≤ 1.87)=0.5+P(0<Z ≤ 1.87)

=0.5+0.4693=0.9693

ExamplesExamples

P(Z<-1.87) =P(Z>1.87) =0.5–0.4693

=0.0307

ExampleExampleData come from distribution: =

0, = 3What proportion fall beyond

X=13?Z = (13-10)/3 = 1

=normsdist(1) or table 0.158715.9% fall above 13


Example dataExample data::Mean of data is 100

Standard deviation of data is 15


The data are normally distributed with mean 100 The data are normally distributed with mean 100 and standard deviation 15. Find the probability and standard deviation 15. Find the probability that a randomly selected data between 100 and that a randomly selected data between 100 and 115115


(100 115)

(100 100 100 115 100)

100 100 100 115 100(

15 15 15(0 1) .3413

P X

P X

XP

P Z

Say we have GRE scores are normally distributed Say we have GRE scores are normally distributed with mean 500 and standard deviation 100. Find with mean 500 and standard deviation 100. Find the probability that a randomly selected GRE score the probability that a randomly selected GRE score is greater than 620is 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


620 5001.2

100z

homeworkhomework : :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?


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

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